Nonlinear Option Pricing: Stochastic Control & BSDEs

Early Exercise Option Pricing in a Nutshell Stochastic control BSDE Nonlinear Option Pricing Julien Guyon École nationale des ponts et chaussées Institut polytechnique de Paris Master 2 Probabilités et Finance Sorbonne Université et École polytechnique Julien Guyon Nonlinear Option Pricing ULMM UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM The field of mathematics is very wide and it is not easy to predict what happens next, but I can tell you it is alive and well. Two general trends are obvious and will surely persist. In its pure aspect, the subject has changed, much for the better I think, by moving to more concrete problems. In both its pure and applied aspects, an equally beneficial shift to nonlinear problems can be seen. Most mathematical questions suggested by Nature are genuinely nonlinear, meaning very roughly that the result is not proportional to the cause, but varies with it as the square or the cube, or in some more complicated way. The study of such questions is still, after two or three hundred years, in its infancy. Only a few of the simplest examples are understood in any really satisfactory way. I believe this direction will be a principal theme in the future. — Henry P. McKean, Some Mathematical Coincidences (May 2003) Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Syllabus The classical curriculum of mathematical finance programs generally covers the link between linear second order parabolic partial differential equations (PDEs) and stochastic differential equations (SDEs), resulting from Feynman-Kac’s formula. However, many challenges faced by today’s practitioners involve nonlinear PDEs. The aim of this course is to provide you with the mathematical tools and numerical methods required to tackle these issues, and illustrate the methods with practical case studies like: American option pricing, uncertain volatility, uncertain mortality, different rates for borrowing and lending, calibration of models to market smiles, credit valuation adjustment (CVA), transaction costs, illiquid markets, super-replication under delta and gamma constraints, etc. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Syllabus We will spend some time on the theory: optimal stopping, stochastic control, backward stochastic differential equations (BSDEs), McKean SDEs and the particle method, branching diffusions. But the main focus will deliberately be on ideas and numerical examples, which we believe help a lot in understanding the tools and building intuition. PDE methods suffer from the curse of dimensionality. Since most quantitative finance problems are high-dimensional, we will mostly focus on simulation-based methods (a.k.a. Monte Carlo algorithms). This course exposes the students with a wide variety of Machine Learning techniques, old and new, including parametric regression, nonparametric regression, neural networks, kernel trick, etc. These techniques allow us to compute some quantities (such as conditional expectations) that are key ingredients of the nonlinear Monte Carlo algorithms. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Syllabus Homeworks will allow you to check your understanding of the course by solving exercises inspired by our experience as quantitative analysts. You are expected to use the Python programming language to complete the 4 homeworks. Final exam. The main reference for this course is the monograph Nonlinear Option Pricing by Julien Guyon and Pierre Henry-Labordère. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Mathematics Julien Guyon and Pierre Henry-Labordère Nonlinear Option Pricing Nonlinear Option Pricing copy to come Nonlinear Option Pricing Guyon and Henry-Labordère K16480 K16480_Cover.indd 1 Julien Guyon Nonlinear Option Pricing 4/11/13 8:36 AM UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Prerequisites We assume familiarity with classical stochastic analysis (Brownian motion, Itô formula, stochastic differential equations) and Black-Scholes option pricing. The students needing a reminder on these matters could consult: Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, Springer-Verlag, 1991. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 5th ed., Springer-Verlag, 1998. We will cover linear option pricing: self-financing strategies, arbitrage, equivalent martingale measures (risk-neutral measures), market completeness, super-replication, Black-Scholes PDE, Feynman-Kac theorem (link between parabolic second order linear PDEs and SDEs). Regarding the new tools introduced, the course will be self-contained. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Numerical tools: Finite difference schemes vs Monte Carlo methods Problems described by a nonlinear PDE ∂t u(t, x) + H(t, x, u(t, x), Du(t, x), D2 u(t, x)) = 0, x ∈ Ω, t ∈ [0, T ), (1) u(T, x) = g(x) Ω: open domain of Rn Du: gradient vector of the solution u with respect to spatial variables x D2 u: Hessian matrix of the solution u with respect to spatial variables x In one dimension: ∂t u(t, x) + H(t, x, u(t, x), ∂x u(t, x), ∂x2 u(t, x)) = 0, x ∈ Ω, t ∈ [0, T ), u(T, x) = g(x) No analytical solution. Need for numerical approximation: Deterministic methods: finite difference schemes, finite elements Probabilistic methods (based on simulation): Monte Carlo Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Finite difference schemes Localize the problem on a bounded domain Discretize the solution u on a space–time grid Gh = ∆t{0, 1, . . . , nT } × ∆xZn where h = (∆t, ∆x) and nT ∆t = T : uki = u(tk , xi ), ∀(tk , xi ) ∈ Gh Replace the differentiation operators ∂t , D, and D2 by their finite difference approximations. In one dimension, one can use Julien Guyon Nonlinear Option Pricing ∂tfwd u(tk , xi ) = Dfwd u(tk , xi ) = Dbwd u(tk , xi ) = Dcen u(tk , xi ) = (D2 )cen u(tk , xi ) = uk+1 − uki i ∆t uki+1 − uki ∆x uki − uki−1 ∆x uki+1 − uki−1 2∆x uki+1 + uki−1 − 2uki ∆x2 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Finite difference schemes PDE (1) can then be approximated by a (nonlinear) algebraic equation with unknowns uki . By using centered discrete differentiation operators, an explicit scheme that uses centered finite differences reads uk+1 − uki i +H(tk+1 , xi , uk+1 , Dcen u(tk+1 , xi ), (D2 )cen u(tk+1 , xi )) = 0, i ∆t ∀(tk+1 , xi ) ∈ Gh The implicit scheme reads uk+1 − uki i + H(tk , xi , uki , Dcen u(tk , xi ), (D2 )cen u(tk , xi )) = 0, ∆t ∀(tk , xi ) ∈ Gh \{t = T } A θ-scheme reads uk+1 − uki i + θH(tk+1 , xi , uk+1 , Dcen u(tk+1 , xi ), (D2 )cen u(tk+1 , xi )) i ∆t + (1 − θ)H(tk , xi , uki , Dcen u(tk , xi ), (D2 )cen u(tk , xi )) = 0 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Monte Carlo methods Curse of dimensionality: Finite diff schemes only work in small dimension No more than 3 space variables (assets and path-dependent variables) Need to turn to simulation-based methods (Monte Carlo methods) Problem American options Chooser options Uncertain Lapse and Mortality Uncertain Volatility Different rates for borrowing/lending Calibration of models to market smiles CVA Numerical tool ML techniques, duality ML techniques, duality ML techniques, BSDEs Parameterization, ML techniques, BSDEs ML techniques, BSDEs Particle method, ML techniques, Malliavin ML techniques, BSDEs, branching diffusions ML techniques will mostly be used to estimate conditional expectations E[Y |X = x]. They include: nearest neighbors, kernel regression, parametric regression, neural networks, random forests, kernel trick (+ cross-validation...). In this course we will apply them to simulated data, but they can be readily applied to data science problems. Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Models of financial markets A filtered probability space Ω, (Ft )0≤t≤T , Phist Phist is the historical or real probability measure under which we model our market. A market model is defined by an n-dimensional stochastic differential equation (SDE) dXti = bi (t, Xt ) dt + d X σi,j (t, Xt ) dWtj , i ∈ {1, . . . , n} j=1 and by another positive stochastic process Bt , called the money-market account, representing the value of cash, which satisfies dBt = rt Bt dt, B0 = 1 i.e., Z t Bt = exp 0 rt is the short term interest rate. Julien Guyon Nonlinear Option Pricing rs ds (2) Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Models of financial markets We set Z u rs ds Dtu := Bt Bu−1 = exp − t which is the discount factor from date u to date t. Throughout the course, we will denote by Ỹt := D0t Yt the discounted value of any price process Yt . Certain market components X i may not be sold or bought in the market, such as the short term interest rate, or a stochastic volatility. Throughout this course, a market component X i that can be sold and bought in the market is called an “asset.” Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Early Exercise Problems Pricing and hedging of American options If you want a happy ending, that depends, of course, on where you stop your story. — Orson Welles Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM European, American and Bermudan options A European option can be exercised only at the expiration date T , VtE = EQ [Dt,T FT |Ft ] An American option may be exercised anytime on or before the expiration date T , VtA = sup EQ [Dt,τ Fτ |Ft ] τ ∈Tt,T conditional on the option not being exercised at or before t, where Tt,T denotes the set of all stopping times τ such that τ ∈ [t, T ]. A Bermudan option can be exercised on certain pre-defined discrete set of dates t1 < · · · < tN = T , VtB = sup EQ [Dt,τ Fτ |Ft ] τ ∈TD conditional on the option not being exercised at or before T t, where TD denotes the set of all stopping times τ ∈ {t1 , . . . , tN } [t, T ]. VtA ≥ VtB ≥ VtE Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM American Call and Put - No dividend Assume positive interest rates. Martingale condition yields EQ Dt,T ST |Ft = St American Call Early exercise of an American call option leads to Loss of the time value of the option (from volatility) Loss of the time value of strike (from discounting) Never optimal to exercise an American call option early. h i + EQ Dt,T (ST − K)+ Ft ≥ EQ Dt,T ST − Dt,T K|Ft + = St − Dt,T K ≥ (St − K)+ American Put Early exercise of an American put option leads to Loss of the time value of the option (from volatility) Gain on the time value of strike (from discounting) Early exercise could be beneficial when the option is deeply in the money (time value of the option is small compared to the gain of the time value of the strike) Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM American Call and Put - Dividend Continuous Dividend Yield EQ [Dt,T ST |Ft ] = e−q(T −t) St , q>0 American Call Early exercise of an American call option could be beneficial if the dividend yield is large enough compared to the time value of the option (from volatility) and the time value of strike (from discounting) lost American Put Large dividend yield benefits continuation Discrete Dividend Std − − Std + = D > 0, D = dividend, td = ex-date American Call The optimal exercise time of an American call should be either right before the ex-date or at the final maturity American Put It is never optimal to exercise an American put right before the ex-date Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM American Call and Put - Optimal Exercise Boundary Example. American Put, S = 100, K = 100, T = 1, σ = 25%, r = 8% I. no dividend II. $1 cash dividend paid at t = 0.1, 0.35, 0.6, 0.85 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM American Call and Put Put-Call parity does not hold for American options. Most listed options on individual stocks are American options with physical delivery Most listed options on stock indices are European options with cash settlement Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Markovian Case - European options Assume that there exists a (deterministic) function r such that rt = r(t, Xt ) and g such that FT = g(XT ), then i h RT u(t, x) = EQ e− t r(s,Xs )ds g(XT ) Xt = x Black-Scholes pricing PDE ∂t u(t, x) + Lu(t, x) − r(t, x)u(t, x) = 0, u(T, x) = g(x). where L is the Ito generator of X: L= n X n bi (t, x)∂i + i=1 d 1 X X σi,k (t, x)σj,k (t, x)∂ij 2 i,j=1 k=1 In dimension one (n = d = 1), L = b(t, x) Julien Guyon Nonlinear Option Pricing ∂ 1 ∂2 + σ 2 (t, x) 2 ∂x 2 ∂x UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Markovian Case - American options Consider an American option that pays g(Xt ) if exercised at time t. Let u(t, x) be the value of the American option at time t if Xt = x and the option has not been exercised yet. Then i h Rτ u(t, x) = sup EQ e− t r(s,Xs )ds g(Xτ ) Xt = x τ ∈Tt,T where Tt,T is the collection of all stopping times τ such that t ≤ τ ≤ T . Since t ∈ Tt,T , u(t, x) ≥ g(x). As long as u(t, Xt ) > g(Xt ), it is not optimal to exercise, and the optimal stopping time τ ∗ is τ ∗ = inf {s ≥ t|u(s, Xs ) = g(Xs )} . Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Markovian Case - American options - Variational Inequality Variational Inequality max (∂t u(t, x) + Lu(t, x) − r(t, x)u(t, x), g(x) − u(t, x)) = 0 u(T, x) = g(x) Let J = ∂t + L − r. The option is worth at least its exercise value: g(x) ≤ u(t, x) Rt The supermartingale property of e− 0 r(s,Xs )ds u(t, Xt ): J u(t, x) ≤ 0 J u = 0 in the continuation region and g = u in the exercise region: J u(t, x) (g(x) − u(t, x)) = 0 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Markovian Case - American options - Variational Inequality Formal proof of variational inequality Assume that u is a smooth solution of the variational inequality max (∂t u(t, x) + Lu(t, x) − r(t, x)u(t, x), g(x) − u(t, x)) = 0 we shall show that u solves the optimal stopping problem, i.e. u(t, Xt ) = sup EQ [Dt,τ g(Xt )|Ft ] . τ ∈Tt,T First, ∂t u + Lu − ru ≤ 0 implies that D0,t u(t, Xt ) is a super-martingale so that for any τ ∈ Tt,T , by the optional sampling theorem and the fact g(Xτ ) ≤ u(τ, Xτ ), we have EQ [D0,τ g (Xτ ) |Ft ] ≤ EQ [D0,τ u(t, Xτ )|Ft ] ≤ D0,t u(t, Xt ) Taking the supremum over all τ ∈ Tt,T , we obtain sup EQ [Dt,τ g(Xt )|Ft ] ≤ u(t, Xt ). τ ∈Tt,T Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Markovian Case - American options - Variational Inequality Formal proof of variational inequality (continued) On the other hand, define τ ∗ = inf {s ≥ t|u(s, Xs ) = g(Xs )}. Then D0,τ ∗ u (τ ∗ , Xτ ∗ ) = D0,t u(t, Xt ) Z τ∗ Z τ∗ + D0,s (∂t u(s, Xs ) + Lu(s, Xs ) − r(s, Xs )u(s, Xs )) ds + · · · dWs t t ∗ ∗ The definition of τ guarantees that u (τ , Xτ ∗ ) = g (Xτ ∗ ) and ∂t u(s, Xs ) + Lu(s, Xs ) − r(s, Xs )u(s, Xs ) = 0 for all s ∈ [t, τ ∗ ], so that u(t, Xt ) = EQ [Dt,τ ∗ g(Xτ ∗ )|Ft ] i.e. the supremum over Tt,T is attained by the optimal stopping time τ ∗ . Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Markovian Case - Bermudan options Assume the option can only be exercised on a selected number of dates t1 , . . . , t N : h Rτ i u(t, x) = sup EQ e− t r(s,Xs )ds g(Xτ ) Xt = x τ ∈T{t ,...,t 1 N }∩[t,T ] Between two dates ti+1 and ti : ∂t u(t, x) + Lu(t, x) − r(t, x)u(t, x) = 0 At date ti (Jump condition): + u(t− i , x) = max(u(ti , x), g(x)) Optimal stopping time τ ∗ = inf{s ∈ {t1 , . . . , tN } ∩ [t, T ] | u(s, Xs ) = g(Xs )} Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Finite Difference Scheme Finite difference scheme can be easily adapted to price options with early exercises features. Consider a Bermudan option with exercise dates t1 < · · · < tN = T and payoff function g(Xt ) Let u(T, x) = g(x) and then proceed backward Between ti and ti+1 , solve the PDE ∂t u + Lu − ru = 0 by a classic finite difference scheme (fully explicit, fully implicit and Crank-Nicholson) − + Replace u(t+ i , x) by u(ti , x) = max u ti , x , g(x) and continue the backward induction from ti with the new terminal condition u(t− i , x). American options can be priced by applying the jump condition to every point in the time grid of the finite difference scheme. Usually only first-order in time. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Bermudan Options and Discrete Time Snell Envelope Definition (Snell Envelope (Discrete Time)) Given a discrete-time stochastic process {Zk }1≤k≤N , the process Uk defined by Uk = sup E [Zτ |Fk ] , k = 1, . . . , N τ ∈Tk,N is called the Snell envelope of {Zk }. Consider a Bermudan option with exercise dates t1 < · · · < tN and payoffs Ft1 , . . . , FtN . Let Vtk be the value of the option at time tk if it has not been exercised yet. Vtk = D0,tk Vtk = sup τ ∈T{t ,...,t } N k sup τ ∈T{t ,...,t } N k EQ [Dtk ,τ Fτ |Ftk ] EQ [D0,τ Fτ |Ftk ] . The discounted value process {D0,tk Vtk }N k=1 is the Snell envelope of the discounted payoff {D0,tk Ftk }N . k=1 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Recursive Scheme of Building Discrete Time Snell Envelope The Snell envelope {Uk } can be constructed explicitly by the following recursive scheme: Uk = sup E [Zτ |Fk ] , k = 0, . . . , N (3) τ ∈Tk,N if and only if ( UN = Z N Uk = max{Zk , E [Uk+1 |Fk ]} k = 0, . . . , N − 1 (4) Proof. First assume Uk is given by (3). Obviously, UN = ZN . Let ∗ τk+1 = argmaxτ ∈Tk+1,N E [Zτ |Fk+1 ] be the optimal stopping time. Then h h i i Fk+1 Fk E [Uk+1 |Fk ] = E E Zτ ∗ k+1 h i = E Zτ ∗ Fk ≤ sup E [Zτ |Fk ] = Uk k+1 τ ∈Tk,N ∗ The inequality holds since τk+1 ∈ Tk,N . Obviously, Zk ≤ Uk since k ∈ Tk,N . Hence max{Zk , E [Uk+1 |Fk ]} ≤ Uk . On the other hand, for any τ ∈ Tk,N , Zτ = Zk 1{τ =k} + Zτ ∨(k+1) 1{τ >k} . Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Recursive Scheme of Building Discrete Time Snell Envelope E [Zτ |Fk ] = Zk 1{τ =k} + E Zτ ∨(k+1) |Fk 1{τ >k} = Zk 1{τ =k} + E E Zτ ∨(k+1) |Fk+1 |Fk 1{τ >k} ≤ Zk 1{τ =k} + E [Uk+1 |Fk ] 1{τ >k} ≤ max{Zk , E [Uk+1 |Fk ]} Thus Uk = sup τ ∈Tk,N E [Zτ |Fk ] ≤ max{Zk , E [Uk+1 |Fk ]}. Next assume that Uk is given by the recursive scheme (4). Clearly Uk is a supermartingale that dominates Zk . Let τk∗ = inf{n ≥ k; Zn = Un }. Then U(n+1)∧τ ∗ − Un∧τ ∗ = 1n+1≤τ ∗ (Un+1 − Un ) = 1n+1≤τ ∗ (Un+1 − E [Un+1 |Fn ]) k k k k h i ∗ Note that {n + 1 ≤ τk } ∈ Fn . Hence E U(n+1)∧τ ∗ Fn = Un∧τ ∗ , i.e. Un∧τ ∗ is a k k ki h i h martingale so that Uk = E Uτ ∗ Fk = E Zτ ∗ Fk . Furthermore, for ∀τ ∈ Tk,N , k k E [Zτ |Fk ] ≤ E [Uτ |Fk ] ≤ Uk . □ Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Discrete-Time Snell Envelope and Bermudan Options Corollary. Given {Zn }n≤N , the Snell envelope {Uk } is the smallest supermartingale dominating {Zn }. For ∀k ≤ N , the optimal stopping time τk∗ is given by τk∗ = inf{n ≥ k; Zn = Un } and the stopped process {Un∧τk∗ }n≥k is a martingale. Bermudan Option Let Vtk be the price of a Bermudan option at time tk provided the option has not been exercised before tk . Then {D0,tk Vtk } is the smallest supermartingale that dominates the discounted payoff {D0,tk Ftk } and can be constructed by the recursive scheme: VtN = FtN Vti = max Fti , EQ [Dti ti+1 Vti+1 |Fti ] If the option has not been exercised before tk , the optimal stopping time τk∗ is τk∗ = inf{tj ≥ tk | Vtj = Ftj } The stopped value process {D0,tj ∧τk∗ Vtj ∧τk∗ }j≥k is a martingale. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Monte Carlo Methods Why do we need it? Curse of dimensionality for finite difference methods when the number of variables exceeds three. Multi-asset options Path-dependent payoff Stochastic volatility/stochastic interest rates Why is it difficult? Need to estimate the continuation value EQ Dti ,ti+1 Vi+1 Fti . Naive nested Monte Carlo is explosive. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Backward Induction First, we approximate American options by Bermudan options with exercise dates t1 , t2 , . . . , tN = T and payoff Ft1 , Ft2 , . . . , FtN . Let Vti be the value of the option at time ti . At the final expiry tN = T , VtN = FtN Then we proceed by backward induction Vti = max EQ Dti ,ti+1 Vti+1 |Fti , g(Xti ) , i = 1, . . . , N − 1 The key is to determine the continuation value Cti = EQ Dti ,ti+1 Vti+1 |Fti Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Regression algorithms Continuation value at ti : Cti = EQ Dti ,ti+1 Vti+1 |Fti (A) Let τi+1 be the optimal stopping time at ti+1 . Then Vti+1 = EQ Dti+1 ,τi+1 Fτi+1 |Fti+1 so that Cti = EQ Dti ,τi+1 Fτi+1 |Fti (B) The conditional expectation can be estimated from the cross-sectional information in the simulated paths. Tsitsiklis-Van Roy (2001) uses (A) and regress V̂ (ti+1 , Xti+1 ) on some functions of Xti Longstaff-Schwartz (2001) uses (B) and regress the sum of subsequent realized cash flows on some functions of Xti (more robust, more accurate) Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM The Tsitsiklis-Van Roy algorithm Bermudan option with exercise dates t1 < · · · < tN and payoff Ft1 , . . . , FtN . Let V̂ti (ω) denote the approximation to Vti (ω) on each path. Procedure Simulate paths until maturity T = tN Let V̂tN = FtN For i = N − 1, N − 2, . . . , 1, estimate the continuation value Ĉti = EQ [Dti ti+1 V̂ti+1 |Fti ] and then let V̂ti = max(Fti , Ĉti ) Estimate V0 by V̂0 = EQ [D0t1 V̂t1 ] where EQ is replaced by the average value over all simulated paths Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM The Longstaff-Schwartz algorithm Procedure Simulate paths until maturity T = tN Let τ̂N = tN For i = N − 1, . . . , 1, Ĉti = EQ [Dti τ̂i+1 Fτ̂i+1 |Fti ] ti if Ĉti ≤ Fti τ̂i = τ̂i+1 otherwise Eventually, estimate the optimal strategy by τ̂ = τ̂1 and the price V0 by V̂0 = EQ [D0τ̂ Fτ̂ ] where EQ is replaced by the average value over all simulated paths Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Tsitsiklis-Van Roy v.s. Longstaff-Schwartz Both Tsitsiklis-Van Roy (TVR) and Longstaff-Schwartz (LS) estimate the continuation value from the cross-sectional information in the simulated paths by regression. TVR uses the fitted values from the previous step in the regression at current step whereas LS uses subsequent realized cash flows. The continuation estimate Ĉti is used directly in building value function Vti in TVR, but only used in making exercise decision in LS. The regression error from estimating Ĉti leads to error in V̂ti in TVR, but error in exercise decision in LS. In both algorithms the errors propagate through the backward induction. In general the error propagation is worse in TVR. In LS, one only needs to estimate the continuation value where option is in the money. In general, the LS algorithm is more accurate and robust than the TVR algorithm. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Tsitsiklis-Van Roy v.s. Longstaff-Schwartz - Example Example: Bermudan put option, quarterly exercise, Black-Scholes model, F (S) = (K − S)+ . S0 = 100, σ = 0.2, r = 0.1, q = 0.02, K = 100, T = 1. PDE: Continuation Value and Optimal Exercise Boundary from PDE Continuation v.s. Exercise Optimal Exercise Boundary 16 1.00 Exercise Value Continuation Value t=0.25 Continuation Value t=0.50 Continuation Value t=0.75 14 12 0.75 10 t 8 0.50 6 4 0.25 2 0 85 90 95 S 100 105 110 0.00 90 92 94 S 96 98 100 Monte Carlo: estimate of continuation value is obtained by quadratic polynomial fit Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Tsitsiklis-Van Roy - Example Monte Carlo paths 160 140 St 120 100 80 0.00 0.25 ti = 0.25 Exercise Value F(Sti) Continuation Value C(Sti) V(St) = max(C(Sti), F(Sti)) e r(ti + 1 ti)V(Sti + 1) 25 20 15 10 20 Julien Guyon Nonlinear Option Pricing 100 Sti 110 120 130 25 20 15 10 10 0 Exercise Value F(Sti) Continuation Value C(Sti) V(St) = max(C(Sti), F(Sti)) e r(ti + 1 ti)V(Sti + 1) 30 15 0 1.00 ti = 0.75 Exercise Value F(Sti) Continuation Value C(Sti) V(St) = max(C(Sti), F(Sti)) e r(ti + 1 ti)V(Sti + 1) 25 5 90 0.75 ti = 0.50 30 5 80 0.50 t 5 0 80 100 120 Sti 140 160 80 100 Sti 120 140 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Longstaff-Schwartz - Example Monte Carlo paths 160 140 St 120 100 80 0.00 0.25 ti = 0.25 Exercise Value F(Sti) Continuation Value C(Sti) e r( i + 1 ti)F(S i + 1) Exercise Region 25 20 0.50 t 0.75 ti = 0.50 30 ti = 0.75 Exercise Value F(Sti) Continuation Value C(Sti) e r( i + 1 ti)F(S i + 1) Exercise Region 25 25 20 15 15 10 10 10 5 5 5 0 0 80 Julien Guyon Nonlinear Option Pricing 90 100 Sti 110 120 130 Exercise Value F(Sti) Continuation Value C(Sti) e r( i + 1 ti)F(S i + 1) Exercise Region 30 20 15 1.00 0 80 100 120 Sti 140 160 80 100 Sti 120 140 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Longstaff-Schwartz - Example Continuation values can be estimated only where the option is in the money. Longstaff-Schwartz ti = 0.50 Continuation Value C(Sti) Exercise Value F(Sti) e r( i + 1 ti)F(S i + 1) Exercise ITM e r( i + 1 ti)F(S i + 1) Continue ITM e r( i + 1 ti)F(S i + 1) Continue OTM 40 30 20 10 0 60 Julien Guyon Nonlinear Option Pricing 80 100 S(ti) 120 140 160 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Convexity of max(x, a) and Jensen’s Bias Example.PLet X1 , . . . , Xn be independent samples of a random variable X and X̄n = n1 n i=1 Xi be the sample mean. Assume that E [X] = a = 0.8. Then E max a, X̄n ≥ max(a, E [X]) = a. 1.4 1.2 y=x y=a y=max(x,a) pdf of Xn 1.0 0.8 0.6 0.4 0.2 0.0 0.0 Julien Guyon Nonlinear Option Pricing 0.2 0.4 0.6 x 0.8 1.0 1.2 1.4 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Bias and Lower Bound Pricing There are both bias and variance from the estimation of the continuation value Ĉti , which lead to bias in the estimation of the final price V̂0 . The variance in Ĉti creates high bias in V̂0 due to Jensen’s bias. The Jensen’s bias diminishes as the number of paths goes to infinity. The bias in Ĉti typically leads to the bias of the same direction in V̂0 in the TVR algorithm. The bias in Ĉti leads to a sub-optimal strategy, which produces low bias in V̂0 in the Longstaff-Schwartz algorithm. Typically, both Tsitsiklis-Van Roy and Longstaff-Schwartz algorithms have undetermined bias. Lower Bound (or Low-biased) pricing requires a two-step procedure. 1 Run TVR or Longstaff-Schwartz algorithm with p1 paths and obtain an estimate of the optimal strategy τ̂1 : τ̂1 = 2 Julien Guyon Nonlinear Option Pricing inf j=1,...,N {tj |Ĉtj <= Ftj } Simulate p2 new independent paths that are stopped according to τ̂1 . UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Lower Bound Pricing - Example Example: Independent Monte Carlo simulation to get lower bound price with exercise strategy given by the Longstaff-Schwartz algorithm. 160 150 140 St 130 120 110 100 90 80 0.00 Julien Guyon Nonlinear Option Pricing 0.25 0.50 t 0.75 1.00 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Conditional Expectation E [Y |X] 1 2 (Best Predictor). If Y ∈ L2 , f∗ (X) ≜ E [Y |X] minimizes the mean squared error E (Y − f (X))2 among all f s.t. f (X) ∈ L2 . Therefore f ∗ (X) can be interpreted as a projection of Y on the space of all functions of X and is the best predictor of Y (with the smallest mean square error) given the value of X. If both X and Y are continuous random variables, then Z E [Y |X = x] = yfY |X (y|x)dy where the conditional density fY |X is the ratio of the joint density fX,Y and marginal density fX , fY |X (y|x) = Julien Guyon Nonlinear Option Pricing fX,Y (x, y) . fX (x) UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Conditional Expectation Estimation - Parametric Regression We approximate E [Y |X] by restricting the subspace of functions of X to a smaller subspace of functions of some parametric form {f (x; α), α ∈ A}: h i α∗ = argminα∈A E (Y − f (X; α))2 In particular, we may choose f (x; α) = basis functions {fk }n k=1 , i.e. Y ≈ n X αk fk (X) + ϵ and E [Y |X] ≈ f (X; α∗ ) Pn k=1 αk fk (x) to be the linear sum of where ϵ is independent noise and E [Y |X] ≈ k=1 n X αk fk (X) k=1 Given N observations ((x1 , y1 ) , . . . , (xN , yN )) of X and Y , finding E [Y |X] comes down to solving least square problem minα∈Rn ∥Aα − y∥2 , where   A= f1 (x1 ) .. . f1 (xN ) ··· .. . ···  fn (x1 )  ..  . Julien Guyon Nonlinear Option Pricing and fn (xN ) −1 α∗ = A T A AT y.  y1  .  y= .  .  yN UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Conditional Expectation Estimation - Parametric Regression Select simple regressors and avoid over fitting problem. Select regressors similar to the payoff function. Piecewise polynomials can be used as basis functions, e.g., piecewise-linear functions, regression splines. Use numpy.polyfit(x, y) for polynomial regression and numpy.linalg.lstsq(A, b) for general parametric regression. Avoid overfitting with regularization (Ridge Regression or Tikhonov regularization): minn ∥Aα − y∥22 + λ∥α∥22 , λ > 0 α∈R λ is a hyperparameter (remaining fixed during optimization/training). The optimal solution α∗ is given by −1 α∗ = AT A + λI AT y. Normalize variables beforehand if necessary Penalty can be applied to selected variables Use sklearn.linear model.Ridge for ridge regression Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Conditional Expectation Estimation - Polynomial Regression Polynomial basis: f1 (x) = 1, f2 (x) = x, f3 (x) = x3 , . . . , fn+1 (x) = xn Polynomial regression with higher order polynomials are more likely to overfit; Its behavior near the boundaries tends to be erratic and extrapolation can be dangerous. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE Conditional Expectation Estimation - Piecewise Linear Fit For given knots x1 < · · · < xn , the basis functions are f1 (x) = 1, f2 (x) = x − x1 f2+i (x) = (x − xi )+ , Julien Guyon Nonlinear Option Pricing i = 1, . . . , n ULMM UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE Conditional Expectation Estimation - Piecewise Linear Fit Tikhonov Regularization Penalties are applied to α2+i , i = 1, . . . , n. Julien Guyon Nonlinear Option Pricing ULMM UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Conditional Expectation Estimation - Regression Cubic Splines For given knots x1 < · · · < xn , the basis functions are f1 (x) = 1, f2 (x) = x − x1 , f3 (x) = (x − x1 )2 , f4 (x) = (x − x1 )3 f4+i (x) = (x − xi )3+ , Penalties are applied to α4+i , i = 1, . . . , n. Julien Guyon Nonlinear Option Pricing i = 1, . . . , n UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Kernel Density Estimation (KDE) n 1 X x − xi fˆh (x) = K nh i=1 h Julien Guyon Nonlinear Option Pricing Z with K ≥ 0 and K(x)dx = 1 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Kernel Density Estimation (KDE) Commonly used kernel functions: Epanechnikov K(x) = 34 1 − x2 for −1 ≤ x ≤ 1 and 0 elsewhere; 15 Quartic K(x) = 16 (1 + x)2 (1 − x)2 for −1 ≤ x ≤ 1 and 0 elsewhere; 1 2 Gaussian K(x) = √12π e− 2 x for all x. Bandwidth Selection: Balance the bias-variance tradeoff. Large h gives lower variance but higher bias. Silverman’s rule of thumb: h= 4σ̂ 5 3N 51 1 ≈ 1.06σ̂N − 5 where σ̂ 2 is the sample variance Optimal (in integrated mean square error terms) if Gaussian kernel is used to estimate Gaussian density. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Kernel Density Estimation (KDE) - Commonly Used Kernels Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Conditional Expectation Estimation - Nonparametric Regression Nadaraya-Watson Kernel Regression (Locally weighted average) PN E[Y |X = x] ≈ Pi=1 N Kh (x − xi )yi i=1 Kh (x − xi ) where K is the kernel, h > 0 is the bandwidth and Kh (x) = K(x/h)/h. Derivation. Z E [Y |X = x] = y fX,Y (x, y) dy fX (x) Use kernel density estimation for fX and fX,Y with kernel Kh : n fX (x) ≈ 1X Kh (x − xi ), n i=1 n fX,Y (x, y) ≈ 1X Kh (x − xi )Kh (y − yi ) n i=1 Z and (assuming that K is centered: Z yKh (y − yi )dy = yi . Julien Guyon Nonlinear Option Pricing yKh (y)dy = 0) the fact □ Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Conditional Expectation Estimation - Nonparametric Regression Local Linear Regression The locally weighted linear regression solves a separate weighted least squares problem at each target point x, min α,β N X Kh (x − xi ) [yi − α − βxi ]2 i=1 Then the estimate is α̂ + β̂x, where α̂ and β̂ are dependent on x. The locally-weighted averages can be badly biased on the boundaries. This bias can be removed by local linear regression to first order. Cross validation can be used for bandwidth selection. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell NW v.s. Local Linear Regression Julien Guyon Nonlinear Option Pricing Stochastic control BSDE ULMM UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Artificial Neutral Networks Feedforward Neutral Network Inputs: a0 = x ∈ Rp Hidden layers Outputs ai = fi (Wi ai−1 + bi ) , i = 1, . . . , N − 1 y = aN = WN aN −1 + bN ∈ Rq The nonlinear function fi is called activation function. Some common choices are sigmoid, hyperbolic tangent, ReLu and ELU functions. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Universal Approximation Theorem Theorem Let ϕ be a non-constant bounded and monotone-increasing continuous function on R. Given any continuous function f on [0, 1]n and ϵ > 0, there exists an integer m and real constants βi , bi and wij , with i = 1, . . . , m and j = 1, . . . , n, such that the function F defined by ! m n X X F (x1 , . . . , xn ) = βi ϕ wij xj + bi i=1 j=1 is a uniform ϵ-approximation to f : |F (x1 , . . . , xn ) − f (x1 , . . . , xn )| < ϵ for all x1 , . . . , xn . Any continuous function can be approximated to arbitrary accuracy by a neural network with a single hidden layer, provided the hidden layer is sufficiently wide. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Artificial Neutral Networks A neural network learns the basis functions instead of using fixed ones. Much of the power of neutral network comes from the repeated composition of simpler functions to represent a more complex function. A loss function needs to be specified to train the model. For regression tasks, a typical choice is the mean squared error. Gradient with respect to the network parameters can be computed efficiently by backpropagation. Then the network parameters can be updated by a gradient-based optimizer such as stochastic gradient descent. A neural network typically has a huge number of parameters. Regularization techniques are helpful to avoid overfitting. The network parameters are initialized randomly (with variances properly chosen) to break the symmetry and avoid the exploding/vanishing gradient problems. It is also important to fine-tune the hyperparameters. Popular deep learning libraries include TensorFlow, CNTK, Theano and PyTorch. Keras is a high-level API with multibackend support. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Primal and Dual Problems Primal Problem (maximizing over stopping times) sup EQ [Dt,τ Fτ |Ft ] (5) τ ∈Tt,T Dual Problem (minimizing over martingales) [Rogers 2002; Haugh & Kogan 2004] inf EQ sup (Dt,s Fs − Ms ) Ft . (6) M ∈Mt,0 t≤s≤T where Mt,0 denotes the set of all right-continuous martingales (Ms )s∈[t,T ] with Mt = 0. Let (Us )s∈[t,T ] be the Snell envelope of discounted payoff (Dt,s Fs )s∈[t,T ] . Then the optimal martingale M ∗ is the martingale part of the Doob-Meyer decomposition of (Us )s∈[t,T ] , minus its value at time t. sup EQ [Dt,τ Fτ |Ft ] = τ ∈Tt,T Julien Guyon Nonlinear Option Pricing inf M ∈Mt,0 EQ sup (Dt,s Fs − Ms ) Ft t≤s≤T Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Primal and Dual Problems Proof. For any τ ∈ Tt,T and M ∈ Mt,0 , it follows from the optional sampling theorem that EQ [Mτ |Ft ] = Mt = 0 Therefore EQ [Dt,τ Fτ |Ft ] = EQ [Dt,τ Fτ − Mτ |Ft ] + EQ [Mτ |Ft ] " # ≤ EQ sup (Dt,s Fs − Ms ) Ft t≤s≤T so that # " sup EQ [Dt,τ Fτ |Ft ] ≤ τ ∈Tt,T inf M ∈Mt,0 EQ sup (Dt,s Fs − Ms ) Ft . (7) t≤s≤T On the other hand, for any fixed t, let Us = supτ ∈Ts,T EQ [Dt,τ Fτ |Fs ], s ∈ [t, T ], be the Snell envelope of the discounted payoff (Dt,s Fs )s∈[t,T ] . Then (Us )s∈[t,T ] is the smallest supermartingale that dominates (Dt,s Fs )s∈[t,T ] . In particular, it has unique Doob-Meyer decomposition Us = Ms − As , ∀s ∈ [t, T ] where (Ms )s∈[t,T ] is a martingale with Mt = Ut and (As )s∈[t,T ] a predictable increasing process with At = 0. Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Primal and Dual Problems Let Ms∗ = Ms − Mt , s ∈ [t, T ]. Then Dt,s Fs − Ms∗ = Dt,s Fs − Us − As + Mt ≤ Mt = Ut . Since this holds for all s ∈ [t, T ], we must have sup (Dt,s Fs − Ms∗ ) ≤ Ut = t≤s≤T sup EQ [Dt,τ Fτ |Ft ] . τ ∈Tt,T Taking conditional expectation EQ [·|Ft ] and using inequality (7) we obtain " # EQ sup (Dt,s Fs − Ms∗ ) Ft = t≤s≤T sup EQ [Dt,τ Fτ |Ft ] (8) τ ∈Tt,T which shows that the infimum in (6) is achieved by M ∗ , thus the equivalence of the primal and dual problems. □ Remark. It is easy to see that (8) holds even without expectation EQ [·] on the left-hand side, i.e. sup (Dt,s Fs − Ms∗ ) = t≤s≤T Julien Guyon Nonlinear Option Pricing sup EQ [Dt,τ Fτ |Ft ] τ ∈Tt,T almost surely Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Lower and Upper bounds Let u0 be the price of an American option at time t = 0, i.e., u0 = sup EQ [D0,τ Fτ ] = inf EQ sup (D0,t Ft − Mt ) M ∈M0,0 τ ∈T0,T 0≤t≤T Any stopping time (exercise strategy) τ ∈ T0,T gives a lower bound: EQ [D0,τ Fτ ] ≤ u0 Any martingale (self-financing hedge) M with M0 = 0 gives an upper bound: u 0 ≤ EQ sup (D0,t Ft − Mt ) . 0≤t≤T Primal problem =⇒ Lower bound, Julien Guyon Nonlinear Option Pricing Dual problem =⇒ Upper bound UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Upper bound Suppose an agent sells an American option at time 0 and let M be the value of the self-financing hedge less its initial value, then EQ sup (D0,t Ft − Mt ) 0≤t≤T gives the maximum exposure from the short option position and its hedge on average. Example 1. Let Mt ≡ 0 (no hedge), i.e. u0 ≤ EQ sup (D0,t Ft ) . 0≤t≤T Example 2. In case of American put option, let Mt be the European put price (discounted to time 0) with the same final maturity and strike less the initial price. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Upper Bound - Examples of Different Hedges Bermudan put, monthly exercises, Black-Scholes model S0 = 100, σ = 0.2, r = 0.1, q = 0.02, K = 100 No Hedge: St 120 115 110 105 100 95 90 85 80 Optimal Exercise Region D0, tFt Mt (Mt 0) 17.5 sup(D0, tFt Mt) = 10.62 t sup(D0, tFt Mt) = 0.00 t sup(D0, tFt Mt) = 17.33 15.0 12.5 t 10.0 7.5 5.0 2.5 0.0 0 Julien Guyon Nonlinear Option Pricing 1/12 2/12 3/12 4/12 5/12 6/12 t 7/12 8/12 9/12 10/12 11/12 1 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Upper Bound - Examples of Different Hedges Hedge with a European Put: St 120 Optimal Exercise Region 110 100 90 80 D0, tFt (solid) and Mt (dashed) 15 10 5 0 5 D0, tFt Mt sup (D0, tFt Mt) = 5.98 t sup (D0, tFt Mt) = 4.33 t sup (D0, tFt Mt) = 8.37 8 6 t 4 2 0 0 Julien Guyon Nonlinear Option Pricing 1/12 2/12 3/12 4/12 5/12 6/12 t 7/12 8/12 9/12 10/12 11/12 1 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Upper Bound - Examples of Different Hedges Optimal Hedge: St 120 Optimal Exercise Region 110 100 90 80 D0, tFt (solid) and Mt (dashed) 15 10 5 0 5 D0, tFt Mt 5 4 3 2 sup (D0, tFt Mt) = 5.14 t sup (D0, tFt Mt) = 5.14 t sup (D0, tFt Mt) = 5.14 1 t 0 0 Julien Guyon Nonlinear Option Pricing 1/12 2/12 3/12 4/12 5/12 6/12 t 7/12 8/12 9/12 10/12 11/12 1 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Upper Bound - Examples of Different Hedges Histogram of supt (D0,t Ft − Mt ): No Hedge (Mt 0) 0 Julien Guyon Nonlinear Option Pricing 10 20 30 Hedge with European Put 40 50 5 6 7 8 9 Optimal Hedge 10 4.0 4.5 5.0 5.5 6.0 Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Doob-Meyer Decomposition - Discrete Time Theorem (Doob Decomposition) Let Un be an adpated and integrable process. Then there exists a unique decomposition Un = Mn − An , where Mn is a martingale and An is predictable process with A0 = 0. In particular, Un is a supermartingale if and only if An is increasing; Un is a submartingale if and only if An is decreasing. Proof. Put M0 = U0 and A0 = 0. For n ≥ 0, let Mn+1 − Mn = Un+1 − E [Un+1 |Fn ] An+1 − An = −E [Un+1 |Fn ] + Un . Then obviously Mn is a martingale and An ∈ Fn−1 . Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Dual Method: Broadie-Andersen algorithm Consider a Bermudan option that pays Fti if exercised at ti , i = 1, . . . , N . Step 1 First run a primal algorithm with n1 paths (e.g. the Longstaff-Schwartz algorithm) to obtain a sequence of (nearly) optimal stopping times τi , i = 1, . . . , N . Let V be the (discounted) value process stopped by τi , i.e. Vti = EQ [ D0,τi Fτi | Fti ] We shall use Vt as an approximation to the Snell envelope St of D0,t Ft and then extract the martingale part of Vt as an approximation to the optimal martingale Mt∗ , which is the martingale part of St . Step 2 Simulate a new set of n2 independent paths. For eachof these paths and at each exercise date ti , we need to estimate Vti and EQ Vti+1 |Fti . Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Dual Method: Broadie-Andersen algorithm If τi indicates continuation, i.e. τi > ti , then simulate nc independent subpaths starting from (ti , Xti ) and stopped by τi nc 1 X (j) (j) D Fτ Vti = EQ Vti+1 |Fti ≈ nc j=1 0,τi i If τi indicates exercise, i.e. τi = ti , then Vti = D0,ti Fti , but we still need to estimate EQ Vti+1 |Fti . Simulate ne independent subpaths starting from (ti , Xti ) and stopped by τi+1 , ne 1 X (j) (j) D EQ Vti+1 Fti ≈ Fτ ne j=1 0,τi+1 i+1 Step 3 Build the martingale M : M0 = 0 and Mti+1 = Mti + Vti+1 − EQ Vti+1 |Fti and then obtain the upper bound price EQ max (D0,ti Fti − Mti ) 1≤i≤N Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Dual method: Broadie-Andersen algorithm The nested simulations in Step 2 are not recursive, i.e., no further simulations are run on each subpath. The number of simulated paths is at most n1 + n2 × (N − 1) × max (nc , ne ) For the martingale (Mti ) generated from a nearly optimal strategy, max1≤i≤N (D0,ti Fti − Mti ) has very small variance. Therefore, in most cases, even small values of n2 give accurate results. A large duality gap is usually an indication of poor exercise rules from the primal algorithm (often caused by regression problems, such as bad selection of basis functions, over-fitting, etc.) The nested Monte Carlo generates independent sampling error with zero mean. However, applying the sup operator subsequently produces a high-biased estimator of the upper bound. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE Option Pricing in a Nutshell Julien Guyon Nonlinear Option Pricing ULMM UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Option Pricing in a Nutshell In this chapter: We recall classical results on stochastic representations of solutions of linear parabolic PDEs. This allows us to revisit key notions of option pricing and set our notations. Most of the material presented here is standard and can be found in various classical references. Yet we also highlight several important notions which are not often covered: super-replication pricing in incomplete models Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM 1. The superreplication paradigm Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Models of financial markets A filtered probability space Ω, (Ft )0≤t≤T , Phist Phist is the historical or real probability measure under which we model our market. A market model is defined by an n-dimensional stochastic differential equation (SDE) dXti = bi (t, Xt ) dt + d X σi,j (t, Xt ) dWtj , i ∈ {1, . . . , n} (9) j=1 and by another positive stochastic process Bt , called the money-market account, representing the value of cash, which satisfies dBt = rt Bt dt, B0 = 1 i.e., Z t Bt = exp rs ds 0 rt is the short term interest rate. It is adapted to Ft , which is the (natural) filtration generated by the d-dimensional uncorrelated standard Brownian motion {Wtj }1≤j≤d . Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Models of financial markets In order to ensure that SDE (9) admits a unique strong solution (see e.g., Karatzas), we assume that b and σ satisfy: Assum(SDE): The functions b and σ are Lipschitz-continuous in x uniformly in t, and satisfy a linear growth condition: there exists a positive constant C such that for all t ≥ 0, x, y ∈ Rn , |b(t, x) − b(t, y)| + |σ(t, x) − σ(t, y)| ≤ C|x − y| |b(t, x)| + |σ(t, x)| ≤ C (1 + |x|) We set Dtu := Bt Bu−1 = exp Z u − rs ds t which is the discount factor from date u to date t. Throughout the course, we will denote by Ỹt := D0t Yt the discounted value of any price process Yt . Certain market components X i may not be sold or bought in the market, such as the short term interest rate, or a stochastic volatility. Throughout this course, a market component X i that can be sold and bought in the market is called an “asset.” Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Self-financing portfolios Let us assume that we have a portfolio consisting of m assets, say Xt1 , . . . , Xtm , and the money-market account Bt . It is convenient to use the notation X 0 for B. The portfolio at a time t is composed of ∆it assets Xti and ∆0t units of Xt0 (cash). The ∆it ’s must be Ft -measurable, i.e., we cannot look into the future. The portfolio value πt is πt := m X ∆it Xti (10) i=0 As time passes, we can readjust the allocations ∆it , but no cash is ever injected into or removed from the portfolio: between t and t + dt, the variation in the portfolio value is only due to the variation of the values of the assets, i.e., m X (11) dπt = ∆it dXti i=0 We then speak of a self-financing portfolio. Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Self-financing portfolios In terms of discounted values, this reads dπ̃t = m X i=0 ∆it dX̃ti = m X ∆it dX̃ti because for any price process Yt , dỸt = D0t (dYt − rt Yt dt), so Z t m Z t X ∆is dX̃si = π0 + ∆s · dX̃s π̃t = π0 + i=1 (12) i=1 0 (13) 0 where · denotes the usual scalar product in Rm .1 As a technical condition, we need to introduce: Definition (∆t , 0 ≤ t ≤ T ) defines an admissible portfolio if π̃t is bounded from below for all t Phist -a.s., i.e., there exists M ∈ R such that Phist (∀t ∈ [0, T ], π̃t ≥ M ) = 1 1 The stochastic integral is well-defined with the condition is then a local martingale. Julien Guyon Nonlinear Option Pricing Rt 0 (∆is )2 d⟨X⟩s < ∞ Phist -a.s. π̃t Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Arbitrage and arbitrage-free models An arbitrage is a self-financing strategy that is worth zero initially and yields a positive gain without any risk: Definition A self-financing admissible portfolio is called an arbitrage if the corresponding value process πt satisfies π0 = 0 and πT ≥ 0 Phist − a.s and Phist (πT > 0) > 0 Arbitrageurs are a special kind of traders. Their role is precisely to detect and take full advantage of arbitrage opportunities as soon as they appear in the market. This impacts market prices: arbitrage opportunities tend to disappear as soon as they arise. Absence of arbitrage opportunities is therefore a natural modeling assumption. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Arbitrage and arbitrage-free models The next lemma gives a sufficient condition under which we exclude arbitrage opportunities in our market model: Lemma (Sufficient condition excluding arbitrage) Suppose there exists a measure Q on (Ω, FT ) such that Q ∼ Phist and such that, for all asset X i , the discounted price process {X̃ti }t∈[0,T ] is a local martingale with respect to Q. Then the market {Xt }t∈[0,T ] has no arbitrage. P is equivalent to Q (denoted by P ∼ Q) if and only if ∀A ∈ FT , P(A) = 0 ⇐⇒ Q(A) = 0. Throughout this course, unless stated otherwise, martingales and local martingales are considered with respect to the filtration (Ft )0≤t≤T . Note that the assumption of this lemma bears only on assets X i only, not on non-tradable components of X, such as instantaneous interest rates, instantaneous stochastic volatility, etc. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Arbitrage and arbitrage-free models Proof. Let us suppose that there exists an arbitrage. We have Z t m Z t X π̃t = π0 + ∆is dX̃si = π0 + ∆s · dX̃s i=1 0 0 and, for all asset X i , the discounted price process {X̃ti }t∈[0,T ] is a local martingale with respect to Q. Therefore π̃t is a local martingale w.r.t. Q. Moreover, ∆ defines an admissible portfolio, so π̃t is a lower bounded local martingale w.r.t. Q. By a standard result, π̃t is thus a Q-supermartingale and hence EQ [π̃T ] ≤ π̃0 = 0. But since π̃T ≥ 0 Phist -a.s. and Q ∼ Phist , we have π̃T ≥ 0 Q-a.s. Hence π̃T = 0 Q-a.s., so π̃T = 0 Phist -a.s., which yields πT = 0 Phist -a.s. This contradicts the fact that Phist (πT > 0) > 0. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Arbitrage and arbitrage-free models This simple lemma invites one to introduce the notion of an equivalent local martingale measure (in short ELMM): Definition (Equivalent local martingale measure) Any measure Q ∼ Phist on (Ω, FT ) such that, for all asset X i , the discounted price process {X̃ti }t∈[0,T ] is a local martingale w.r.t. Q is called an equivalent local martingale measure (ELMM). An ELMM is also called a risk-neutral measure. The previous lemma now reads: if there exists an ELMM, then the market has no arbitrage opportunities. Throughout this course we will always assume that we are in a situation where there exists at least one ELMM. Let us consider an asset X i . Under an ELMM Q, {X̃ti }t∈[0,T ] is an (Ft )-local martingale, hence has zero drift. As a consequence, the drift of the asset X i under an ELMM is rt Xti : Z t dXti = exp rs ds (dX̃ti + rt X̃ti dt) = rt Xti dt + (· · · ) · dWt 0 This is not the case for non-tradable components of the market. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Super-replication Let us assume that, at time t, we buy and delta-hedge a European option written on m assets, say Xt1 , . . . , Xtm , with maturity T and payoff FT , at the price z. In general, the payoff FT is a function of the paths (Xti , 0 ≤ t ≤ T ) followed by the prices of the m assets between times 0 and T . The final value of the buyer’s portfolio, discounted at time 0, is m Z T X π̃TB = −D0t z + ∆is dX̃si + D0T FT i=1 t Z T = −D0t z + ∆s · dX̃s + D0T FT t We then define the buyer’s super-replication price at time t as the greatest price z s.t. the value of the buyer’s portfolio π̃TB is Phist -a.s. nonnegative: Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Super-replication Definition (Buyer’s price) n Bt (FT ) = sup z ∈ Ft there exists an admissible portfolio ∆ such that Z T o π̃TB := −D0t z + ∆s · dX̃s + D0T FT ≥ 0 Phist − a.s. (14) t The price z must be Ft -measurable, denoted by z ∈ Ft , i.e., we cannot look into the future. Similarly, we can define the seller’s super-replication price as: Definition (Seller’s price) n St (FT ) = inf z ∈ Ft there exists an admissible portfolio ∆ such that Z T o π̃TS := D0t z + ∆s · dX̃s − D0T FT ≥ 0 Phist − a.s. (15) t Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Super-replication Theorem (Arbitrage-free bounds) Assume that there exists an ELMM Q ∼ Phist . Then Bt (FT ) ≤ EQ [DtT FT |Ft ] ≤ St (FT ) Proof. From the definition of Bt (FT ), we assume that there exists an admissible portfolio ∆ such that Z T −D0t z + ∆s · dX̃s + D0T FT ≥ 0 Phist − a.s. t Rt s · dX̃s is a lower bounded Q-local martingale, hence a Q-supermartingale, so 0 ∆ R EQ [ tT ∆s · dX̃s |Ft ] ≤ 0 and we deduce that D0t z ≤ EQ [D0T FT |Ft ] i.e., z ≤ EQ [DtT FT |Ft ] Taking the supremum over z, we get Bt (FT ) ≤ EQ [DtT FT |Ft ]. The inequality EQ [DtT FT |Ft ] ≤ St (FT ) can be derived similarly. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Attainable payoff These bounds can be sharpened by assuming that the market is complete. In order to define what it means for a market to be complete, we need to introduce the notion of attainable payoff: Definition (Attainable payoff) A payoff FT is said to be attainable (at time 0) if there exists an admissible RT portfolio ∆ and a real number z such that z + 0 ∆s · dX̃s − D0T FT = 0 R t Phist -a.s. and 0 ∆s · dX̃s is a (true) Q-martingale with Q ∼ Phist . Stated otherwise, a payoff FT is said to be attainable if we can generate the wealth FT at time T from an initial wealth z by trading only in the underlying assets and the cash in a self-financing way. The portfolio ∆ and the real number z are unique because if RT z ′ + 0 ∆′s · dX̃s − D0T FT = 0, then Z T (∆′s − ∆s ) · dX̃s = z − z ′ 0 and since X̃ is a (nontrivial) Q-local martingale, this yields ∆′s = ∆s , hence z ′ = z. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Complete markets As a consequence, there is a unique fair price at t = 0 for an attainable payoff, and this price is z. Rt The assumption that 0 ∆s · dX̃s is a (true) Q-martingale guarantees that z = EQ [D0T FT ], so we have a formula to compute the price z from the payoff FT . Definition (Complete market) A market is said to be complete when every payoff is attainable at time 0. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Complete markets Theorem For a complete market, for any ELMM Q and any payoff FT , we have Bt (FT ) = St (FT ) = EQ [DtT FT |Ft ] Proof. By completeness, we can find a real number z and an admissible portfolio ∆ such that R z + 0T ∆s · dX̃s − D0T FT = 0 Phist -a.s., hence Q-a.s. As a consequence, Z T ∆s · dX̃s − D0T FT = 0 Q−a.s. D0t zt + t −1 with zt = D0t R R z + 0t ∆s · dX̃s being Ft -measurable. Since 0t ∆s · dX̃s is a Q-martingale, we have D0t zt = EQ [D0T FT |Ft ], i.e., zt = EQ [DtT FT |Ft ], whence Bt (FT ) ≥ EQ [DtT FT |Ft ]. Combined with the reverse inequality, we get Bt (FT ) = EQ [DtT FT |Ft ]. We proceed similarly for the other equality. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Super-replication in incomplete markets With incomplete markets or markets with friction, perfect replication is no longer possible. In the theorem below, we state that the buyer’s super-replication price is given by the infimum of expectations of the discounted payoff over the set ELMM of equivalent local martingale measures: Theorem (Super-replication price (El Karoui-Quenez, Kramkov)) The buyer’s super-replication price and the seller’s super-replication price are given by Bt (FT ) = inf Q∈ELMM St (FT ) = sup Q∈ELMM Julien Guyon Nonlinear Option Pricing EQ [DtT FT |Ft ] EQ [DtT FT |Ft ] UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Characterization of complete markets Corollary (Complete market) A market is complete if and only if there exists a unique ELMM. Proof. Assume the market is complete. Let Q0 be an ELMM. For any payoff FT , Q0 B0 (FT ) = S0 (FT ) = E [D0T FT ] = Q Q inf E [D0T FT ] = sup E [D0T FT ]. Q∈ELMM Q∈ELMM 0 Hence, for all FT , EQ [D0T FT ] = EQ [D0T FT ] for all Q ∈ ELMM , so that ELMM = {Q0 }. Conversely, if 0 0 ELMM = {Q }, then B0 (FT ) = S0 (FT ) = EQ D0T FT . As a consequence, there exist two admissible portfolios ∆B and ∆S (one for the buyer and one for the seller) such that Phist -a.s., i.e., Q0 -a.s. Q0 −E [D0T FT ] + Z T B ∆t · dX̃t + D0T FT 0 Z T 0 Q S E [D0T FT ] + ∆t · dX̃t − D0T FT 0 ≥ 0 ≥ 0 R S Summing those inequalities, this means that Q0 -a.s. 0T (∆B t + ∆t ) · dX̃t ≥ 0, and, because the discounted prices of all assets S Q0 -a.s. Eventually, we have that (X̃ti ) are Q0 -local martingales, this yields ∆B = −∆ t t 0 R 0 hist -a.s., and the market is complete. EQ [D0T FT ] + 0T ∆S t · dX̃t − D0T FT = 0 Q -a.s., hence P Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Complete models versus incomplete models From the proof of the super-replication price theorem, the market model (9) is complete if and only if there exists a unique Ft -adapted vector λt ∈ Rd s.t. ∀i ∈ asset, bi (t, Xt ) − rt Xti = d X σi,j (t, Xt )λjt (16) j=1 The unique ELMM Q is then given by d RT j Y j j 2 1 RT dQ = e− 0 λt dWt − 2 0 (λt ) dt hist dP |F T j=1 The unique arbitrage-free price is Bt (FT ) = St (FT ) = EQ [DtT FT |Ft ] Julien Guyon An inspection of (16) reveals that if the market is complete, then the rank of σ(t, Xt ) is equal to d a.s. (which implies that #assets ≥ d). In the case where #assets < d, the market cannot be complete. Moreover, if the number of assets coincides with the number of Brownian motions, i.e., #assets = d, and (σi,j (t, Xt ))i∈asset,1≤j≤d is invertible, then the market is complete. Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Complete models versus incomplete models Examples of complete models that are commonly used by practitioners include Dupire’s local volatility model (see Dupire) Libor market models with local volatilities, e.g., BGM with deterministic volatilities (see Brace-Gatarek-Musiela) and Markov functional models (see Hunt-Kennedy-Pelsser). Common examples of incomplete models are stochastic volatility models (in short SVMs). Here #assets < d. An example of stochastic volatility model is the Heston model. The dynamics of the underlying, denoted by Xt , reads under a risk-neutral measure Q0 ∼ Phist as p √ dXt (17) = rt dt + Vt ρ dWt2 + 1 − ρ2 dWt1 Xt √ 2 dVt = −k(Vt − m) dt + σ Vt dWt (18) with Wt1 , Wt2 , two uncorrelated standard Q0 -Brownian motions. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Complete models versus incomplete models Vt is the instantaneous variance, it is not a tradable instrument. The only asset is Xt . Its drift under any ELMM is rt Xt . The incompleteness of such a model can be detected as the drift term in Vt can be arbitrarily modified with a change of measure from Q0 to Q1 ∼ Phist : dQ1 dQ0 |F = T 2 RT j Y j 2 j 1 RT e− 0 λt dWt − 2 0 (λt ) dt j=1 such that ρλ2t + p 1 − ρ2 λ1t = 0 (19) From Girsanov’s theorem, Equation (19) guarantees that the drift of Xt under Q1 is still rt Xt , i.e., that Q1 is still an ELMM. Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Complete models versus incomplete models We can show that the seller’s super-replication (undiscounted) price at t = 0 of a European vanilla payoff FT := g(XT ) in such an SVM is g conc (X0 ) with g conc the concave envelope of g, i.e., the smallest concave function that is greater than or equal to g. Similarly, the buyer’s super-replication (undiscounted) price at t = 0 of a European vanilla payoff g(XT ) in such an SVM is g conv (X0 ) with g conv the convex envelope of g, i.e., the largest convex function that is smaller than or equal to g. For example, for a call option g(x) = (x − K)+ , g conc (x) = x and g conv (x) = (x − K)+ . This simple example illustrates the main drawback of the super-replication framework: the buyer’s and seller’s super-replication prices are not close and therefore cannot be used in practice.2 2 This drawback can be circumvented by adding extra market instruments, such as vanilla options, in the superhedging strategy. This leads to tight bounds and a nice generalization of the optimal transport theory (see Beiglböck-Henry-Labordère-Penkner, Galichon-Henry-Labordère-Touzi) where now the transport is performed along a martingale measure. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Pricing in practice In practice, the seller’s price at time t is computed by picking out a particular ELMM Q: ut := EQ [DtT FT |Ft ] Under Q, the drift for an asset X i (20) is fixed to bi (t, Xt ) = rt Xti . In an incomplete market, Q does not necessarily achieve the supremum in the super-replication price theorem, and we lose the superhedging strategy paradigm. Selling options becomes a risky business. In practice, picking a particular ELMM simplifies a lot the pricing problem: it becomes a linear problem: the price of the (European) payoff FT1 + FT2 equals the sum of the prices of the (European) payoffs FT1 and FT2 . Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM 2. Stochastic representation of solutions of linear PDEs Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM The Cauchy problem We will always assume that there exists (deterministic) function r such that R rt = r(t, Xt ). Dt1 t2 = exp − tt2 r(s, Xs ) ds 1 We denote by L the Itô generator of X defined under a risk-neutral measure Q: L= n X n bi (t, x)∂i + i=1 d 1 X X σi,k (t, x)σj,k (t, x)∂ij 2 i,j=1 k=1 (21) If X i is an asset, bi (t, x) = r(t, x)xi . If there exists g such that FT = g(XT ), we speak of a vanilla option. In such a case, by the Markov property of X, Z T ut = EQ exp − r(s, Xs ) ds g(XT ) Ft =: u(t, Xt ) t is a function u of (t, Xt ); the discounted price process D0t ut = D0t EQ [DtT FT |Ft ] = EQ [D0T FT |Ft ] is a Q-martingale. Then application of Itô’s lemma to R a straightforward D0t u(t, Xt ) = exp − 0t r(s, Xs ) ds u(t, Xt ) shows that the pricing function u, if smooth enough, satisfies the second order parabolic linear PDE: ∂t u(t, x) + Lu(t, x) − r(t, x)u(t, x) = 0 with the terminal condition u(T, x) = g(x). Julien Guyon Nonlinear Option Pricing (22) Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM The Cauchy problem Consider the case where there exists f, g such that the payoffs are g(XT ) at maturity T and f (t, Xt ) dt between t and t + dt (0 ≤ t ≤ T ). f : stream of continuous cash flows; think of daily, weekly, or monthly coupons. By the Markov property of X, Z T Z s Z T ut = EQ exp − r(s, Xs ) ds g(XT ) + exp − r(a, Xa ) da f (s, Xs ) ds Ft t t t is a function u of (t, Xt ); the discounted price process Z T Z t D0s f (s, Xs ) ds Ft D0s f (s, Xs ) ds = D0t EQ DtT FT + D0t ut + 0 0 Z T Q =E D0s f (s, Xs ) ds Ft D0T FT + 0 is a Q-martingale. R Then a straightforward application of Itô’s lemma to D0t u(t, Xt ) + 0t D0s f (s, Xs ) ds shows that the pricing function u, if smooth enough, satisfies the second order parabolic linear PDE: ∂t u(t, x) + Lu(t, x) − r(t, x)u(t, x) + f (t, x) = 0 (23) with the terminal condition u(T, x) = g(x). f (t, x) is called “source term.” The case with dividend/repo yield q(t, Xt ). Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM The Cauchy problem Conversely, a solution u to (23) admits the stochastic representation (20) as stated by the Feynman-Kac theorem: Theorem (Feynman-Kac, see e.g., Karatzas) Let b, σ satisfy Assum(SDE), r be uniformly bounded from below and f have quadratic growth in x uniformly in t. Let u ∈ C 1,2 ([0, T ) × Rd ) ∩ C 0 ([0, T ] × Rd ) with quadratic growth in x uniformly in t and solution to the parabolic PDE: ∂t u(t, x) + Lu(t, x) − r(t, x)u(t, x) + f (t, x) = 0 with terminal condition u(T, x) = g(x). Then u admits the stochastic representation Z T Rs RT u(t, x) = EQ e− t r(u,Xu )du f (s, Xs )ds + e− t r(s,Xs )ds g(XT ) Xt = x t Think of f as intermediate payoffs. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Path-dependent options Path-dependent payoffs are those payoffs for which one cannot write FT = g(XT ), i.e., whose value depends on the asset values not only at maturity T , but also at previous dates t < T . For instance, with one asset Xt = Xt1 : Asian options: FT = g 1 T RT 0 Xt dt Barrier and lookback options: FT = g mint∈[0,T ] Xt , maxt∈[0,T ] Xt , XT Cliquet options: FT = g XTi /XTi−1 , 1 ≤ i ≤ N In some cases, one can write SDEs for the path-dependent variables and add them to the SDEs describing the market X so that one can write FT = g(XT′ ) for an enlarged market X ′ , and the option price satisfies a PDE in the variables x′ . For instance, the Rprice of an Asian option satisfies t a PDE in the variables (x, a) where At = 0 Xs ds and dAt = Xt dt. Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Path-dependent options For a general path-dependent payoff FT = g(Xt1 , . . . , Xtn ), depending on the spot values at the observation dates t1 < · · · < tn = T , the PDE, depending on the past spot values Z 1 , . . . , Z n , can be written as ∂t ui (t, x, Z 1 , . . . , Z i−1 ) + Lui (t, x, Z 1 , . . . , Z i−1 ) = 0, ∀t ∈ [ti−1 , ti ) with the matching conditions at the dates t1 , . . . , tn , un (tn , x, Z 1 , . . . , Z n−1 ) = g(Z 1 , . . . , Z n−1 , x) 1 i−1 ui (t− ) i , x, Z , . . . , Z = 1 i−1 ui+1 (t+ , x) i , x, Z , . . . , Z The final price is u1 (0, X0 ). Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Path-dependent options - An example Example. Consider a path-dependent payoff at time tn written as Xt1 + · · · + Xtn g n Introduce an auxiliary variable I such that the pricing function u(t, x, I) satisfies ∂t ui (t, x, I) + Lu(t, x, I) = 0, ti−1 < t < ti , i = 2, . . . , n with terminal condition at tn , x u(tn , x, I) = g I + n and jump condition at ti , i = 1, . . . , n − 1, x + u(t− . i , x, I) = u ti , x, I + n Discretize I-values into m different values, and solve m different one-dimensional PDEs backward in time and exchange information with each other at each ti by the jump condition. Note that in the first time interval 0 < t < t1 , the auxillary variable I is not needed and the pricing function u simply satisfies ∂t u(t, x) + Lu(t, x) = 0, Finally the price at time 0 is u(0, X0 ). Julien Guyon Nonlinear Option Pricing 0 < t < t1 . UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Derivatives with Default Risk We model the time of default, τ , to be the first jump time of a Poisson process, with deterministic intensity λt . That is, for any t ≥ 0, we have P [ τ ∈ (t, t + dt)| τ > t] = λt dt (24) Let Φ(t) = P [τ ≤ t] be the CDF of τ . Then it follows from (24) that Φ′ (t)dt = λt dt 1 − Φ(t) Solving this ODE gives us the following: Rt P [τ ≤ t] = 1 − e− 0 λs ds Rt P [τ > t] = e− 0 λs ds Rt P [τ ∈ (t, t + dt)] = λt e− 0 λs ds dt RT P [ τ > T | τ > t] = e− t λs ds , P [ τ ∈ (s, s + ds)| τ > t] = λs e Julien Guyon Nonlinear Option Pricing R − ts λa da T >t ds, s>t Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Derivatives with Default Risk Consider European payoff g(XT ) at time T where the underlying asset Xt is given by dXt = [r (t, Xt ) − q (t, Xt )] dt + σ (t, Xt ) dWt . Xt If the counter-party defaults at time τ before T , the contract pays uD (τ, Xτ ). The default time τ is assumed to be independent from X. The fair value of this contract at time t, provided that the counter-party has not defaulted yet at time t, is u(t, x) = E Q Q Q e R R − T λ ds − tT r(s,Xs )ds g (XT ) e t s h E E =E R R − tT r(s,Xs )ds − τ r(s,Xs )ds g(XT )1τ ≥T + e t uD (τ, Xτ )1τ <T =E Q e e Q h e Xt = x + R − tτ r(s,Xs )ds uD (τ, Xτ ) 1τ <T R − tT (r(s,Xs )+λs )ds g (XT ) + Xt = x, τ > t Z T e (Xs )t≤s≤T , τ > t i i Xt = x, τ > t R − ts (r(ν,Xν )+λν )dν λs uD (s, Xs )ds Xt = x t Thus, u satisfies the PDE ∂u ∂u 1 2 ∂2u +(r(t, x) − q(t, x)) x + σ (t, x)x2 2 +λt (uD (t, x) − u(t, x))−r(t, x)u = 0 ∂t ∂x 2 ∂x Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Selected references Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices: A mass transport approach, Finance and Stoch., 17(3):477–501, 2013. Bick, A.: Quadratic-variation-based dynamic strategies, Management Science, 41:722–732, 1995. Brace, A., Gatarek, D., Musiela, M.: The market model of interest rate dynamics, Mathematical Finance, 7(2):127–154, 1997. Dupire, B.: Pricing with a smile, Risk magazine, 7:18–20, 1994. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Selected references El Karoui, N., Quenez, M.-C.: Dynamic programming and pricing of contingent claims in an incomplete market, SIAM Journal on Control and Optimization archive, 33(1), 1995. Galichon, A., Henry-Labordère, P., Touzi, N.: A stochastic control approach to no arbitrage bounds given marginals, with an application to lookback options, to appear in Ann. Appl. Probab., available at http:// ssrn.com/abstract=1912477, 2011. Hunt, P., Kennedy, J., Pelsser, A.: Markov-functional interest rate models, Finance and Stoch., 4(4):391–408, 2000. Kramkov, D.: Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probab. Theory Related Fields, 105:459–479, 1996. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Stochastic optimal control and the Hamilton-Jacobi-Bellman PDE Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Optimal control problems and HJB PDE: summary Controlled Diffusion: dXtα,i = bi (t, Xtα , αt ) dt + d X σi,j (t, Xtα,i , αt ) dWtj , i ∈ {1, . . . , n} j=1 u(t, x) = EQ sup (αs ,s∈[t,T ]) adapted, αs ∈A Z T f (s, Xsα , αs )ds + g(XTα ) Xtα = x t For a fixed control α: ∂t u(t, x) + Lα u(t, x) + f (t, x, α) = 0 u(T, x) = g(x) Hamilton-Jacobi-Bellman PDE: ∂t u(t, x) + sup {f (t, x, α) + Lα u(t, x)} = 0 = g(x) α∈A u(T, x) Many nonlinear problems that have recently arisen in quantitative finance are described by HJB PDEs. Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Stochastic optimal control problems (Ω, (Ft )0≤t≤T , Q) denotes a probability space equipped with a d-dimensional Brownian motion W ; (Ft ) denotes the natural filtration of W. We introduce the following SDE dXtα,i = bi (t, Xtα , αt ) dt + d X σi,j (t, Xtα , αt ) dWtj , i ∈ {1, . . . , n} (25) j=1 where the drift b and the volatility σ depend on t, Xtα and a control parameter αt , which is an Ft -adapted process valued in a domain A, not necessarily compact. We denote At,T = {(αs )t≤s≤T adapted | ∀s ∈ [t, T ], αs ∈ A} Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Stochastic optimal control problems Controlled diffusion process dXtα,i = bi (t, Xtα , αt ) dt + d X σi,j (t, Xtα , αt ) dWtj , i ∈ {1, . . . , n} j=1 with Ito generator La = n X n bi (t, x, a)∂i + i=1 d 1 X X σi,k (t, x, a)σj,k (t, x, a)∂ij 2 i,j=1 k=1 In order to ensure that SDE (26) admits a strong solution, we assume that b, σ satisfy: Assum(SDEα ): b and σ are Lipschitz-continuous functions in x uniformly in t and α, and satisfy a linear growth condition: |b(t, x, α) − b(t, y, α)| + |σ(t, x, α) − σ(t, y, α)| ≤ C|x − y| |b(t, x, α)| + |σ(t, x, α)| ≤ C (1 + |x|) for every t ≥ 0, x, y ∈ Rn and α ∈ A where C is a positive constant. This condition ensures that " # EQ sup |Xsα |2 < ∞ 0≤s≤T Julien Guyon Nonlinear Option Pricing (26) Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Stochastic optimal control problems A standard stochastic optimal control problem consists of maximizing a reward function (or minimizing a cost function) J given by Z T J(t, x, α) := EQ f (s, Xsα , αs ) ds + g(XTα ) Xtα = x t with respect to the control α ∈ At,T : Z T u(t, x) := sup EQ f (s, Xsα , αs ) ds + g(XTα ) Xtα = x α∈At,T (27) t where the initial condition is Xtα = x. In order to ensure that the reward function is finite, we assume that the supremum is taken over the subset A∗t,T of At,T satisfying Z T EQ |f (s, Xsα , αs )| ds + |g(XTα )| < ∞ (28) t Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Stochastic optimal control problems: standard form (f = 0) By augmenting the state variables Xtα with the path-dependent continuous variable dZt = f (t, Xtα , αt ) dt, Z0 = 0 our stochastic control problem can be written without loss of generality as u(t, x̃) = sup EQ [g̃(X̃Tα )|X̃tα = (x, z)] − z α∈A∗ t,T with X̃ α = (X α , Z) and g̃(x̃) = g(x) + z. We shall appeal to this standard form (i.e., f = 0) u(t, x) := sup EQ [g(XTα )|Xtα = x] (29) α∈A∗ t,T to review results on stochastic control. Assum(g): g has quadratic growth, i.e., there exists C ≥ 0 such that for all x ∈ Rn , |g(x)| ≤ C 1 + |x|2 . From the integrability result (26), this assumption on g implies that A∗t,T = At,T . Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Bellman’s principle Bellman’s principle, also known as dynamic programming principle (DPP): u(t, x) = α EQ [u(t + h, Xt+h )|Xtα = x] sup (30) α∈At,t+h Bellman’s principle also holds for any stopping time τ with t ≤ τ ≤ T , Q−a.s. u(t, x) = sup EQ [u(τ, Xτα )|Xtα = x] α∈At,τ Proof. (i). Using iterated conditional expectation, one gets h i u(t, x) = sup EQ EQ [ g (XTα )| Ft+h ] Xtα = x α∈At,T By the Markov property of X α and the definition of u, one has α α ≤ u t + h, Xt+h , EQ [ g (XTα )| Ft+h ] = EQ g (XTα )| Xt+h Hence u(t, x) ≤ sup α∈At,t+h Julien Guyon Nonlinear Option Pricing α α EQ u t + h, Xt+h Xt = x . ∀α ∈ At,T . Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Bellman’s principle (ii). For any α ∈ At,t+h , there is an αϵ ∈ At+h,T such that h i ϵ α α u t + h, Xt+h ≤ EQ g XTα Xt+h + ϵ. We glue together α and αϵ : ( α̃ϵs = αs αϵs if s ∈ [t, t + h) . if s ∈ [t + h, T ] Then α̃ϵs ∈ At,T and h h i i α ϵ α α EQ u t + h, Xt+h Xt = x ≤ EQ EQ g XTα Xt+h Xtα = x + ϵ h h i i ϵ = EQ EQ g XTα Ft+h Xtα = x + ϵ h h i i ϵ = EQ EQ g XTα̃ Ft+h Xtα = x + ϵ h i ϵ = EQ g XTα̃ Xtα = x + ϵ ≤ u(t, x) + ϵ. Letting ϵ → 0, we obtain the reverse inequality. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Formal derivation of the HJB PDE Let us take an arbitrary constant control αs = a with a ∈ A during the interval [t, t + h]. From the Bellman principle, we get a u(t, x) ≥ EQ [u(t + h, Xt+h )|Xta = x] α By applying Itô’s lemma to u(t + h, Xt+h ) (u should be smooth enough, 1,2 n u ∈ C ([0, T ) × R )), we get Z t+h u(t, x) ≥ EQ u(t, x) + (∂s + La ) u(s, Xsa ) ds + M Xta = x t R t+h where M := t Du(s, Xsa )σ(s, Xsa , a) dWs and La is the Itô generator associated to Xt (26) controlled by the constant a ∈ A. By assuming that M is not only a local martingale but a true martingale, we obtain Z t+h Q a E (∂s + L ) u(s, Xs ) ds Xt = x ≤ 0 t Dividing by h > 0, letting h → 0, and permuting the limit and the expectation using the dominated convergence theorem, we finally get ∂t u(t, x) + La u(t, x) ≤ 0, Julien Guyon Nonlinear Option Pricing ∀a ∈ A (31) Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Formal derivation of the HJB PDE Then we take the optimal control α∗ that realizes the supremum in Equation (30). By following a similar route, we get ∗ ∂t u(t, x) + Lαt u(t, x) = 0 (32) By combining Equations (31) and (32), we get that u, as given by Equation (29), is a solution to the following parabolic second order fully nonlinear PDE, the so-called HJB equation: ∂t u(t, x) + sup La u(t, x) = 0, u(T, x) = g(x) (33) a∈A The interpretation is clear: For a deterministic constant “control” a, the linear PDE reads − ∂t u(t, x) = La u(t, x), u(T, x) = g(x) At each time, we actually have the possibility to choose a in A so as to maximize u(0, ·). Since u is known at the final date, a ∈ A must be chosen so that − ∂t u is maximized. This yields − ∂t u(t, x) = sup La u(t, x), a∈A which is Equation (33). Julien Guyon Nonlinear Option Pricing u(T, x) = g(x) Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM HJB PDE with source term f and discount factor For our initial stochastic control problem with a source term (27), the HJB PDE reads ∂t u(t, x) + sup {La u(t, x) + f (t, x, a)} = 0, u(T, x) = g(x) (34) a∈A When we include a discount factor: h RT α u(t, x) = sup EQ g(XTα )e− t r(s,Xs ,αs ) ds + α∈At,T Z T i Rs α e− t r(u,Xu ,αu )du f (s, Xsα , αs ) ds Xt = x (35) t the HJB PDE reads ∂t u(t, x) + sup {La u(t, x) + f (t, x, a) − r(t, x, a)u(t, x)} = 0 (36) = g(x) a∈A u(T, x) Note that if the control can only be chosen at discrete dates, the HJB equation is not applicable because it assumes that the control can be chosen continuously in time. In these cases, we must rely on the (discrete) Bellman’s dynamic programming equation. Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Derivation of HJB PDE directly with source term f Bellman’s dynamic programming principle: Z t+h α f (s, Xsα , αs ) ds + u(t + h, Xt+h ) Xtα = x u(t, x) = sup EP αs∈[t,t+h] ∈A t For any constant control a on [t, t + h], Z t+h a u(t, Xta ) ≥ EQ f (s, Xsa , a) ds + u(t + h, Xt+h ) Xta = x t Z t+h = EQ f (s, Xsa , a) ds + u(t, Xta ) + t Z t+h (∂s + La ) u(s, Xsa ) ds + M Xta = x t Z t+h 0 ≥ EQ f (s, Xsa , a) ds + t Z t+h (∂s + La ) u(s, Xsa ) ds Xta = x t Letting h → 0, one gets f (t, x, a) + (∂t + La ) u(t, x) ≤ 0. For the optimal control α∗ , the inequality becomes equality: ∗ f (t, x, α∗t ) + ∂t + Lαt u(t, x) = 0 ∂t u(t, x) + sup {f (t, x, α) + Lα u(t, x)} = 0 α∈A Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Verification Theorem (Verification Theorem) Let U ∈ C 1,2 ([0, T ) × Rn ) ∩ C 0 ([0, T ] × Rn ) be a function with quadratic growth in x, uniformly in t, and for all (t, x) ∈ [0, T ) × Rn , there exists α∗ (t, x) ∈ A such that ∗ −∂t U (t, x) − sup {Lα U (t, x)} = −∂t U (t, x) − Lα U (t, x) = 0 α∈A ∗ and that the SDE with the control α∗ admits a strong solution Xtα . We also assume that U (T, ·) = g. Then U = u in [0, T ] × Rn . For any admissible control α, Itô’s lemma gives Z T U (T, XTα ) = U (t, Xtα ) + (∂s U + Lα U (s, Xsα )) ds + martingale. t Taking expectation E [ ·| Xtα = x] yields U (t, x) ≥ supα E g(XTα ) Xtα = x . i h ∗ ∗ Setting α = α∗ one has U (t, x) = E g(XTα ) Xtα = x . Thus U = u. For non-smooth solutions, consider viscosity solutions. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Example 1: The uncertain volatility model Here the nonlinearity arises as a result of modeling choices. We model the asset Xt (n = d = 1) by a positive local Itô (Ft , Q)-martingale (zero interest rates, repos, and dividends for simplicity): dXt = σt Xt dWt The volatility σt is unspecified for the moment. In the UVM introduced in Avellaneda-Levy-Paras and Lyons (1995), the volatility is uncertain. As a minimal modeling hypothesis, we only assume that: the volatility is valued in a compact interval [σ, σ], and it cannot look into the future (i.e., it is an Ft -adapted process). Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM The uncertain volatility model Consider an option delivering a payoff FT at maturity T which is a function of the asset path (Xt , 0 ≤ t ≤ T ). Then the time-t value ut of the option is (worst case pricing): ut = sup EQ [FT |Ft ] (37) [t,T ] Here, sup[t,T ] means that the supremum is taken over all (Ft )-adapted processes (σs )t≤s≤T such that for all s ∈ [t, T ], σs belongs to the domain [σ, σ]. We cannot control the volatility of the underlying. However, the pricing formulation is equivalent to that of the value function in a stochastic control problem. Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM The uncertain volatility model: vanilla options For vanilla payoffs FT = g(XT ), the price u(t, x) is solution to a nonlinear PDE: ∂t u(t, x) + 1 sup σ 2 x2 ∂x2 u(t, x) = 0, 2 σ∈[σ,σ] u(T, x) = g(x) Taking the supremum over σ, we get a fully nonlinear PDE called the Black-Scholes-Barenblatt equation (in short BSB) ∂t u(t, x) + 2 1 2 x Σ ∂x2 u(t, x) ∂x2 u(t, x) 2 u(T, x) (t, x) ∈ [0, T ) × R∗+ = 0, = g(x), x ∈ R∗+ with Σ(Γ) = σ1Γ<0 + σ1Γ≥0 . Fully nonlinear PDE: the nonlinearity affects the 2nd order derivative ∂x2 u. If we consider a convex payoff, u coincides with the Black-Scholes price with the upper volatility σ. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Example 2: The uncertain mortality model for reinsurance deals Examples of reinsurance deals: GMxB deals. GMxB stands for GMIB (Guaranteed Minimum Investment Benefit) or GMDB (Guaranteed Minimum Death Benefit). Let us ignore the usual lapse feature of GMxB deals. Then those deals consist of only two payoffs: at maturity T , the seller of the option pays the payoff g(XT ) with XT the value of an asset; in the case where the insurance subscriber dies before maturity, the seller pays at time t the payoff gD (Xt ) to the beneficiaries of the insurance policy. The index D stands for default. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM The uncertain mortality model for reinsurance deals Like in credit modeling, we model the time to default τ (here, the time of death) by the first time of default of a Poisson process with intensity λt (the mortality rate). In the uncertain mortality model, this rate λt is assumed to be uncertain and to range into the interval [λ(t), λ(t)]. Also, the rate cannot look into the future, i.e., it must be adapted. The fair value of this option is then given by (worst case pricing): u(t, x) = sup EQ [g(XT )1τ ≥T + gD (Xτ )1τ <T |Xt = x] λ∈Λt,T where Λt,T is the set of all adapted processes λ such that for all s ∈ [t, T ], λs ∈ [λ(s), λ(s)]. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM The uncertain mortality model for reinsurance deals u is solution to a nonlinear PDE: ∂t u(t, x) + Lu(t, x) + sup λ (gD (x) − u(t, x)) = 0 λ∈[λ(t),λ(t)] which is equivalent to ∂t u(t, x) + Lu(t, x) + Λ (t, gD (x) − u) = 0 with the terminal condition u(T, x) = g(x) and Λ defined by λ(t)y if y ≥ 0 Λ(t, y) = λ(t)y if y < 0 Semilinear PDE: the nonlinearity does not affect the 2nd order derivative ∂x2 u. Here the nonlinearity affects only u. No analytical solution. How to numerically compute u? PDE solver useful only when X has dimension 1 or 2. We will discuss Monte Carlo methods (Least Squares Monte Carlo, BSDEs). Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Example 3: Portfolio optimization Merton problem (a genuine stochastic control problem): consider an investor who can invest over the horizon [0, T ] in two assets: a risk-free asset Bt with rate of return r > 0 and a risky asset St (“stock”): dBt = rBt dt dSt = µSt dt + σSt dWt Let XtB denote the investor’s wealth in the bank at time t ≥ 0, XtS the wealth in stock, Xt = XtB + XtS the investor’s total wealth, and πt = XtS /Xt the proportion of wealth invested in stock. The investor has some initial capital X0 = x > 0 to invest and there are no further cash injections. The investor can decide at each time t ≥ 0 how much money to hold in stock: by choosing, at time t, the value πt the investor controls the evolution of his/her wealth in the future. The investor’s goal is to maximize expected total utility at a final time T : J(t, x, π) = E [g(XT )|Xt = x] , u(t, x) = sup π∈A(t,x) u is solution to a nonlinear PDE. Julien Guyon Nonlinear Option Pricing J(t, x, π) UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Portfolio optimization: Exercise 1 S What are the quantities ∆B t and ∆t of risk-free asset and risky asset held by the investor at time t? Show that, given a portfolio process π, the wealth of the investor evolves according to the SDE dXtx,π = (r + πt (µ − r)) Xtx,π dt + σπt Xtx,π dWt 2 Write down the HJB PDE for this problem. 3 Show that it reads ∂t v + rx∂x v + π ∗ (µ − r)x∂x v + with π ∗ (t, x) = − Julien Guyon Nonlinear Option Pricing 1 ∗ 2 2 2 2 (π ) σ x ∂x v 2 v(T, x) (µ − r)x∂x v(t, x) σ 2 x2 ∂x2 v(t, x) = 0 = g(x) (38) Early Exercise Option Pricing in a Nutshell Stochastic control BSDE Backward Stochastic Differential Equations I find it hard to focus looking forward. So I look backward. — Iggy Pop Julien Guyon Nonlinear Option Pricing ULMM UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Classical Feynman-Kac Formula Linear Parabolic PDE: ∂t u(t, x) + Lu(t, x) + a(t, x)u(t, x) + c(t, x) = 0, u(T, x) = g(x) d where (t, x) ∈ [0, T ] × R and L= d X i=1 d bi (t, x)∂i + r 1 X X σi,k σj,k ∂i,j 2 i,j=1 k=1 We consider a d-dimensional Itô process defined by the (forward) SDE dXt = b(t, Xt ) dt + σ(t, Xt ) dWt where b ∈ Rd , σ ∈ Rd×r and W is an r-dimensional standard Brownian motion. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Let Yt = u(t, Xt ). Applying Itô’s lemma gives dYt = (∂t u + Lu) dt + σ(t, Xt )T Dx u(t, x) · dWt = (−a(t, Xt )u(t, Xt )dt − c(t, Xt )) dt + σ(t, Xt )T Dx u(t, Xt ) · dWt . and therefore Rt e 0 a(s,Xs )ds Yt + Z t Rs c(s, Xs )e 0 a(v,Xv )dv ds is a martingale. 0 Classical Feynman-Kac " RT u(t, x) = E g(XT )e t a(r,Xr )dr + Z T Rr c(r, Xr )e t a(s,Xs )ds dr Xt = x t Provides probabilistic representation of solutions of linear parabolic PDEs. Julien Guyon Nonlinear Option Pricing # Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM First-Order BSDE - Motivation Semilinear Parabolic PDE: ∂t u(t, x) + Lu(t, x) + f t, x, u(t, x), σ(t, x)T Dx u(t, x) = 0, u(T, x) = g(x) dYt = (∂t u + Lu) dt + σ(t, Xt )T Dx u(t, Xt ) · dWt = −f t, Xt , Yt , σ(t, Xt )T Dx u(t, Xt ) dt + σ(t, Xt )T Dx u(t, Xt ) · dWt . Let Zt = σ(t, Xt )T Dx u(t, Xt ), dYt = −f (t, Xt , Yt , Zt ) dt + Zt · dWt with the terminal condition YT = g (XT ) Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM First-Order BSDE - Definition Definition (First-Order BSDE) A solution to a (Markovian) first-order BSDE is a couple (Y, Z) of (Ft )-adapted Itô processes taking values in R × Rr and satisfying dYt = −f (t, Xt , Yt , Zt ) dt + Zt · dWt with the terminal condition YT = g(XT ). The above equation means that Z T Z T Yt = g(XT ) + f (s, Xs , Ys , Zs ) ds − Zs · dWs t t The deterministic function f is called the driver or generator. We emphasize that the diffusion term Z is a part of the solution. They provide a probabilistic representation of solutions of semi-linear parabolic PDEs, generalizing the Feynman-Kac formula. Interpretation: payoff g(XT ), price at time t, Yt , delta strategy Zt , source term f (t, Xt , Yt , Zt ). Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM First-Order BSDE - Existence and Uniqueness Theorem (Pardoux-Peng) Assume that f : [0, T ] × Rd × R × Rr → R is a deterministic function satisfying the Lipschitz condition |f (t, x, y1 , z1 ) − f (t, x, y2 , z2 )| ≤ K (|y1 − y2 | + |z1 − z2 |) for some constant K independent of (y1 , y2 ) ∈ R and (z1 , z2 ) ∈ Rr , that Z T E |f (t, Xt , 0, 0)|2 dt < ∞ 0 2 d and that g ∈ L (Ω, F, R ). Then there is a unique adapted solution (Y, Z) to the 1-BSDE satisfying Z T sup E Yt2 + E |Zt |2 dt < ∞ 0≤t≤T Julien Guyon Nonlinear Option Pricing 0 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Martingale Representation Theorem Example (Martingale Representation Theorem) We consider the case where f does not depend on Y and Z, dYt = −f (t, Xt )dt + Zt · dWt . Rt Then Yt + 0 f (s, Xs )ds is an (Ft )-martingale with terminal value R g(XT ) + 0T f (s, Xs ) ds at t = T . " # Z Z t Yt + T f (s, Xs ) ds Ft =: Mt f (s, Xs ) ds = E g(XT ) + 0 0 According to the martingale representation theorem, there exists a unique Z s.t. Z t Mt = M0 + Zs · dWs . 0 " Define Yt := Mt − Rt 0 f (s, Xs ) ds = E # g(XT ) + RT t f (s, Xs ) ds Ft . Then Y is adapted and YT = g(XT ), dYt = Zt · dWt − f (t, Xt )dt. The pair (Y, Z) gives the solution to the BSDE. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Linear First-Order BSDE Example (Linear 1-BSDE) dYt = − (at Yt + bt · Zt + ct ) dt + Zt · dWt , YT = g (XT ) where at , bt , and ct are deterministic functions of time. Let dΓt = Γt (at dt + bt · dWt ), Γ0 = 1. Z t Z t Z t 1 |br |2 dr + ar dr br · dWr − Γt = exp 2 0 0 0 Applying Itô’s lemma to Γt Yt , we obtain d (Γt Yt ) = −ct Γt dt + Γt (Zt + bt Yt ) dWt . R Then Γt Yt + 0 cr Γr dr is a Ft -martingale with terminal value g(XT )ΓT + 0T cr Γr dr h i R at t = T . Define Mt := E g(XT )ΓT + 0T cr Γr dr Ft and Yt by # " Z Z Rt Yt := Γ−1 t t Mt − cr Γr dr 0 = Γ−1 t E g(XT )ΓT + T cr Γr dr Ft . t The existence and uniqueness of Zt is guaranteed from the martingale representation theorem applied to Mt . Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM First-Order BSDE - Numerical Simulation Divide (0, T ) into subintervals (ti−1 , ti ), 1 ≤ i ≤ n, and set ∆ti = ti − ti−1 , ∆Wti = Wti − Wti−1 , and ∆ = maxi ∆ti Simulating the forward process X gives us values of XT = Xtn and YT = Ytn = g(Xtn ). Then we proceed backward Z ti Z ti Yti − Yti−1 = − f (s, Xs , Ys , Zs ) ds + Zs · dWs ti−1 ti−1 Discretize the integral Yti − Yti−1 ≈ −f ti−1 , Xti−1 , Yti−1 , Zti−1 ∆ti + Zti−1 · ∆Wti Taking conditional expectation Ei−1 := E ·|Fti−1 Yti−1 = Ei−1 [Yti ] + f (ti−1 , Xti−1 , Yti−1 , Zti−1 )∆ti 1 Ei−1 [Yti ∆Wti ] Zti−1 = ∆ti Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM First-Order BSDE - Numerical Simulation Implicit scheme: Use regression to estimate the conditional expectations Ei−1 [Yti ] and Ei−1 [Yti ∆Wti ], and then solve, say, by a root-finding routine, for Yti−1 . Explicit schemes: Yti−1 = Ei−1 [Yti + f (ti , Xti , Yti , Zti )∆ti ] or Yti−1 = Ei−1 [Yti ] + f (ti−1 , Xti−1 , Ei−1 [Yti ], Zti−1 )∆ti θ-schemes Yti−1 = Ei−1 [Yti ] + θEi−1 [f (ti , Xti , Yti , Zti )] ∆ti + (1 − θ) f ti−1 , Xti−1 , Yti−1 , Zti−1 ∆ti or Yti−1 = Ei−1 [Yti ] + θf (ti , Xti , Ei−1 [Yti ] , Zti ) ∆ti + (1 − θ) f ti−1 , Xti−1 , Yti−1 , Zti−1 ∆ti Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM First-Order BSDE - Numerical Simulation - Example Example (CVA pricing) The semilinear PDE arising in the pricing of counterparty risk in case of risky close-out is ∂t u + Lu + β(u+ − u) = 0, u(T, x) = g(x) The corresponding BSDE is dYt = −β(Yt+ − Yt ) dt + Zt · dWt , YT = g(XT ) This can be solved numerically using either of the following schemes: Yti−1 = 1 Ei−1 [Yti ] 1 + β∆ti 1Ei−1 [Yti ]≤0 or Yti−1 = Ei−1 [Yti + β∆ti (Yt+ − Yti )] i or Yti−1 = Ei−1 [Yti ] + β∆ti (Ei−1 [Yti ]+ − Ei−1 [Yti ]) or θ-schemes. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Second Order BSDE - Motivation Consider the fully nonlinear parabolic PDE ∂t u(t, x) + f (t, x, u, Dx u, Dx2 u) = 0, u(T, x) = g(x) where (t, x) ∈ [0, T ) × Rd . Assume that dXt = σ(t, Xt )dWt Let Yt = u(t, Xt ), Zt = Dx u(t, Xt ), Γt = Dx2 u(t, Xt ), and apply Itô’s formula dYt = 1 −f (t, Xt , Yt , Zt , Γt ) + tr σ(t, Xt )σ(t, Xt )T Γt dt + Zt · σ(t, Xt ) dWt 2 dZt = (∂t + L)Dx u(t, Xt ) dt + Γt σ(t, Xt )dWt Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Second Order BSDE - Definition Definition (Second order BSDE) Let (Yt , Zt , Γt , αt )t∈[0,T ] be a quadruple of (Ft )-adapted processes taking values in R, Rd , S d , and Rd respectively.a We call (Y, Z, Γ, α) a solution to a (Markovian) 2-BSDE if dYt = −f (t, Xt , Yt , Zt , Γt ) dt + Zt ◦ σ(t, Xt ) dWt dZt = αt dt + Γt σ(t, Xt ) dWt where YT = g(XT ). The use of the Stratonovich product ◦ is only for convenience and can be replaced by an Itô integral 1 ′ dYt = −f (t, Xt , Yt , Zt , Γt ) + tr σ(t, Xt )σ(t, Xt ) Γt dt+Zt ·σ(t, Xt ) dWt 2 a S d stands for the space of d-dimensional symmetric matrices. They provide a probabilistic representation of solutions of fully non-linear parabolic PDEs, generalizing the Feynman-Kac formula. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE Second Order BSDE - Numerical Simulation Following the same discretization scheme as in 1-BSDE: Ytn = g(Xt∆n ) Ztn = Dg(Xtn ) Yti−1 = Ei−1 [Yti ] + f (ti−1 , Xti−1 , Yti−1 , Zti−1 , Γti−1 ) 1 − tr[σ(ti−1 , Xti−1 )σ(ti−1 , Xti−1 )T Γti−1 ] ∆ti 2 −1 1 σ(ti−1 , Xti−1 )T Ei−1 [Yti ∆Wti ] Zti−1 = ∆ti 1 Γti−1 = Ei−1 [Zti ∆Wt′i ]σ(ti−1 , Xti−1 )−1 ∆ti Julien Guyon Nonlinear Option Pricing ULMM UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Uncertain Lapse and Mortality Model Yes, man is mortal, but that would be only half the trouble. The worst of it is that he’s sometimes unexpectedly mortal — there’s the trick! — Mikhail Bulgakov Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Reinsurance Deals Reinsurance deals are sold by banks to insurance companies that need to hedge their book of variable annuities. A typical reinsurance deal consists of three payoffs: At maturity, in the case where the insurance subscriber is still alive, the issuer delivers a put on the net asset value (NAV) X of the underlying fund, whose strike is denoted Kmat . umat (x) = (Kmat − x)+ In the case where the insurance subscriber dies before maturity, the issuer delivers a put on X whose strike is denoted KD . uD (t, x) = (KD − x)+ Each month, the issuer receives a constant fee. All three payoffs are canceled as soon as the insurance subscriber lapses, which he or she can do whenever he or she wants: uL (t, x) = 0 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Deterministic lapse and mortality model Assumptions We model death (D) and lapse (L) as independent default events, jointly independent of the path (Xt , 0 ≤ t ≤ T ) of the NAV of the underlying fund. They are supposed to be first jump times of two independent Poisson L processes, τ D and τ L , with respective deterministic intensities λD t and λt . Once one of these defaults occurs, the product is canceled, and the NAV is unchanged. The payoffs umat (·), uD (t, ·) and uL (t, ·) are functions of the underlying asset price at the time (Markov payoff). For simplicity, we assume the underlying asset has constant volatility σ, and interest and repo rates are zero. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Deterministic lapse and mortality model The fair value of this contract is Z T h mat α1τ D ≥s 1τ L ≥s ds (XT )1τ D ≥T 1τ L ≥T − u(t, x) = EQ t,x u t D D + u (τ , Xτ D )1τ D <T 1τ D ≤τ L + uL (τ L , Xτ L )1τ L <T 1τ L <τ D i h RT D RT L mat mat (XT ) (XT )1τ D ≥T 1τ L ≥T = e− t λs ds− t λs ds EQ EQ t,x u t,x u Z T Z T Rs D Rs L = −α e− t λv dv e− t λv dv ds α1 1 ds EQ − D L t,x τ ≥s τ ≥s t t EQ t,x h EQ t,x h D D L L u (τ , Xτ D )1τ D <T 1τ D ≤τ L u (τ , Xτ L )1τ L <T 1τ L <τ D i where Et,x [·] = E [ ·| Xt = x, τ > t]. Julien Guyon Nonlinear Option Pricing i i = EQ t,x = EQ t,x Z T Rs D Rs L − t λv dv − t λv dv uD (s, Xs )λD e ds s e t Z T t Rs L Rs D − t λv dv − t λv dv uL (s, Xs )λL e ds se Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Deterministic lapse and mortality model Summing up, we get h RT D L i − t (λs +λs )ds mat u(t, x) = EQ u (XT ) + t,x e Z T R Q dv D D L L +λL − ts (λD ) v v Et,x −α + u (s, Xs ) λs + u (s, Xs ) λs ds e t By the Itô formula, we have the linear PDE ∂t u + 1 2 2 2 σ x ∂x u − α + λD uD − u + λ L uL − u = 0 t t 2 with the terminal condition u(T, x) = umat (x). Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Deterministic lapse and mortality model The P&L of the issuer’s delta-hedged position incurred between t and t + dt, averaged over mortality and lapse events EQ [ dP&Lt | Xs , 0 ≤ s ≤ T, and no mortality or lapse has occured before t] 1 2 2 2 L = 1 − λD dt − λ dt σ X ∂ u(t, X )dt + (∆ − ∂ u(t, X )) dX −∂ u(t, X )dt − t t t x t t t t t x 2 + αdt D − λD t dt u (t, Xt ) − u(t, Xt ) L − λL t dt u (t, Xt ) − u(t, Xt ) + O dt2 Choose ∆ = ∂x u(t, Xt ) to hedge the risk from the underlying asset Mortality and lapse risks are hedged in average. The total P&L per contract vanishes in the limit by the strong law of large numbers when the number of contracts tends to infinity. Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Deterministic lapse and mortality model - Monte Carlo I. Direct simulation of default times L D For simplicity, assume λD and λL respectively. t and λt are constants λ 1 2 3 Simulate N random future market paths, denote by Xtp , 0 ≤ t ≤ T , the underlying asset price at time t on path p; For each path p, simulate exponential random variables τ D,p and τ L,p with rates λD and λL respectively; For each path p, compare τ D,p , τ L,p and T , and then compute the payoff as follows: if T is the smallest, payoff is umat T, XTp − αT ; if τ D is the smallest, payoff is uD τ D,p , XτpD,p − ατ D,p ; if τ L is the smallest, payoff is uL τ L,p , XτpL,p − ατ L,p ; 4 Averaging payoffs across all N paths gives an estimate of the fair value of the contract at time zero. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Deterministic lapse and mortality model - Monte Carlo II. Averaging over default events Simulate N random future market paths at dates 0 = t0 , · · · , tn = T . On each path p, denote by Xtpk the underlying asset price at time tk , Ctpk (or more precisely, our estimate of) the sum of all future cashflows from tk onwards, averaging over death and lapse events, provided that no death or lapse has occurred up to time tk . p mat 2 At tn , Ct = u tn , Xtpn n 3 At tk with 0 ≤ k < n, L p Ctpk = exp −λD tk+1 ∆tk+1 exp −λtk+1 ∆tk+1 Ctk+1 1 − α∆tk+1 L + 1 − exp −λL exp −λD tk+1 , Xtk+1 tk+1 ∆tk+1 tk+1 ∆tk+1 u uD tk+1 , Xtk+1 + 1 − exp −λD tk+1 ∆tk+1 4 Averaging Ctp0 across all N paths gives an estimate of the fair value of the contract at time zero. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Uncertain Lapse and Mortality Model (ULMM) L Assume that λD t and λt are unknown but adapted and belong to a moving D L D L L corridor, i.e. λ (t) ≤ λD t ≤ λ (t) and λ (t) ≤ λt ≤ λ (t). h RT D L − t (λs +λs )ds mat u(t, x) = sup EQ u (XT ) + t,x e λD ,λL Z T Rs D L L L e− t (λv +λv )dv −α + uD (s, Xs ) λD s + u (s, Xs ) λs ds t HJB equation: ∂t u + 1 2 2 2 σ x ∂x u − α 2 + ΛD t, uD − u uD − u + ΛL t, uL − u uL − u = 0 with terminal condition u(T, x) = umat (x), where ( D λ (t) D Λ (t, y) = λD (t) ( L λ (t) L Λ (t, y) = λL (t) if y ≥ 0 otherwise if y ≥ 0 otherwise Julien Guyon The lowest price guaranteeing that the conditional expectation of the global P&L of Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Uncertain Lapse and Mortality Model (ULMM) The insurance subscriber can always lapse. Assume that the fee α matches exactly the managing fee that the subscriber pays to the insurance company. A rational policyholder should lapse as soon as uL > u. A GMxB deal is therefore an American option. Why not price it as such? It is always possible to price reinsurance deals as American options, but the resulting price is often far from the market quotes. In fact, it has been observed that, so far, policyholders do L not exercise optimally, which is taken into account through the λ function. L The λ function accounts for the fact that policyholders may not lapse when they should. The λL function accounts for the fact that policyholders may L lapse when they should not. Choosing λ = +∞ and λL = 0 boils down to pricing the GMxB as an American option: 1 max ∂t u + σ 2 x2 ∂x2 u − α + ΛD t, uD − u uD − u , uL − u = 0 2 with terminal condition u(T, x) = umat (x). Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Numerical results 45678 86 30 !2"10 !2"0 1222 32 10 12 0 2 02 #0 #12 Julien Guyon Nonlinear Option Pricing $2 %2 &2 '2 122 *+ 112 132 1(2 1)2 102 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Longstaff-Schwartz-like method (G., 2007) 1 Simulate N1 random future market paths at dates t0 , · · · , tn = T . On each path p, denote by Xtpk the underlying asset price at time tk D,p λL,p tk and λtk our estimates of the optimal lapse and death rates at time tk Ctpk our estimate of the sum of all future cashflows from tk onwards, averaging over death and lapse events with lapse and death rates λL,p tj and λD,p for tj > tk , provided that no death or lapse has occurred up to time tj tk . 2 At date tn = T , Ctpn = umat tn , Xtpn L tn , uL tn , Xtpn − uptn λL,p tn = Λ D λD,p tn , uD tn , Xtpn − uptn tn = Λ Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Longstaff-Schwartz-like method (G., 2007) 3 At date tk with 0 = k < n, D,p p Ctpk = exp −λL,p tk+1 ∆tk+1 exp −λtk+1 ∆tk+1 Ctk+1 − α∆tk+1 L + 1 − exp −λL,p exp −λD,p tk+1 , Xtpk+1 tk+1 ∆tk+1 tk+1 ∆tk+1 u + 1 − exp −λD,p uD tk+1 , Xtpk+1 tk+1 ∆tk+1 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Longstaff-Schwartz-like method (G., 2007) 4 Find a functional estimate û tk , Xtpk of EQ Ctpk Xtpk by parametric or non-parametric regression. Update the mortality and lapse rates at tk by ( λL,p tk = λD,p tk = ( L λ (tk ) λL (tk ) D λ (tk ) λD (tk ) if uL tk , Xtpk − û tk , Xtpk ≥ 0 otherwise if uD tk , Xtpk − û tk , Xtpk ≥ 0 otherwise p D,p With these definitions of λL,p tk and λtk , we can now estimate the Ctk−1 at tk−1 , etc. This is how the backward induction works. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Longstaff-Schwartz-like method 5 Simulate N2 new independent paths to obtain a lower-bound price. At date tn = T , Ctpn = umat tn , Xtpn At date tk , k < n, D,p p Ctpk = exp −λL,p tk+1 ∆tk+1 exp −λtk+1 ∆tk+1 Ctk+1 − α∆tk+1 L + 1 − exp −λL,p exp −λD,p tk , Xtpk+1 tk+1 ∆tk+1 tk+1 ∆tk+1 u + 1 − exp −λD,p uD tk+1 , Xtpk+1 tk+1 ∆tk+1 D,p where λL,p (1 ≤ p ≤ N2 ) are determined by comparing tk and λtk p L D u tk , Xtk and u tk , Xtpk with the functional estimate û(tk , Xtpk ) where û was computed during step 4. Averaging Ctp0 across all N2 paths to get a lower-bound estimate. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM BSDE for ULMM ∂t u + 1 2 2 2 σ x ∂x u − α 2 + ΛD t, uD − u uD − u + ΛL t, uL − u uL − u = 0 The nonlinear part of f involves u but not ∂x u, ∂x2 u, we will use 1-BSDEs. u(0, x) can be represented by the solution Y0x to the 1-BSDE dYt = −f (t, Xt , Yt , Zt ) dt + Zt dWt with the terminal condition YT = umat (XT ), where dXt = σXt dWt , X0 = x and f (t, x, y, z) = −α + ΛD t, uD (t, x) − y uD (t, x) − y + ΛL t, uL (t, x) − y uL (t, x) − y Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM BSDE for ULMM Yt∆ = umat (Xt∆n ) n Implicit Scheme ∆ Yt∆ = EQ i−1 [Yti ] − α∆ti i−1 + ΛD ti−1 , uD (ti−1 , Xt∆i−1 ) − Yt∆ uD (ti−1 , Xt∆i−1 ) − Yt∆ ∆ti i−1 i−1 + ΛL ti−1 , uL (ti−1 , Xt∆i−1 ) − Yt∆ uL (ti−1 , Xt∆i−1 ) − Yt∆ ∆ti i−1 i−1 Explicit Scheme h Q Yt∆ = E Yt∆ − α∆ti i−1 i i−1 D ∆ ∆ + ΛD ti , uD (ti , Xt∆i ) − Yt∆ u (t , X ) − Y ∆ti i t t i i i i L ∆ ∆ + ΛL ti , uL (ti , Xt∆i ) − Yt∆ u (t , X ) − Y ∆t i i t t i i i Note: No need to compute Zt∆i−1 as f does not depend on z. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Numerical results 45678 86 30 !"# !$ 32 10 12 0 2 02 %0 %12 Julien Guyon Nonlinear Option Pricing &2 '2 112 )* 1(2 102 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Numerical results Optimal lapse frontier in the time-spot plane. Left: using MC. Right: using PDE 14 Julien Guyon Nonlinear Option Pricing 012345678946 194 327914 32814 012346789 47 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Numerical results Optimal mortality frontier in the time-spot plane. Left: using MC. Right: using PDE 43 Julien Guyon Nonlinear Option Pricing 0123456378219362 43 1345 1345 012345637921 3620 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE Uncertain Volatility Model Il n’est pas certain que tout soit incertain.3 — Blaise Pascal 3 “It is not certain that everything is uncertain.” Julien Guyon Nonlinear Option Pricing ULMM UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM The 1D Uncertain Volatility Model (UVM) Avellaneda-al (1995), Lyons (1995) dXt = σt Xt dWt , σt ∈ [σ, σ], Vt = sup E[g(XT )|Ft ] [t,T ] where the supremum is taken over all (Fs )-adapted processes (σs )t≤s≤T such that for all s ∈ [t, T ], σs ∈ [σ, σ]. Worst-case scenario for the sell-side HJB: Vt = u(t, Xt ) with ∂t u(t, x) + 1 sup x2 σ 2 ∂x2 u(t, x) = 0 2 σ Black-Scholes-Barenblatt (BSB) PDE: ∂t u(t, x) + 2 1 2 x Σ ∂x2 u ∂x2 u(t, x) = 0, 2 (t, x) ∈ [0, T ) × R∗+ with u(T, x) = g(x) and Σ(Γ)2 = σ 2 1Γ≥0 + σ 2 1Γ<0 . Julien Guyon Nonlinear Option Pricing (39) Early Exercise Option Pricing in a Nutshell Stochastic control BSDE Analogies Transaction cost [Leland]: Same equation with r 2 kσ √ sign(Γ) Σ(Γ)2 = σ 2 + π δt Pricing in illiquid market [Willmott-al]: Same equation with Σ(Γ)2 = Julien Guyon Nonlinear Option Pricing σ2 (1 − ϵSΓ)2 ULMM UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Solving the pricing equation 1D Black-Scholes-Barenblatt (BSB) PDE: ∂t u(t, x) + 1 2 x Σ(∂x2 u)2 ∂x2 u(t, x) = 0, 2 (t, x) ∈ [0, T ) × R∗+ with u(T, x) = g(x) and Σ(Γ)2 = σ 2 1Γ≥0 + σ 2 1Γ<0 No analytical solution Finite-difference schemes Monte Carlo compulsory when the number of variables (underlyings and path-dep variables) is 3 or more Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Robust super-replication The buyer’s and seller’s super-replication prices have been defined w.r.t. the historical measure Phist (the definition involves the space of ELMM ∼ Phist ), meaning that the volatility process of the asset is fixed. In practice, the volatility process is unknown. In the uncertain volatility framework, we consider instead the set P of all measures Q under which Xt is a local martingale such that σ2 ≤ d⟨ln X⟩t ≡ σt2 ≤ σ 2 dt The seller’s super-replication price in the UVM is defined by n St (FT ) = inf z ∈ Ft there exists an admissible portfolio ∆ such that Z T o S πT ≡ z + ∆s dXs − FT ≥ 0 Q−a.s ∀Q ∈ P (40) t Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Robust super-replication n St (FT ) = inf z ∈ Ft there exists an admissible portfolio ∆ such that Z T o πTS ≡ z + ∆s dXs − FT ≥ 0 Q−a.s ∀Q ∈ P t Theorem St (FT ) = sup EQ [FT |Ft ] = sup E[FT |Ft ] Q∈P [t,T ] This theorem justifies our definition of the UVM price. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Robust super-replication: proof Proof. We assume that FT = g(XT ) with g ∈ Cb3 (R+ ) and σ < ∞. (i) We set z = u(t, Xt ) = supQ∈P EQ [g(XT )|Ft ] and ∆s = ∂x u(s, Xs ) with u the solution of PDE (39). As u ∈ C 1,2 ([0, T ) × R+ ), Itô’s lemma gives Z T Z T 1 − ∂s u(s, Xs ) − Xs2 σs2 ∂x2 u(s, Xs ) ds z+ ∆s dXs − g(XT ) = 2 t t Using the PDE (39), we get Z T Z 1 T 2 2 Xs ∂x u(s, Xs )(Σ(∂x2 u(s, Xs ))2 − σs2 ) ds ≥ 0 z+ ∆s dXs − g(XT ) = 2 t t 2 Q-a.s. for all Q ∈ P as Σ ∂x2 u(s, Xs ) − σs2 ∂x2 u(s, Xs ) ≥ 0 for all adapted σs such that for all s ∈ [t, T ], σs ∈ [σ, σ]. This implies that St (g(XT )) ≤ u(t, Xt ) ≡ supQ∈P EQ [g(XT )|Ft ] as we have obtained that the delta-hedging strategy (z = u(t, Xt ), ∆s = ∂x u(s, Xs )) super-replicates the payoff. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Robust super-replication: proof (ii) (Weak duality) Let z be such that there exists an admissible portfolio ∆ such that Z T z+ ∆s dXs − g(XT ) ≥ 0 Q−a.s ∀Q ∈ P t By taking the conditional expectation we obtain z ≥ EQ [g(XT )|Ft ] for all RT Q ∈ P. We have used that the local martingale t ∆s dXs bounded from below is a supermartingale. This implies that St (g(XT )) ≥ sup EQ [g(XT )|Ft ] Q∈P Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM The multidim uncertain volatility model Vt = sup E[FT |Ft ], dXtα = σtα Xtα dWtα , dWtα dWtβ = ραβ t dt [t,T ] Supremum taken over all (Fs )-adapted processes (ξs )t≤s≤T ≡ ((σsα , ραβ s )1≤α<β≤d )t≤s≤T such that for all s ∈ [t, T ], ξs belongs to some compact domain D We will consider domains D of the form D = [σ, σ] when d = 1, and D = [σ 1 , σ 1 ] × [σ 2 , σ 2 ] × [ρ, ρ] when d = 2 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Pricing vanilla options Consider vanilla payoff FT = g(XT ). We have ut = u(t, Xt ) where u(·, ·) is the unique (viscosity) solution with quadratic growth of the nonlinear PDE (t, x) ∈ [0, T ) × (R∗+ )d ∂t u(t, x) + H(x, Dx2 u(t, x)) = 0, (41) with the terminal condition u(T, x) = g(x) and the Hamiltonian d H(X, Γ) = X αβ α β α β αβ 1 max ρ σ σ X X Γ 2 (σα ,ραβ )1≤α<β≤d ∈D α,β=1 If g ∈ C 3 ((R∗+ )d ) and g, Dx g have quadratic growth, then u ∈ C 1,2 ([0, T ] × (R∗+ )d ). Julien Guyon Nonlinear Option Pricing (42) Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Pricing vanilla options - Computation of the Hamiltonian When d = 2, the Hamiltonian reads H(X, Γ) = max (σ 1 ,σ 2 ,ρ)∈D 2 1 2 2 1 1 2 σ (X 1 )2 Γ11 + σ X 2 Γ22 + ρσ 1 σ 2 X 1 X 2 Γ12 2 2 where D = σ 1 , σ 1 × σ 2 , σ 2 × ρ, ρ . The optimal correlation is bang-bang: R(Γ12 ) = ρ1Γ12 <0 + ρ1Γ12 ≥0 . Then the problem becomes a two-dimensional quadratic form maximization under a double inequality constraint, which can be solved by first freezing σ 2 and maximize in σ 1 and then compute the maximum over σ 2 . When d > 2, maximize in ρij first and then solve the quadratic programming problem. Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Pricing path-dependent options We consider the price of an option that depending on path-dependent variables whose values can change continuously. The Hamiltonian may not involve only the gammas Example: if u(t, x, v) depends on the continuously compounded realised variance v, 1 2 2 1 2 2 H(x, ∂x2 u, ∂v u) = max σ 2 x ∂x u + ∂v u = Σ2 x ∂x u + ∂v u σ≤σ≤σ 2 2 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM The BSDE approach: Probabilistic representation of PDEs The classical link between Monte Carlo and finite-difference methods as stated by the Feynman-Kac formula is only valid for linear second-order parabolic PDEs Cheridito-Touzi-Soner-Victoir provide a stochastic representation for solutions of fully nonlinear parabolic PDEs by introducing a new class of BSDEs, the so-called second-order BSDEs Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM 2-BSDE [Cheridito-al] Let (Yt , Zt , Γt , αt )t∈[0,T ] a quadruple of Ft -adapted processes taking values in R, Rd , S d (space of d-dimensional symmetric matrices) and Rd respectively. Then, we call (Y, Z, Γ, α) a solution of a 2-BSDE if dXt = σ(t, Xt )dWt dYt = −f (t, Xt , Yt , Zt , Γt )dt + Zt ◦ σ(t, Xt )dWt dZt = αt dt + Γt σ(t, Xt )dWt YT = g(XT ) (43) where ◦ is the Stratonovich integral dYt The use of the Stratonovich product is only for convenience and can be replaced by an Itô integral 1 ′ −f (t, Xt , Yt , Zt , Γt ) + tr σ(t, Xt )σ(t, Xt ) Γt dt + Zt .σ(t, Xt ) dWt = 2 Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Generalization of the Feynman-Kac formula Let us consider the fully nonlinear parabolic PDE ∂t u(t, x) + f (t, x, u, ∂x u, ∂x2 u) = 0 u(T, x) = g(x) (44) We recall Theorem Let u be a function smooth (enough to apply Itô’s formula) satisfying the above PDE. Then (Yt = u(t, Xt ), Zt = ∂x u(t, Xt ), Γt = ∂x2 u(t, Xt ), αt = (∂t + L) ∂x u(t, Xt )) is a solution of the 2-BSDE (43). Y = price, Z = delta, Γ = gamma Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Discretization scheme for 2-BSDEs Numerical simulation of 2-BSDEs = a way to build approximations of solutions to PDE (44) Proceeding by analogy with 1-BSDEs, we obtain the following (implicit) discretization scheme for 2-BSDEs (Cheridito et al.): Scheme 2-BSDE: Julien Guyon Nonlinear Option Pricing Yt∆ n = g(Xt∆n ) Zt∆n = Dg(Xt∆n ) Yt∆ i−1 = Zt∆i−1 = Γ∆ ti−1 = Ei−1 [Yt∆ ] + f (ti−1 , Xt∆i−1 , Yt∆ , Zt∆i−1 , Γ∆ ti−1 ) i i−1 1 − tr[σ(ti−1 , Xt∆i−1 )σ(ti−1 , Xt∆i−1 )′ Γ∆ ti−1 ] ∆ti 2 −1 1 σ(ti−1 , Xt∆i−1 )′ Ei−1 [Yt∆ ∆Wti ] i ∆ti 1 Ei−1 [Zt∆i ∆Wt′i ]σ(ti−1 , Xt∆i−1 )−1 ∆ti (45) Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Discretization scheme for 2-BSDEs One can also consider explicit schemes like h Yt∆ = Ei−1 Yt∆ + f (ti , Xt∆i , Yt∆ , Zt∆i , Γ∆ ti ) i i i−1 − i 1 tr[σ(ti , Xt∆i )σ(ti , Xt∆i )′ Γ∆ ti ] ∆ti 2 (46) or Yt∆ = Ei−1 [Yt∆ ] + f (ti−1 , Xt∆i−1 , Ei−1 [Yt∆ ], Zt∆i−1 , Γ∆ ti−1 ) i i i−1 − 1 tr[σ(ti−1 , Xt∆i−1 )σ(ti−1 , Xt∆i−1 )′ Γ∆ ti−1 ] ∆ti 2 or linear combinations of those (θ-schemes). Julien Guyon Nonlinear Option Pricing (47) Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Discretization scheme for 2-BSDEs The quantity P&L × ∆ti where P&L ≡ f (ti−1 , Xt∆i−1 , Yt∆ , Zt∆i−1 , Γ∆ ti−1 ) i−1 1 − tr[σ(ti−1 , Xt∆i−1 )σ(ti−1 , Xt∆i−1 )′ Γ∆ (48) ti−1 ] 2 is the so-called gamma-theta P&L between ti−1 and ti . Note that PDE (44) does not depend on the function σ (volatility of forward process X), which can be chosen arbitrarily. This scheme requires a final condition for Ztn = Dg(Xtn ). Since the payoffs usually considered in finance are not smooth, this scheme may perform poorly. We now suggest a new scheme in the case of the UVM. Using an induction argument, one can show that the random variables ∆ Yt∆ , Zt∆i , and Γ∆ ti are deterministic functions of Xti . Hence the i conditional expectations above can be replaced by Julien Guyon Nonlinear Option Pricing Ei−1 [Yt∆ ∆Wti ] i = E[Yt∆ ∆Wti |Xt∆i−1 ] i Ei−1 [Zt∆i ∆Wt′i ] = E[Zt∆i ∆Wt′i |Xt∆i−1 ] UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Discretization scheme for the BSB 2-BSDE Scheme 2-BSDE for the UVM (dropping superscript ∆ for clarity): t −(σ̂ α )2 2i +σ̂ α Wtα Xtαi = X0α e i , E[∆Wtαi ∆Wtβi ] = ρ̂αβ ∆ti Ytn = g(Xtn )  d X (49)  1  ∆ti ρ̂αβ σ̂ α σ̂ β Xtαi−1 Xtβi−1 Γαβ ti−1 2 α,β=1 " # −1 α β α α (∆ti )2 σ̂ α σ̂ β Xtαi−1 Xtβi−1 Γαβ = E Y U U − ∆t ρ̂ − ∆t σ̂ U δ X ti−1 ti i αβ i ti ti ti αβ ti−1 Yti−1 = E[Yti |Xti−1 ] + f (Xti−1 , Γti−1 ) − with Utαi ≡ Pd β −1 β=1 ρ̂αβ ∆Wti We have introduced explicitly the Malliavin weight for a log-normal diffusion with volatility σ̂ and correlation ρ̂ Analogy with American options: Yti−1 = max EQ i−1 [Yti ], g(Xti−1 ) Julien Guyon Nonlinear Option Pricing Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Discretization Scheme for the BSB 2-BSDE in 1D Scheme 2-BSDE for the UVM in 1D: Xti Ytn = = Yti−1 = = (∆ti )2 σ̂ 2 Xt2i−1 Γti−1 Julien Guyon Nonlinear Option Pricing = 2 ti X0 e−σ̂ 2 +σ̂Wti g(Xtn ) (50) 1 2 2 E[Yti |Xti−1 ] + f (Xti−1 , Γti−1 ) − σ̂ Xti−1 Γti−1 ∆ti 2 1 2 2 E[Yti |Xti−1 ] + (Σ(Γti−1 ) − σ̂ )Xt2i−1 Γti−1 ∆ti 2 E Yti ∆Wt2i − ∆ti (1 + σ̂∆Wti ) |Xti−1 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Malliavin Weights for Black-Scholes Delta and Gamma i h 1 2 Let h(x) = E g xeσWT − 2 σ T . Then i h 1 2 1 h′ (x) = E g xeσWT − 2 σ T WT σxT h i 1 2 1 ′′ h (x) = 2 2 2 E g xeσWT − 2 σ T WT2 − T (1 + σWT ) σ x T Proof. Let n(y) = √ 1 2πT y2 e− 2T be the probability density function of WT . Then Z 1 2 h(x) = g xeσy− 2 σ T n(y)dy. R Z 1 2 1 2 d σy− 1 σ2 T 1 2 g xe n(y)dy h (x) = g xeσy− 2 σ T eσy− 2 σ T n(y)dy = dy σx R R Z Z 1 2 1 2 1 y 1 = − g xeσy− 2 σ T n′ (y)dy = g xeσy− 2 σ T n(y)dy σx R σx R T h i 1 2 1 = E g xeσWT − 2 σ T WT σxT The expression for h′′ can be proved similarly. ′ Z Julien Guyon Nonlinear Option Pricing ′ UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Interpretation Consider the case where i = n: since we simulated lognormal X’s, the price Ytn−1 is the sum of the Black-Scholes price EQ [Ytn |Xtn−1 ] and the gamma-theta P&L correction   d X 1 αβ α β α β αβ f (Xtn−1 , Γtn−1 ) − ρ̂ σ̂ σ̂ Xtn−1 Xtn−1 Γtn−1  ∆tn (51) 2 α,β=1 This last term requires that we estimate the gamma at time tn−1 . We use the Black-Scholes gamma, as given by the last equation in (49), which uses the Malliavin weight for the lognormal diffusion. In dimension 1, the gamma-theta P&L correction reads 1 1 f (Xtn−1 , Γtn−1 ) − σ̂ 2 Xt2n−1 Γtn−1 ∆tn = Xt2n−1 (Σ(Γtn−1 )2 − σ̂ 2 )Γtn−1 ∆tn 2 2 with Σ(Γ)2 = σ 2 1Γ≥0 + σ 2 1Γ<0 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM The algorithm (G. and H.-L., 2008) Simulate N1 replications of X with a log-normal diffusion. Apply the backward algorithm using a regression approximation. Simulate N2 independent replications of X using the gamma functions Γti = ϕ(ti , Xti ) which are the result of the regression at the previous step. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Numerical experiment: Call on one underlying (XT − 100)+ At-the money call option valued using the BSDE approach with σ̂ = 0.15. The true price is u = 7.97 ∆ M1 12 13 14 15 16 17 1/2 Price 8.04 8.06 8.00 8.00 8.00 8.00 1/4 Price 8.53 8.29 8.00 7.87 7.89 7.86 1/8 Price 9.28 8.72 8.02 7.79 7.78 7.68 At-the money call option valued using the BSDE approach with σ̂ = 0.15 and an independent second MC run. The true price is u = 7.97 ∆ M1 12 13 14 15 16 17 1/2 Price 7.93 7.94 7.95 7.95 7.96 7.96 1/4 Price 7.88 7.90 7.92 7.93 7.95 7.94 1/8 Price 7.53 7.93 7.58 7.60 7.96 7.96 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Numerical experiments T = 1, ti = i/n, ∆ = 1/n For each asset α, X0α = 100, σ α = 0.1, σ α = 0.2 and σ̂ α = 0.15 We compare with a parametric approach (not presented in this course) Param: In the gamma calibration stage, we pick N1 = 2M1 with M1 = 12, and the N1 replications of X use a time step tk+1 − tk = 1/100 BSDE: We allow M1 to vary from 12 to 17 Pricing stage: the N2 = 215 replications of X use a time step ∆2 = 1/400 Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Call spread:(XT − 90)+ − (XT − 110)+ The true price (PDE) is uPDE = 11.20 Param: We pick θi ∈ R and λti (X; θi ) = σ if θi − ln(X/X0 ) ≥ 0, = σ otherwise BSDE: we use non-parametric regressions ∆ 1/2 1/4 1/8 Param 11.19 11.19 11.18 M1 Price Price Price 12 11.08 11.01 10.74 13 11.07 11.12 10.55 14 11.06 11.06 10.73 Algo/σ̂ Param BSDE Black-Scholes 2% 11.19 9.36 10.00 5% 11.19 10.72 9.97 10% 11.19 11.01 9.76 15% 11.19 11.12 9.52 Julien Guyon Nonlinear Option Pricing 15 11.06 11.07 11.01 20% 11.19 11.07 9.30 16 11.06 11.11 11.04 30% 11.19 11.00 8.87 17 11.06 11.11 11.11 50% 11.14 10.81 8.06 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Digital option: 100 × 1XT ≥K The true price (PDE) is uPDE = 63.33 Param: same parameterization as for the call spread BSDE: we use non-parametric regressions ∆ 1/2 1/4 1/8 Param 63.13 63.14 62.68 Julien Guyon Nonlinear Option Pricing M1 Price Price Price 12 62.83 62.53 60.06 13 62.83 62.86 59.16 14 62.74 62.77 60.56 15 62.75 62.35 60.59 16 62.75 62.45 60.94 17 62.74 62.43 60.53 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM √ Call Sharpe: (XT − 100)+ / VT VT = T1 A1t = 2 P12 Xtl ln is the monthly realized volatility. l=1 Xt X l−1 ln {l|tl ≤t} Xtl Xtl−1 2 , A2t = Xsup{l|tl ≤t} tl , u(t, Xt , A1t , A2t ) The true price (PDE) is uPDE = 58.4 1 2 2 Param: We take θ√ i = (θi , θi ) ∈ R and pick 1 2 1 γti (X, A; θi ) = θi A + θi − ln(X/X0 ) BSDE: we assume as a first approximation that Ei−1 [·] ≡ E[·|Xti−1 , A1ti−1 , A2ti−1 ] ≃ E[·|Xti−1 ]. The latter is computed using a one dimensional non-parametric regression ∆ 1/2 1/4 1/12 Julien Guyon Nonlinear Option Pricing Param 54.98 55.55 54.32 M1 Price Price Price 12 47.73 46.93 48.03 13 47.18 47.34 49.26 14 48.82 48.01 49.78 15 48.09 48.92 50.87 16 48.10 48.67 51.11 17 48.01 49.38 51.66 18 48.09 49.44 52.12 Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM UVM Outperformer option: (XT1 − XT2 )+ Two underlyings and no uncertainty on correlation Outperformer option with 2 uncorrelated assets valued using σ̂ i , i = 1, 2 and an independent MC. The true price is u = 11.25 Param: We choose θi = (θi1 , θi2 ) ∈ R2 , σt1i (X; θi ) = σ 1 if θi1 − ln(X 1 /X01 ) ≥ 0, = σ 1 otherwise, and σt2i (X; θi ) = σ 2 if θi2 − ln(X 2 /X02 ) ≥ 0, = σ 2 otherwise BSDE: Basis functions = {1, X1 , X2 }: ∆ Param M1 12 13 1/2 11.26 Price 11.24 11.24 1/4 11.26 Price 9.65 10.11 1/8 11.26 Price 9.17 9.45 1/12 11.26 Price 9.17 9.67 Julien Guyon Nonlinear Option Pricing 14 11.24 10.04 9.14 9.47 15 11.24 10.16 9.26 9.38 16 11.24 10.28 9.40 9.32 17 11.24 9.69 9.47 9.84 Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Outperformer option: (XT1 − XT2 )+ BSDE: Basis functions = {1, X1 , X2 , X12 , X22 , X1 X2 }: ∆ Param M1 12 13 14 15 1/2 11.26 Price 11.09 11.15 11.15 11.15 1/4 11.26 Price 10.95 11.07 11.10 11.13 1/8 11.26 Price 10.35 10.71 10.88 10.94 1/12 11.26 Price 10.39 10.68 10.74 10.91 Julien Guyon Nonlinear Option Pricing 16 11.15 11.16 11.07 11.02 17 11.15 11.19 11.14 11.13 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Outperformer option: (XT1 − XT2 )+ ρ = −0.5. The true price is u = 13.75 BSDE: Basis functions = {1, X1 , X2 , X12 , X22 , X1 X2 } ∆ 1/2 1/4 1/8 1/12 Julien Guyon Nonlinear Option Pricing Param 13.77 13.74 13.74 13.75 M1 Price Price Price Price 12 13.58 13.13 12.45 12.52 13 13.66 13.48 12.87 12.82 14 13.68 13.50 13.12 12.98 15 13.66 13.58 13.20 13.27 16 13.66 13.64 13.48 13.44 17 13.65 13.68 13.63 13.59 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Outperformer spread option:(XT2 − 0.9XT1 )+ − (XT2 − 1.1XT1 )+ ρ = −0.5. The true price is uPDE = 11.41 Param: We choose θi = (θi1 , θi2 ) ∈ R2 , σt1i (X; θi ) = σ 1 if θi1 − ln(X 2 /X 1 ) ≥ 0, = σ 1 otherwise, and σt2i (X; θi ) = σ 2 if θi2 − ln(X 2 /X 1 ) ≥ 0, = σ 2 otherwise BSDE: Basis functions = {1, X1 , X2 , X12 , X22 , X1 X2 }: ∆ 1/2 Param 11.37 M1 Price 12 11.07 13 11.07 X2 14 11.10 15 11.11 16 11.09 17 11.10 15 11.26 11.27 16 11.27 11.29 17 11.27 11.28 X3 BSDE: Basis functions = {X1 , X2 , X21 , X22 }: 1 ∆ 1/2 1/4 Param 11.37 11.37 Julien Guyon Nonlinear Option Pricing M1 Price Price 12 11.24 10.85 13 11.25 11.12 14 11.26 11.20 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Options with two underlyings and uncertainty on correlation Outperformer spread option ρ = −0.5, ρ = 0.5 and ρ̂ = 0. The (2d) PDE price is 12.83 Param: We choose θi = (θi1 , θi2 , θi3 ) ∈ R3 , σt1i (X; θi ) = σ 1 if θi1 − ln(X 2 /X 1 ) ≥ 0, = σ 1 otherwise, σt2i (X; θi ) = σ 2 if θi2 − ln(X 2 /X 1 ) ≥ 0, = σ 2 otherwise, and ρti (X; θi ) = ρ if −θi3 + ln(X 2 /X 1 ) ≥ 0, = ρ otherwise 2 2 2 3 ) BSDE: We use the basis functions = {1, X 1 , X 2 , (XX 1) , (X } (X 1 )2 ∆ 1/2 1/4 Param 12.50 12.67 Julien Guyon Nonlinear Option Pricing M1 Price Price 12 12.37 11.58 13 12.40 11.79 14 12.41 12.38 15 12.41 12.44 16 12.39 12.44 17 12.38 12.41 UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Comments Results depend greatly on the choice of regressors which may require numerical experimentations and good understanding of the financial derivatives under consideration. This is a common feature in the pricing of American options Recent advances based on deep neural networks: Deep BSDE method. For semilinear PDEs, approximate the delta Zt as a function of Xt using neural nets, then simulate Yt in a purely forward fashion based on this parameterization of Z, and use stochastic gradient descent to minimize E (YT − g(XT ))2 . See Han, Jentzen, E:Solving high-dimensional partial differential equations using deep learning, PNAS 115(34):8505–8510, 2018. Combination of the BSDE and parametrical approaches The payoffs g that we will use in our numerical experiments below do not satisfy the regularity condition g ∈ C 3 under which existence and uniqueness the BSB BSDE hold. However, even for these non-smooth payoffs, our discretization scheme seems numerically convergent Remark about the generation of the first N1 paths Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Selected references Andersen, L. (1999): A Simple Approach to the pricing of Bermudan Swaptions in the Multi-Factor Libor Market Model : Journal of Computational Finance, 3:5–32. Avellaneda, M., Levy, A., Paras, A. (1995): Pricing and hedging derivative securities in markets with uncertain volatilities : Journal of Applied Finance, Vol 1. Bouchard, B., Touzi, N. (2004): Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications, 111, 175–206. Cheridito, P., Soner, M., Touzi, N., Victoir, N. (2007): Second Order Backward Stochastic Differential Equations and Fully Non-Linear Parabolic PDEs, Communications on Pure and Applied Mathematics, Volume 60, Issue 7, 1081–1110. Julien Guyon Nonlinear Option Pricing UVM Early Exercise Option Pricing in a Nutshell Stochastic control BSDE ULMM Selected references El Karoui, N., Peng, S., Quenez, M.C. (1997): Backward Stochastic Differential Equations in Finance, Mathematical Finance, Vol. 47, No. 1, 1–77. Guyon, J., Henry-Labordère, P. (2011) : Uncertain Volatility Model: A Monte Carlo approach, Journal of Computational Finance, 14(3):37–71. Lyons, T.J. (1995): Uncertain volatility and the risk-free synthesis of derivatives, Applied Mathematical Finance, 2(2):117–133. Pardoux, E., Peng, S. (1992): Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations, Lecture Notes in CIS, vol. 176. Springer-Verlag, 200–217. Julien Guyon Nonlinear Option Pricing UVM

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