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A Game Theoretic Approach to Robust Option Pricing Peter DeMarzo, Stanford Ilan Kremer, Stanford Yishay Mansour, TAU Finance: Efficient markets, Option Pricing, Black-Scholes-Merton Approachability: Hannan-Blackwell Universal Portfolios: Cover et al. Game theoryCalibration, regret matching Example 1: Approachability You repeatedly predict the outcome of a coin toss. The coin need not be a fair coin. What success rate can you guarantee in the long run? Simple strategy: Predict Heads with probability 50% and obtain 50% success rate. Learning Strategy: At each point in time guess heads if there were more heads in the past, otherwise guess tails. In the limit the success rate is max{p,1-p} where p is the fixed parameter of the coin. Q: What if the coin can change arbitrarily from period to period? A: You can still get an equivalent performance!! lim n {realized success rate} max pn* ,1 pn* a.s. 0 Example 2: Competitive Analysis (Regret Minimization) When you go gambling each minute you choose which slot machine to use. There are N different machines, some machines may be better. You see the payoff of machines even if you did not use them. Goal: In the long run obtain an average payoff that is no worse than the best individual machine ex-post. (No Regret) Q: What if payoffs are not stationary (so a machine which has a high payoff may deteriorate over time)? Regret/Approachability Introduced by Blackwell and Hannan in the late 50s, rediscovered and used in: • Computer Science- online algorithms • Statistic and Information Theory • Game Theory- Calibration, dynamic foundation for Nash/correlated equilibrium • With the exception of work on ‘Universal Portfolios’ not incorporated into finance. Regret Minimization • Dynamic optimization under uncertainty without a prior. • Worst case analysis but specified in relative rather than absolute terms (as in Gilboa and Schmeidler) Insights/Results • Minimizing Regret can be expressed as robust upper bound for option pricing. Describe trading strategies that are based on approachability and the bounds/regret they imply for call option with different strike prices. • The optimal robust upper bound can be expressed as a value of a zero sum game. Provide a numerical solution and conjecture about closed form solutions. Bounds are not wide and resemble empirical patterns. Options: Basics • What is an option: A right to buy a stock at a given price strike price = K At a given date (European) CT duration = T • Option payoff: Max{ 0 , ST -K } St = Stock price at time t • This talk: For Simplicity: zero interest rate no dividends K ST Option Pricing: Arbitrage Bounds • Upper Bound [Merton 1973] Current stock price: S1 • Lower bounds [Merton 1973] Always positive At worst, payoff is zero. Stock versus strike price: S1 – K • Claim these bounds are tight • Proof: Assume a huge change in first period… • Better pricing needs more assumptions! Example 1- Binomial model • Suppose the risky asset can take only two values $0.8 0.5(0.8 Stock Bond $1 $1 $1 $1.2 1.2) - 0.4 (1 Call (K=1) $? $1 1) $0 = (0 Option price is 0.5-0.4=0.1 $0.2 0.2) Example II: Black & Scholes • Extend the tree to many periods The limit is continuous time • Black and Scholes continuous prices and complete markets A specific stochastic model: random walk + drift Regret Regret- There is a given strategy and a set of alternative strategies. Regret is defined as the difference/ratio between the performance of the given strategy and the ex-post optimal strategy among the alternatives. Regret guarantee- A lower bound for regret that holds even in the worst case scenario. This guarantee may be conditional on some restricted set of possible scenarios. For the purpose of this talk we ignore any behavioral aspects. We do not argue that people behave according to our measure of regret or that they should behave in this way!!! We consider a specific regret measure to allow us to derive pricing bounds and compare them to the existing literature. Regret and Financial markets I have $100 which equals the price of IBM. Should I buy one share of IBM or get a risk free asset? Alternatives: IBM, risk free asset Ex-post, compare to: Max{IBM, risk free asset} Loss: ratio Linking Regret to Options Note that holding Treasuries plus at-themoney call option on IBM leads to no regret: Payoff = Max{IBM, risk-free asset} Thus, regret minimizing trading strategies have implications for option values. Regret and option pricing- An Example • Suppose we measure regret by looking at the ratio of our performance to the best asset ex-post. In addition, suppose that the current IBM share price is $100 and the risk free interest rate is zero. Your goal is to minimize regret as compared to the best asset ex-post. • Suppose we have a trading strategy such that if we start with $100 then at time T our payoff will always exceed max{80,0.8ST}, where ST denotes IBM share price at time T. Hence, the regret is guarantee is 20%. • We later describe how one can construct such strategies and know only focus on the implications to option pricing. • By scaling we conclude that starting with $125 our strategy would have a payoff that exceeds max{100, ST}. • max{100, ST} is like $100 plus a call option with strike $100 • the value of the option is bounded by $25 Model • Discrete-time finite-horizon model t=1..T • A risky asset whose value at time t is St where St=(1+rt)*St-1, where rt≥-1. • In addition agents can borrow and lend at zero interest rates. • Restriction on price paths: (rt…rt) RT, is compact and 0 Example: {r | q(r ) n 2 log(1 r ) qˆ} j j 1 Model- cntd. • A dynamic trading strategy has initial value G0=c. At time t invest a fraction xt in the risky asset and 1- xt in the risk free asset. Zero risk free rate implies that Gt+1=Gt (1+ xt rt); • Definition We say that c=C(K) is an upper bound if there exists a dynamic trading strategy that starts with $c and for all possible price path in its final payoff, GT, satisfies: GT≥max{0,ST-K} (super replication). Blackwell- (recall Example #2) • You repeatedly choose a single action among {1..I} possible alternatives; j,i denotes the payoff of alternative i at time j. • Can use a randomized strategy which is described by a random variable j; j=i implies that you choose alternative i at time j; your time t payoff is given by ,j Suppose Machine #1 pays more Period n+1 expected regret if Machine #1 pays more Play 1 Regret vs. machine #2 Choose the two alternatives with probability proportional to current regret Regret vs. machine #1 Play 2 Period n+1 expected regret i Machine #2 pays more Aggregate Regret so far E ( An 1 2 | t 1 ) | An |2 ( n 1,2 n 1,1 ) 2 2 Finite horizon properties Proposition Conditional on the set of realized payoffs : 2 n 2 E A n (1 j 2 j )2 j 1 Corollary Conditional on the set of realized payoffs : n n E (max i A ) E j , - max i { j ,i } n ,i j 1 j 1 n ( j 1 1 j 2 j ) 2 Asymptotic No-Regret Theorem (Hannan & Blackwell) If payoff are uniformly bounded then there exists a randomized strategy so that : 1 n 1 n limn [ j , j - max i { j ,i }] a.s 0 n j 1 n j 1 Arbitrary starting pointConsider a variant of the previous strategy where instead of starting at (0,0) we start at an arbitrary point (-x,-y) for some non negative x,y Corollary Conditional on the set of realized payoffs : n E( A x, y 2 ) q 2 ( r ) x 2 y 2 2 Useful in improving performance and in the application for different strike prices A trading strategy • Multiplicative model versus additive model: Let 0,t=0, 1,t=ln(1+rt). • Remove randomness: Invest at time t a fraction of xt=E(t) in the risky asset Proposition The payoff of a trading strategy based on the generalized strategy satisfies: Gn max{ (r ), (r ) S n } where (r ) exp( x q 2 (r ) x 2 y 2 ), and (r ) exp( y q 2 (r ) x 2 y 2 ) Application- Upper bounds for at the money options (K=1) Using the same logic as the IBM example if Gn max{ , * Sn } then the bound is So we can choose x=y 1 1 Application- Upper bounds for at the money options (K=1) Restricting price paths: {r | q(r ) qˆ} Using the expressions we derive before we can get an upper bound on the regret. Using the same logic as the example of IBM one gets: Using the basic trading strategy: exp[qˆ ] 1 Using a generalized strategy with optimal starting point (that depends on q̂log ) exp[qˆ / 2] 1 K1 • Choose starting point where x=y+log(K) • That implies: Gn max{ (r ), (r ) Sn } where (r ) / (r ) k • It also implies a bound of 1/-k for the value of an option with a strike k - Borrow $k and invest 1/ in the trading strategy. Optimal bound • Let V(s,2,n) denote the optimal (lowest) upper bound for a call option with a strike k=1 when the current price is S. This is equivalent to having S=1 and arbitrary K. The restriction on the price paths is again: {r | q(r ) } • Let V(s,2) denote the limit as n goes to infinity. Dynamic Programming V ( S , q 2 , n) min , B S B s.t. Se r B V Se r , q 2 r 2 , n 1 for all r such that r 2 q 2 V * ( S , q 2 ) lim n V ( S , q 2 , n) Conjecture q q 1/ q for S 1 q q S V * (S , q2 ) q q S 1/ q S 1 for S 1 q q where q e 1, q 1 e q q Consider small q • Original strategyexp[ qˆ ] 1 qˆ • Optimal starting pointexp[qˆ / 2] 1 qˆ / 2 • Optimal strategy: qˆ / 2 • Black-Scholes qˆ / 2 Example: 20% (vs. BlackScholes) Gradient Strategy Optimal q-Bound Black-Scholes Black-Scholes Implied Volatility of Price Bound 50% 45% 40% 35% 30% 25% 20% 15% 0.75 0.85 0.95 1.05 Strike Price K 1.15 1.25 Approaches to option pricing Black and Scholes: • Continuous price paths • Constant volatility (quadratic variation) Exact replication and pricing No probability or preference assumptions but Strong assumption on allowable paths With jumps & stochastic volatility, exact pricing requires: (i) A probability distribution P over price paths (ii) A utility function, U Our Approach: • No probability or preference assumptions • Constraints on the set of price paths (support of P) • Super-replication Upper Bound for Option Price Relation to Universal Portfolios: • Cover and Ordentlich (1998)- consider the set of constant-rebalanced portfolios. - Provide a close form tight universal regret bound (minmax). This provides a bound for the value of the derivative that pays ex-post the optimal constantrebalanced portfolios. - Universal means that we consider all possible scenarios (?). Call (or put) options - The relevant set of benchmarks is much simpler – buy and hold strategies. - The min-max value is 0.5 (trivial to prove). From option pricing perspective yields an upper bound of the current stock price. - Hence, need to consider a ‘less universal’ approach and the Blackwell strategy is useful. Early empirical work (joint work with Tyler Shumway) S&P 500 options prices from 1/96 to 4/05 from OptionMetrics. Options with 15 to 45 calendar days to maturity.