Lecture 3: Review of mathematical finance and
derivative pricing models
Xiaoguang Wang
STAT 598W
January 21th, 2014
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Outline
1
Some model independent definitions and principals
2
European Options Pricing
Binomial Trees
Black-Scholes Model
3
American Options Pricing
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Outline
1
Some model independent definitions and principals
2
European Options Pricing
Binomial Trees
Black-Scholes Model
3
American Options Pricing
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Self Financing Portfolio
Notations:
N = the number of different types of assets or stocks.
hi (t) = number of shares of type i held during the period [t, t + ∆t).
h(t) = the portfolio [h1 (t), · · · , hN (t)] held during period t.
c(t) = the amount of money spent on consumption per unit time
during the period [t, t + ∆t).
Si (t) = the price of one share of type i during the period [t, t + ∆t).
V (t) = the value of the portfolio h at time t.
Definition
A self-financing portfolio is a portfolio with no exogenous infusion or
withdrawal of money (apart from the consumption term c). It must satisfy
dV (t) = h(t)dS(t) − c(t)dt
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Dividends
Definition
We take as given the processes D1 (t), ·, DN (t), where Di (t) denotes the
cumulative dividends paid to the holder of one unit of asset i during the
interval (0, t]. If Di has the structure
dDi (t) = δi (t)dt
for some process δi , then we say that asset i pays a continuous dividend
yield.
Taking into account the dividend payments, a self-financing portfolio
should now satisfy
dV (t) = h(t)dS(t) + h(t)dD(t) − c(t)dt
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Free of Arbitrage
Definition
An arbitrage possibility on a financial market is a self-financed portfolio
such that
V h (0) = 0
P(V h (T ) ≥ 0) = 1
P(V h (T ) > 0) > 0
We say the market is arbitrage free if there are no arbitrage possibilities.
In most cases, we assume that the market of interest is arbitrage free.
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Martingale measure
Definition
Consider a market model containing N assets S1 , · · · , SN and fix the asset
S1 (in most cases this will be the risk free rate account) as the numeraire
asset. We say that a probability measure Q defined on Ω is a martingale
measure if it satisfies the following conditions:
1. Q is equivalent to P, i.e.
Q∼P
2. For every i = 1, · · · , N, the normalized (discounted) asset price process
Zti =
Sti
St1
is a martingale under the measure Q.
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First Fundamental Theorem
Theorem
Given a fixed numeraire, the market is free of arbitrage if and only if there
exists a martingale measure Q.
Under the martingale measure Q, every discounted (normalized) price
process (either underlying or derivative) is a martingale.
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Complete market
Definition
A contingent claim is a stochastic variable X of the form
X = Φ(ST ),
where the contract function Φ is some given real valued function.
A given contingent claim X is said to be reachable if there exists a
self-financing portfolio h such that
VTh = X , P − a.s.
with probability 1. In that case we say that the portfolio h is a hedging
portfolio or a replicating portfolio. If all contingent claims can be
replicated we say that the market is (dynamically) complete.
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Pricing principal
A reachable claim X must be priced properly such that there is no
arbitrage opportunity caused by this pricing.
Theorem
If a claim X is reachable with replicating (self-financing) portfolio h, then
the only reasonable price process for X is given by
Π(t; X ) = Vth , 0 ≤ t ≤ T
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General pricing formula: martingale approach
Theorem
In order to avoid arbitrage, a contingent claim X must be priced according
to the formula
X
Q
Π(t; X ) = S0 (t)E
|Ft
S0 (T )
where Q is a martingale measure for [S0 , S1 , · · · , SN ], with S0 as the
numeraire.
In particular, we can choose the bank account B(t) as the numeraire.
Then B has the dynamics
dB(t) = r (t)B(t)dt
where r is the (possibly stochastic) short rate process. In this case the
pricing formula above reduces to
h RT
i
Π(t; X ) = E Q e − t r (s)ds X |Ft
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Second Fundamental Theorem
Theorem
Assuming absence of arbitrage, the market model is complete if and only if
the martingale measure Q is unique.
For incomplete market, different choices of Q will generally give rise to
different price process for a fixed claim X . However, if X is attainable
(reachable) then all choices of Q will produce the same price process,
which then is given by
h RT
i
Π(t; X ) = V (t; h) = E Q e − t r (s)ds X |Ft
where h is the hedging portfolio. Different choices of hedging portfolio (if
such exist) will produce the same price process.
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General Pricing formula: PDE approach
Assume the market is complete, under the martingale measure
(risk-neutral) Q, any derivative price process Π(t) must have the following
dynamics
dΠ(t) = r Π(t)dt + σΠ (t)dW (t)
where W is a Q− Wiener process, and σΠ is the same under Q as under P.
Π(t)
is a Q martingale, i.e. it has a zero drift
With Π above, the process B(t)
term.
Assume Π(t) has the form Π(t) = F (t, S(t)), then apply Ito’s formula on
F (t,S(t))
and set the drift term as zero. Then we will come up with a PDF
B(t)
as follows
1
Ft + rxFx + x 2 σ 2 Fx x − rF = 0,
2
F (T , x) = Φ(x)
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Feynman-Kac̆
Assume that F is a solution to the boundary value problem
∂F
∂F
1
∂2F
(t, x) + µ(t, x)
+ σ 2 (t, x) 2 (t, x) = 0
∂t
∂x
2
∂x
F (T , x) = Φ(x)
Assume furthermore that process
σ(s, Xs )
∂F
(s, Xs )
∂x
is in L2 , where X is defined below. Then F has the representation
F (t, x) = Et,x [Φ(XT )]
where X satisfies the SDE
dXs = µ(s, Xs )ds + σ(s, Xs )dWs ,
Xt = x
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Greeks
Let P(t, s) denote the pricing function at time t for a portfolio based on a
single underlying asset with price process St . The portfolio can thus
consist of a position in the underlying asset itself, as well as positions in
various options written on the underlying asset. For practical purpose it is
often of vital importance to have a grip on the sensitivity of P with
respect to the following
Price changes of the underlying asset
Changes in the model parameters
Definition
∆=
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∂P
∂2P
∂P
,Γ =
,ρ =
2
∂s
∂s
∂r
∂P
∂P
,V =
Θ=
∂t
∂σ
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Incomplete market: market price of risk
Assume that a derivative price F has the following dynamics under
measure P:
dF = αF Fdt + σF FdW
Definition
Assume that the market for derivatives is free of arbitrage. Then there
exists a universal process λ(t) called such that, with probability 1, and for
all t, we have
αF (t) − r
= λ(t)
σF (t)
regardless of the specific choice of the derivative F . The λ(t) is called
”market price of risk”, which can be interpreted as the ”risk premium
per unit of volatility”.
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Incomplete Market Pricing formula: PDE approach
Assuming absence of arbitrage, the pricing function F (t, x) of a T -claim
Φ(X (T )) solves the following boundary value problem
Ft (t, x) + AF (t, x) − rF (t, x) = 0,
(t, x) ∈ (0, T ) × R
F (T , x) = Φ(x), x ∈ R
where
1
AF (t, x) = {µ(t, x) − λ(t, x)σ(t, x)}Fx (t, x) + σ 2 (t, x)Fxx (t, x)
2
and we assume that the asset X (t) has the following dynamics under
measure P:
dX (t) = µ(t, X (t))dt + σ(t, X (t))d W̄ (t)
Here W̄ is a standard scalar P-Wiener process.
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Incomplete market pricing: Martingale approach
Theorem
Assuming absence of arbitrage, the pricing function F (t, x) of the T -claim
Φ(X (T )) is given by the formula
Q
F (t, x) = e −r (T −t) Et,x
[Φ(X (T ))]
, where the dynamics of X under the martingale measure Q are given by
dX (t) = {µ(t, X (t)) − λ(t, X (t))σ(t, X (t))}dt + σ(t, X (t))dW (t)
Here W is a Q-Wiener process, and the subscripts t, x indicate as usual
that X (t) = x.
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Outline
1
Some model independent definitions and principals
2
European Options Pricing
Binomial Trees
Black-Scholes Model
3
American Options Pricing
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European Call and Put Options
Definition
A European Call (Put) Option on an underlying asset S with strike
price K and exercise date T is a contract written at time t = 0 with the
following properties:
The holder of the contract has, exactly at time t = T , the right to
buy (sell) one unit of the asset at the strike price K .
The holder of the option has no obligation to buy (sell) the asset.
The pay-off function for a European Call option is
X = Φ(S(T )) = max(S(T ) − K , 0)
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Model Set Up
Discrete time running from t = 0 to t = T , where T is fixed.
There are two underlying assets, a risk-free bond with price process
Bt and a stock with price process St .
Assume a constant deterministic short rate of interest R. And
Bn+1 = (1 + R)Bn , B0 = 1.
Sn+1 = Sn · Zn , S0 = s.
Z0 , · · · , ZT −1 are assumed to be i.i.d. stochastic variables, taking
only two values u and d with probabilities:
P(Zn = u) = pu , P(Zn = d) = pd
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Portfolio Strategy
Definition
A portfolio strategy is a stochastic process
{ht = (xt , yt ); t = 1, · · · , T }
such that ht is a function of S0 , S1 , · · · , St−1 . For a given portfolio
strategy h we set h0 = h1 by convention. The value process corresponding
to the portfolio h is defined by
Vth = xt (1 + R) + yt St
A self-financing portfolio strategy h will satisfy for all t = 0, · · · , T − 1,
xt (1 + R) + yt St = xt+1 + yt+1 St
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No arbitrage and Completeness
Theorem
The condition
d ≤ (1 + R) ≤ u
is a necessary and sufficient condition for absence of arbitrage.
The binomial model defined above is complete.
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Martingale Measure
Theorem
The martingale measure probabilities qu and qd which must satisfy
s=
1
E Q [St+1 |St = s]
1+R
are given by
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qu =
(1 + R) − d
u−d
qd =
u − (1 + R)
u−d
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Pricing formulas
Theorem
The arbitrage free price at t = 0 of a T -claim X is given by
Π(0; X ) =
1
E Q [X ]
(1 + R)T
where Q denotes the martingale measure, or more explicitly
T X
T k T −k
1
Π(0; X ) =
q q
Φ(su k d T −k )
T
k u d
(1 + R)
k=1
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Hedging portfolio
A T -claim X = Φ(ST ) can be replicated using a self-financing portfolio. If
Vt (k) denotes the value of the portfolio at node (t, k) then Vt (k) can be
computed recursively by

V (k) = 1 {q V (k + 1) + q V (k)}
(1)
u t+1
t
d t+1
1+R

VT (k) = Φ(su k d T −k )
(2)
And the portfolio strategy is given by



xt (k) = 1 uVt (k) − dVt (k + 1)

1+R
u−d
1
V
(k
+
1)
− Vt (k)

t


yt (k) = S
u−d
t−1
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(3)
(4)
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Model set up
Definition
The Black-Scholes model consists of two assets with dynamics given by
dB(t) = rB(t)dt
dS(t) = αS(t)dt + σS(t)d W̄ (t)
Definition
A contingent claim with date of maturity (exercise date) T , also called a
T -claim, is ant stochastic variable X ∈ FTS . A contingent claim X is
called a simple claim if it is of the form X = Φ(S(T )).
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Model assumptions
The derivative instrument in question can be bought and sold on a
market.
The market is free of arbitrage.
The price process for the derivative asset is of the form
Π(t; X ) = F (t, S(t))
where F is some smooth function.
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Black-Scholes formula
Theorem
The price of a European call option with strike price K and time of
maturity T is given by the formula Π(t) = F (t, S(t)), where
F (t, s) = sN[d1 (t, s)] − e −r (T −t) KN[d2 (t, s)]
Here N is the cumulative distribution function for the N(0, 1) distribution
and
s
1 2
1
ln
+ r + σ (T − t)
d1 (t, s) = √
K
2
σ T −t
√
d2 (t, s) = d1 (t, s) − σ T − t
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Other European type of derivatives
X = S(T ) − K (Forward contract)
Z T
1
S(t)dt − K , 0
(Asian Option)
X = max
T 0
X = S(T ) − inf S(t)
0≤t≤T
X = 1S(T )>K
(Lookback contract)
(cash-or-nothing call, exotic option)
Down and Out (Barrier Options):
(
Φ(S(T )), if S(t) > L for all t ∈ [0, T ]
X =
0,
if S(t) > L for some t ∈ [0, T ]
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Outline
1
Some model independent definitions and principals
2
European Options Pricing
Binomial Trees
Black-Scholes Model
3
American Options Pricing
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Definitions
Definition
A American Call (Put) Option on an underlying asset S with strike
price K and exercise date T is a contract written at time t = 0 with the
following properties:
The holder of the contract has, up to time t = T , the right to buy
(sell) one unit of the asset at the strike price K .
The holder of the option has no obligation to buy (sell) the asset.
It is well known that for an American call option written on an underlying
stock without dividends, the optimal exercise time τ is given by τ = T .
Thus the price of the American option coincides with the price of the
corresponding European option. But for American put options, it is much
harder and there is even no analytical solution.
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Optimal stopping problem
For an American Call Option it is not hard to understand that the price at
time zero should be of the following form
Π(0) = max E Q [e −r τ max{Sτ − K , 0}]
0≤τ ≤T
where the stock dynamics under the risk neutral measure Q are given by
dSt = rSt dt + σt St dWt
Similarly, for American put, we have
Π(0) = max E Q [e −r τ max{K − Sτ , 0}]
0≤τ ≤T
In order to solve these pricing problems, we need to take use of some
Optimal Stopping Theories.
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Optimal Stopping Theory: definitions
Definition
A nonnegative random variable τ is called an (optional) stopping time
w.r.t. the filtration F = {Ft }t≥0 if it satisfies the condition
{τ ≤ t} ∈ Ft , for allt ≥ 0
In general we refer to an integrable process Z as the ”reward process”
which we want to maximize. Then we can formulate our problem as
max E [Zτ ]
0≤τ ≤T
We say that a stopping time τ̂ ≤ T is optimal (not necessarily exist) if
E [Zτ̂ ] = sup E [Zτ ]
0≤τ ≤T
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Some simple results
Theorem
The following hold:
If Z is a submartingale, then late stopping is optimal, i.e. τ̂ = T .
If Z is supermartingale, then it is optimal to stop immediately, i.e.,
τ̂ = 0.
If Z is martingale, then all stopping times τ with 0 ≤ τ ≤ T are
optimal.
Assume that the process Z has the dynamics
dZt = µt dt + σt dWt
where µ and σ are adpated process and
following hold
If µt ≥ 0, P − a.s. for all t, then Z
If µt ≤ 0, P − a.s. for all t, then Z
If µt = 0, P − a.s. for all t, then Z
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σ is square integrable. Then the
is a submartingale.
is a supermartingale.
is a martingale.
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Some simple results
Theorem
If Z is a martingale and g is convex, then g (Zt ) is a submartingale.
If Z is submartingale and g is convex and increasing, then g (Zt ) is a
submartingale.
If Z is a supermartingale and g is concave and increasing, then g (Zt )
is a supermartingale.
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Snell Envelope
Theorem
Consider a fixed process Y , we say that a process X dominates the process
Y if Xt ≥ Yt P − a.s. for all t ≥ 0.
Assuming that E [Yt ] < ∞, the Snell Envelop S, of the process Y is
defined as the smallest supermartingale dominating Y . More precisely: S
is a supermartingale dominating Y , and if D is another supermartingale
dominating Y , then St ≤ Dt , P − a.s. for all t ≥ 0.
For any integrable semimartingale Y , the Snell Envelope exists.
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Optimal value process: discrete case
Assuming discrete time framework, the optimal value process of Z is
defined by
Vn = sup E [Zτ |Fn ]
n≤τ ≤T
A stopping time which realizes the supremum above is said to be optimal
at n, and it will be denoted as τ̂n .
Theorem
The optimal value process V is the solution of the following backward
recursion
Vn = max{Zn , E [Vn+1 |Fn ]},
VT = ZT
Furthermore, it is optimal to stop at time n if and only if Vn = Zn . If
stopping at n is not optimal, then Vn > Zn , and Vn = E [Vn+1 |Fn ].
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Optimal Stopping Rule
Theorem
An optimal stopping rule τ̂ at time t = 0 is given by
τ̂ = min{n ≥ 0 : Vn = Zn }
For a fixed n an optimal stopping time τ̂n is given by
τ̂ = min{k ≥ n : Vk = Zk }
For any n we have
Vτ̂n = Zτ̂n
Theorem
The optimal value process V is the Snell envelope of the reward process Z .
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Continuous case
Definition
For a fixed (t, x) ∈ [0, T ] × R, and each stopping time τ with τ ≥ t, the
value function J si defined by
J(t, x; τ ) = Et,x [Φ(τ, Xτ )]
The optimal value function V (t, x) is defined by
V (t, x) = sup Et,x [Φ(τ, Xτ )]
t≤τ ≤T
A stopping time which realizes the supremum for V above is called
optimal and will be denoted as τ̂tx or τ̂t without confusion.
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Assumptions
We assume
There exists an optimal stopping time τ̂t,x for each (t, x).
The optimal value function V is ”regular enough”. More precisely we
assume that V ∈ C 1,2 . All processes interested are ”integrable
enough”, in the sense that expected values exist, stochastic integrals
are true (rather than local) martingales, etc.
Then we must have
V (t, x) ≥ Φ(t, x)
V (t, x) ≥ Et,x [V (t + h, Xt+h )]
using the Ito operator
Af (t, x) = µ(t, x)
1
∂2f
∂f
(t, x) + σ 2 (t, x) 2 (t, x)
∂x
2
∂x
then we have
(STAT 598W)
V (t, x) ≥ Φ(t, x)
∂
+ A V (t, x) ≤ 0
∂t
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Stopping Rule
It is optimal to stop at (t,x) if and only if
V (t, x) = Φ(t, x)
in which case
∂
+ A V (t, x) < 0
∂t
It is optimal to continue if and only if
V (t, x) > Φ(t, x),
in which case
∂
+ A V (t, x) = 0
∂t
Thus we can define the continuation region C by
C = {(t, x); V (t, x) > Φ(t, x)}
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Free Boundary value problem
Theorem
Assuming enough regularity, the optimal value function satisfies the
following parabolic equation
∂
+ A V (t, x) = 0, (t, x) ∈ C ,
∂t
V (t, x) = Φ(t, x), (t, x) ∈ ∂C
Generally speaking, there is little hope of having an analytical solution of a
free boundary value problem, so typically one has to resort to numerical
schemes.
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Variational inequalities
Theorem
Given enough regularity, the optimal value function is characterized by the
following relations
V (T , x) = Φ(T , x)
V (t, x) ≥ Φ(t, x), ∀(t, x)
∂
+ A V (t, x) ≤ 0
∂t
∂
max{Φ(t, x) − V (t, x),
+ A V (t, x)} = 0, ∀(t, x)
∂t
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Applicable partial results
Theorem
It is never optimal to stop at a point where
∂Φ
∂Φ
1
∂2Φ
(t, x) + µ(t, x)
(t, x) + σ 2 (t, x) 2 (t, x) > 0
∂t
∂x
2
∂x
Expressed otherwise, we have
∂
+ A Φ(t, x) > 0 ⊆ C
(t, x);
∂t
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American Call Option
The optimal stopping problem becomes
max E Q [e −r τ max{Sτ − K , 0}]
0≤τ ≤T
where the stock price dynamics under the risk neutral measure Q are given
by
dSt = rSt dt + σt St dWt
Thus the reward process Z is
Zt = e −rt max{St − K , 0} = max{e −rt St − e −rt K , 0}
It is not hard to see that Z is a submartingale in this case. Thus the
optimal stopping time should be τ̂ = T .
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American Put Option
The optimal stopping problem becomes
max E Q [e −r τ max{K − Sτ , 0}]
0≤τ ≤T
And the optimal value function is
Q
V (t, x) = sup Et,x
[e −r (τ −t) max{K − Sτ , 0}]
t≤τ ≤T
Under Q we still have
dSt = rSt dt + σt St dWt
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Pricing for American Put
Assume that a sufficiently regular function V (t, s), and an open set C ⊆ R+ × R+ ,
satisfying the following conditions:
C has a continuously differentiable boundary bt , i.e. b ∈ C 1 and (t, bt ) ∈ ∂C .
V satisfies the PDE
∂V
∂V
1
∂2V
+ rs
+ s 2 σ 2 2 − rV = 0, (t, s) ∈ C
∂t
∂s
2
∂s
V satisfies the final time boundary condition
V (T , s) = max[K − s, 0], s ∈ R+
V satisfies the inequality
V (t, s) > max[K − s, 0], (t, s) ∈ C
V satisfies
V (t, s) = max[K − s, 0], (t, s) ∈ C c
V satisfies
V (t, s) = max[K − s, 0], (t, s) ∈ C c
V satisfies the smooth fit condition
∂V
lim
(t, s) = −1, 0 ≤ t ≤ T
s↓b(t) ∂s
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Pricing for American Put
Then the following hold:
V is the optimal value function
C is the continuation region
The optimal stopping time is given by
τ̂ = inf{t ≥ 0; St = bt }
Unfortunately there is no analytical formulas for the pricing function or the
optimal boundary. For practical use, the following alternatives are
frequently used:
Solve the free boundary value problem numerically.
Solve the variational inequalities numerically.
Approximate the Black-Scholes model by a binomial model and
compute the exact binomial American put price.
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Perpetual American Put
A perpetual American Put option is an American put with infinite time
horizon. This option is fairly simple to analyze, since the infinite horizon
and the time invariance of the stock price dynamics implies that the option
price as well as the optimal boundary are constant as functions of running
time.
There exists a constant critical price b such that we exercise the option
whenever St < b. Since in this case V (t, s) is independent of running time
t then we have V (t, s) = V (s), and the free boundary value problem
reduces to the ODE
rs
(STAT 598W)
1
∂2V
∂V
+ s 2 σ 2 2 − rV = 0,
∂s
2
∂s
Lecture 3
s>b
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Perpetual American Put Pricing results
Theorem
For a perpetual American put with strike K , the pricing function V and
the critical price b are given by
γK
1+γ
γ
b
K
, s>b
V (s) =
1+γ s
b=
where
γ=
(STAT 598W)
2r
σ2
Lecture 3
51 / 51