Lecture 6 Black-Scholes PDE

Lecture 6 Black-Scholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1

Pricing function Let the dynamics of underlining S t be given in the risk-neutral measure Q by If the contingent claim X equals ds t = rs t dt + σs t dw t. X = hs T ) for some function h, then the price of X at time t is given by V t = V S t, t), where V s, t) is given by the formula V s, t) = e rt t) E Q hst ) S t = s ). Computational Finance p. 2

Replication strategy We call ψ, φ) the replication strategy for contingent claim X, if the value process is given by where B t is the bank account. V t = ψ t B t + φ t S t, Computational Finance p. 3

Goal Find the function V s, t) : 0, ) [0, T] [0, ). With the function V s, t) we can: compute the price of the contingent claim: at t it equals find the replicating strategy V S t, t). φ t = V s S t, t), ψ t = e rt V S t, t) φ t S t ). Computational Finance p. 4

The Black-Scholes PDE Theorem. If X = hs T ) then there exists a function V : 0, ) [0, T] R such that V t = V S t, t). This function is a solution to the Black-Scholes partial differential equation V s, t) t + rs V s, t) s + 1 2 σ2 s 2 2 V s, t) s 2 rv s, t) = 0 with the terminal condition V s, T) = hs) for any s > 0 and t [0, T]. Computational Finance p. 5

Applications of B-S PDE We can compute V analytically for: vanilla call and put options, binary call and put options, call and put options on the foreign exchange market. We cannot compute analytically but can solve B-S PDE numerically for: options with non-standard payoffs, American call and put options, Asian options, some other complicate contingent claims. Computational Finance p. 6

Derivation of the Black-Scholes PDE V t = V S t, t). Once we are at t, the value V t is no longer random as it is F t measurable. Applying Itô s formula gives dv t = d V S t, t) ) = V S t, t) σs t s + ) d W t rs t V S t, t) s + 1 2 σ2 S 2 t 2 V S t, t) + V S ) t, t) dt. s 2 t The discounted value process ˆV t = e rt V t is a martingale under the risk-neutral measure Q. Computational Finance p. 7

Compute dˆv t dˆv t = d e rt V t ) = e rt σs t V S t, t) s + e rt rs t V S t, t) s + 1 2 σ2 S 2 t ) d W t 2 V S t, t) s 2 + V S t, t) t ) rv S t, t) dt. Observation. ˆV t is a martingale with respect to Q, so the term by dt must be zero. Hence the function V s, t) satisfies rs V s, t) s + 1 2 σ2 s 2 2 V s, t) s 2 + which is the Black-Scholes PDE. V s, t) t rv s, t) = 0, Computational Finance p. 8

Boundary conditions B-S PDE + Conditions = Unique solution Boundary conditions are required to establish uniqueness of the solution to the Black-Scholes PDE. Their role is to impose some economically justified constraints on the solution of the PDE. We have to be able to find conditions without knowing the formula for the function V. Terminal condition: V s, T) = hs), s > 0. Left-boundary: what happens to V s, t) when s approaches 0. Right-boundary: what happens to V s, t) when s approaches. Computational Finance p. 9

Call option For a call option with payoff S T K) + good boundary conditions are: V s, t) 0, V s, t) s, for s very small, for s very large. 1 2 σ2 s 2 2 V s, t) V s, t) + rs s 2 s rv s, t) + V s, t) t = 0, V s, T) = s K) +, s > 0, lim V s, t) = 0, s 0 lim s V s, t) s t [0, T], = 1, t [0, T]. Computational Finance p. 10

Put option For a put option with payoff K S T ) + good boundary conditions are: V s, t) Ke rt t), for s very small, V s, t) 0, for s very large. 1 2 σ2 s 2 2 V s, t) V s, t) + rs s 2 s rv s, t) + V s, t) t = 0, V s, T) = K s) +, s > 0, lim V s, t) = s 0 Ke rt t), lim V s, t) = 0, s t [0, T]. t [0, T], Computational Finance p. 11

General form of the Black-Scholes PDE V s, t) is a unique solution to the Black-Scholes PDE 1 2 σ2 s 2 2 V s, t) V s, t) V s, t) + rs rv s, t) + s 2 s t V s, T) = hs), s > 0, lim V s, t) = r 1t), s 0 lim s V s, t) r 2 s, t) = 1 or t [0, T], lim V s, t) = r 2t), s where r 1 and r 2 are appropriately chosen to match h. = 0, t [0, T], Computational Finance p. 12

Change of variables Consider the following time dependent change of variables: x := log s + r 1 2 σ2 )T t), τ := σ2 2 T t), 2τ rt yx, τ) := e σ 2 ) V 2r x e σ 2 1)τ, T 2τ ) σ 2. The Black-Scholes PDE is transformed into the heat equation y τ x, τ) 2 y x, τ) = 0. x2 Computational Finance p. 13

Properties time t = 0 corresponds to τ = σ2 2 T, time t = T corresponds to τ = 0, terminal condition V s, T) = hs) changes to the initial condition yx, 0) = e rt he x ), unlike s, the variable x is between and, new PDE is much simpler to solve numerically than the Black-Scholes PDE. Computational Finance p. 14

Transformation completed The function y from the previous slide is a unique solution to the following PDE y τ x, τ) 2 y x, τ) = 0, x2 yx, 0) = e rt he x ), x R, 2τ lim yx, τ) = e rt σ 2 ) r 1 T 2τ ), τ x σ 2 lim x yx, τ) 2τ rt e σ 2 ) r 2 e x 2r σ 2 1)τ ) = 1,, T 2τ σ 2 or [0, σ2 2 T ], 2τ lim yx, τ) = e rt σ 2 ) r 2 T 2τ ) x σ 2, τ [0, σ2 2 T ]. Computational Finance p. 15

Dividend-paying assets When assets pay dividend with a constant dividend yield d then V s, t) is a unique solution to the following Black-Scholes PDE V s, T) = hs), s > 0, 1 2 σ2 s 2 2 V s, t) V s, t) + r d)s s 2 s lim V s, t) = r 1t), s 0 lim s V s, t) r 2 s, t) = 1 or t [0, T], rv s, t) + lim V s, t) = r 2t), s where r 1 and r 2 are appropriately chosen to match h. V s, t) t t [0, T], = 0, Computational Finance p. 16

Change of variables The time dependent change of variables for dividend-paying assets: x := log s + r d 1 2 σ2 )T t), τ := σ2 2 T t), 2τ rt yx, τ) := e σ 2 ) V e x 2r d) σ 2 1)τ, T 2τ ) σ 2. Then the Black-Scholes PDE is transformed into the heat equation y τ x, τ) 2 y x, τ) = 0. x2 Computational Finance p. 17

Transformed equation The function y from the previous slide is a unique solution to the following PDE y τ x, τ) 2 y x, τ) = 0, x2 yx, 0) = e rt he x ), x R, 2τ lim yx, τ) = e rt σ 2 ) r 1 T 2τ ), τ x σ 2 lim x 2τ rt e yx, τ) σ 2 ) r 2 e x 2r d) σ 2 or 1)τ, T 2τ σ 2 ) = 1, [0, σ2 2 T ], 2τ lim yx, τ) = e rt σ 2 ) r 2 T 2τ ), τ x σ 2 [0, σ2 2 T ]. Computational Finance p. 18

Another change of variables For some problems the proposed changes of variables are not very convenient because straight lines S = c became time dependent in the new variables. Then the following time independent change of variables can be useful: x := log s, τ := σ2 T t), 2 1 yx, τ) := exp 2 q d 1)x + 1 4 q d 1) 2 + q) τ ) V e x, T 2τ ) σ 2, where q := 2r σ 2, q d := 2r d) σ 2. In this case the Black-Scholes PDE is transformed also into the heat equation y τ x, τ) 2 y x 2 x, τ) = 0. Computational Finance p. 19

Boundary conditions For the change of variables from the previous slide the boundary conditions are as follows yx, 0) = e 1 2 q d 1)x he x ), x R, lim yx, τ) = exp 1 x 4 q d 1) 2 + q ) τ )r 1 T 2τ ), τ σ 2 lim x exp 1 2 q d 1)x + 1 yx, τ) 4 q d 1) 2 + q ) τ [0, σ2 2 T ], ) r 2 ex, T 2τ σ 2 ) = 1, τ [0, σ2 2 T ]. Computational Finance p. 20