Partial Differential Equations in Option Pricing Jean-Pierre Fouque University of California Santa Barbara Special Semester on Stochastics with Emphasis on Finance Tutorial September 5, 2008 RICAM, Linz, Austria 1
PART 1: Review of the Black-Scholes Theory of Derivative Pricing 1.1 Market Model One riskless asset (savings account): dβ t = rβ t dt, (1) where r ≥ 0 is the instantaneous interest rate. Setting β 0 = 1, we have β t = e rt for t ≥ 0. The price X t of the other asset, the risky stock or stock index, evolves according to the stochastic differential equation dX t = µX t dt + σX t dW t , (2) where µ is a constant mean return rate, σ > 0 is a constant volatility and ( W t ) t ≥ 0 is a standard Brownian motion . 2
Risky Asset Price Model Differential form: dX t = µdt + σdW t (3) X t Integral form: � t � t X t = X 0 + µ X s ds + σ X s dW s (4) 0 0 General class of stochastic differential equations driven by a Brownian motion: dX t = µ ( t, X t ) dt + σ ( t, X t ) dW t , (5) or in integral form � t � t X t = X 0 + µ ( s, X s ) ds + σ ( s, X s ) dW s . (6) 0 0 3
Itˆ o’s formula in differential form: dg ( W t ) = g ′ ( W t ) dW t + 1 2g ′′ ( W t ) dt . (7) More generally, when X t satisfies dX t = µ ( t, X t ) dt + σ ( t, X t ) dW t , and g depends also on t , one has ∂ 2 g dg ( t, X t ) = ∂g ∂t ( t, X t ) dt + ∂g ∂x ( t, X t ) dX t + 1 ∂x 2 ( t, X t ) d � X � t , (8) 2 � t 0 σ 2 ( s , X s ) ds is the quadratic variation of the where � X � t = martingale part of X t . In terms of dt and dW t the formula is � ∂g � 2 σ 2 ( t, X t ) ∂ 2 g ∂t + µ ( t, X t ) ∂g ∂x + 1 dt + σ ( t, X t ) ∂g dg ( t, X t ) = ∂xdW t , (9) ∂x 2 where all the partial derivatives of g are evaluated at ( t, X t ). 4
Application to the discounted price g ( t, X t ) = e − rt X t d ( e − rt X t ) − re − rt X t dt + e − rt dX t = e − rt ( − rX t + µ ( t, X t )) dt + e − rt σ ( t, X t ) dW t (10) = ( µ − r ) ( e − rt X t ) dt + σ ( e − rt X t ) dW t . = The discounted price � X t = e − rt X t satisfies the same equation as X t where the return µ has been replaced by µ − r : d � X t = ( µ − r ) � X t dt + σ � X t dW t . (11) Integration by parts formula d ( X t Y t ) = X t dY t + Y t dX t + d � X, Y � t , (12) where the covariation (also called “bracket”) of X and Y is given by d � X, Y � t = σ X ( t, X t ) σ Y ( t, Y t ) dt. 5
Lognormal Risky Asset Price dX t = X t ( µ dt + σ dW t ) gives by Itˆ o’s formula � � µ − 1 2 σ 2 − → d log X t = dt + σdW t � � µ − 1 2 σ 2 log X t = log X 0 + t + σW t − → � � ( µ − 1 2 σ 2 ) t + σ W t X t = X 0 exp . (13) The return X t /X 0 is lognormal and the process ( X t ) is called a geometric Brownian motion . which can also be obtained as a diffusion limit of binomial tree models which arise when the Brownian motion is approximated by a random walk. 6
108 106 Stock Price X t 104 102 100 98 96 94 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time t Figure 1: A sample path of a geometric Brownian motion, with µ = 0 . 15 , σ = 0 . 1 and X 0 = 95 . It exhibits the “average growth plus noise” behavior we expect from this model of asset prices. 7
1.2 Replicating Strategies The Black-Scholes-Merton analysis of a European style derivative yields an explicit trading strategy in the underlying risky asset and riskless bond whose terminal payoff is equal to the payoff h ( X T ) of the derivative at maturity, no matter what path the stock price takes. This replicating strategy is a dynamic hedging strategy since it involves continuous trading, where to hedge means to eliminate risk . The essential step in the Black-Scholes methodology is the construction of this replicating strategy and arguing, based on no-arbitrage , that the value of the replicating portfolio at time t is the fair price of the derivative. We develop this idea now. 8
Replicating Self-Financing Portfolios We consider a European style derivative with payoff h ( X T ). Assume that the stock price ( X t ) follows the geometric Brownian motion model. A trading strategy is a pair ( a t , b t ) of adapted processes specifying the number of units held at time t of the underlying asset and the riskless bond, respectively. �� T � � T 0 a 2 0 | b t | dt are finite so that the We suppose that I E t dt and stochastic integral involving ( a t ) and the usual integral involving ( b t ) are well-defined. The value at time t of this portfolio is a t X t + b t e rt . It will replicate the derivative at maturity if its value at time T is almost surely equal to the payoff: a T X T + b T e rT = h ( X T ) (14) 9
In addition, this portfolio is to be self-financing , � a t X t + b t e rt � = a t dX t + rb t e rt dt , d (15) which implies the self-financing property X t da t + e rt db t + d � a , X � t = 0 . (16) In integral form: � t � t a t X t + b t e rt = a 0 X 0 + b 0 + rb s e rs ds , 0 ≤ t ≤ T . a s dX s + 0 0 In discrete time: a t n X t n +1 + b t n e rt n +1 = a t n +1 X t n +1 + b t n +1 e rt n +1 − → � a t n +1 X t n +1 + b t n +1 e rt n +1 � � a t n X t n + b n e rt n � − � � � e rt n +1 − e rt n � X t n +1 − X t n = a t n + b t n , which in continuous time becomes (15). 10
The Black-Scholes Partial Differential Equation Assume that the price of a European-style contract with payoff h ( X T ) is given by P ( t , X T ) where the pricing function P ( t , x ) is to be determined. Cconstruct a self-financing portfolio ( a t , b t ) that will replicate the derivative at maturity (14). The no-arbitrage condition requires that a t X t + b t e rt = P ( t , X t ) , for any 0 ≤ t ≤ T. (17) Differentiating (17) and using the self-financing property (15) on the left-hand side, Itˆ o’s formula (9) on the right-hand side and equation (2), we obtain � a t µ X t + b t re rt � dt + a t σ X t dW t (18) � ∂ P � ∂ 2 P ∂ P ∂ x + 1 ∂ P 2 σ 2 X 2 = ∂ t + µ X t dt + σ X t ∂ x dW t t ∂ x 2 11
Eliminating risk ( or equating the dW t terms ) gives a t = ∂ P ∂ x ( t , X t ) . (19) From (17) we get b t = ( P ( t , X t ) − a t X t ) e − rt . (20) Equating the dt terms in (18) gives � � ∂ 2 P ∂ P = ∂ P ∂ t + 1 2 σ 2 X 2 P − X t r ∂ x 2 , (21) t ∂ x which needs to be satisfied for any stock price X t . Note that µ disappeared! 12
P ( t, x ) is the solution of the Black-Scholes PDE L BS ( σ ) P = 0 , (22) where the Black-Scholes operator is defined by � � 2 σ 2 x 2 ∂ 2 L BS ( σ ) = ∂ ∂ t + 1 x ∂ ∂ x − · ∂ x 2 + r . (23) Equation (22) is to be solved backward in time with the terminal condition P ( T, x ) = h ( x ), on the upper half-plane x > 0. Knowing P , the portfolio ( a t , b t ) is uniquely determined by (19) and (20). a t is the “Delta” of the portfolio. Only the volatility σ is needed. 13
The Black-Scholes Formula For European call options the Black-Scholes PDE (22) is solved with the final condition h ( x ) = ( x − K ) + . There is a closed-form solution known as the Black-Scholes formula : C BS ( t , x ; K , T ; σ ) = xN ( d 1 ) − Ke − r ( T − t ) N ( d 2 ) , (24) � 2 σ 2 � r + 1 log( x/K ) + ( T − t ) √ d 1 = , (25) T − t σ √ d 1 − σ T − t, d 2 = (26) � z 1 e − y 2 / 2 dy. √ N ( z ) = (27) 2 π −∞ (By direct check or probabilistic derivation later) The Delta hedging ratio a t for a call is given by ∂ C BS = N ( d 1 ) . ∂ x 14
45 40 Call Option Price 35 30 25 20 15 Payoff ( x − K ) + 10 t = 0 price 5 0 60 70 80 90 100 110 120 130 140 Current Stock Price x Figure 2: Black-Scholes call option price C BS (0 , x ; 100 , 0 . 5; 10%) at time t = 0 , with K = 100 , T = 0 . 5 , σ = 0 . 1 and r = 0 . 04 . 15
European put options We have the put-call parity relation C BS ( t , X t ) − P BS ( t , X t ) = X t − Ke − r ( T − t ) , (28) between put and call options with the same maturity and strike price. This is a model-free relationship that follows from simple no-arbitrage arguments. If, for instance, the left side is smaller than the right side then buying a call and selling a put and one unit of the stock, and investing the difference in the bond, creates a profit no matter what the stock price does. Under the lognormal model, this relationship can be checked directly since the difference C BS − P BS satisfies the PDE (22) with the final condition h ( x ) = x − K . This problem has the unique simple solution x − Ke − r ( T − t ) . 16
Using the Black-Scholes formula (24) for C BS and the put-call parity relation (28), we deduce the following explicit formula for the price of a European put option : P BS ( t , x ) = Ke − r ( T − t ) N ( − d 2 ) − xN ( − d 1 ) , (29) where d 1 , d 2 and N are as in (25), (79) and (27) respectively. Other types of options do not lead in general to such explicit formulas. Determining their prices requires solving numerically the Black-Scholes PDE (22) with appropriate boundary conditions . Nevertheless probabilistic representations can be obtained as explained in the following section. In particular American options lead to free-boundary value problems associated with equation (22). 17
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