Lecture 4. Finite difference and finite element methods

Transcription

1 Finite difference and finite element methods Lecture 4

2 Outline Black-Scholes equation

3 From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation V (t, x) = E [e R ] T r(x s)ds t g(x T ) X t = x, (1) where X is the (unique) solution of the stochastic differential equation (the dynamics of the underlying of the option) dx t = b(x t ) dt + σ(x t ) dw t. (2) It turns out: V (t, x) solves a PDE. We need the notion of the so-called infinitesimal generator of the process X.

4 Proposition Let A denote the differential operator which is, for functions f C 2 (R) with bounded derivatives, given by (Af)(x) := 1 2 σ2 (x) xx f(x) + b(x) x f(x). (3) Then, the process M t := f(x t ) t 0 (Af)(X s)ds is a martingale with respect to the filtration of W. The operator A is the infinitesimal generator of the process X, which solves the SDE dx t = σ(x t ) dw t + b(x t ) dt.

5 We need a discounted version of the above Proposition. Proposition Let f C 1,2 (R R) with bounded derivatives in x, let A be as in (3) and assume that r C 0 (R) is bounded. Then the process M t := e R t 0 r(xs)ds f(t, X t ) t is a martingale with respect to the filtration of W. 0 e R s 0 r(xτ )dτ ( t f+af rf)(s, X s )ds We now are able to link the stochastic representation of the option price V (t, x) = E [e R ] T r(x s)ds t g(x T ) X t = x with a parabolic partial differential equation.

6 Theorem Let V C 1,2 (J R) C 0 (J R) with bounded derivatives in x be a solution of t V + AV rv = 0 in J R, V (T, x) = g(x) in R, (4) with A as in (3). Then, V (t, x) can also be represented as V (t, x) = E [e R ] T t r(x s)ds g(x T ) X t = x. (5) Remark The converse of this Theorem is also true. Any V (t, x) as in (5), which is C 1,2 (J R) C 0 (J R) with bounded derivatives in x, solves the PDE (4). This is known as Feynman-Kac Theorem.

7 We apply the Feynman-Kac Theorem to the Black-Scholes model. In the Black-Scholes market with no dividends, the risky asset s spot-price is modelled by a geometric Brownian motion X, i.e., the SDE for this model is as in (2), with coefficients b(x) = rx, σ(x) = σx, where σ > 0 and r 0 denote the (constant) volatility the (constant) interest rate, respectively. Therefore, the SDE is given by (use S instead of X) with infinitesimal generator ds t = rs t dt + σs t dw t, A := 1 2 σ2 s 2 ss + rs s. (6)

8 The Black-Scholes PDE We obtain: the discounted price of a European contract with payoff g(s), i.e., V (t, s) = E[e r(t t) g(s T ) S t = s], is equal to a regular solution V (t, s) of the Black-Scholes equation t V σ2 s 2 ss V + rs s V rv = 0 in [0, T ) R + V (T, s) = g(s) in R +. The infinitesimal generator (and hence the Black-Scholes PDE) degenerates at s = 0. Furthermore, the PDE is backward in time.

9 To obtain a nondegenerate equation, we switch to the log-price process X t = log(s t ) which solves the SDE dx t = ( r 1 2 σ2) dt + σdw t. The infinitesimal generator for this process has constant coefficients: A BS := 1 2 σ2 xx + ( r 1 2 σ2) x. We furthermore change to time-to-maturity t T t, to obtain a forward parabolic problem.

10 The Black-Scholes PDE in log-price Thus, by setting V (t, s) =: v(t t, log s), the BS equation (in real price) satisfied by V (t, s) becomes the BS equation for v(t, x) in log-price t v A BS v + rv = 0 in (0, T ] R V (0, x) = g(e x ) in R, (7) with A BS := 1 2 σ2 xx + ( r 1 2 σ2) x. We next study the variational formulation of (7).

11 Outline Black-Scholes equation

12 Denote by (, ) the L 2 (R)-inner product, i.e., (φ, ψ) := R φψdx. Denote by a BS (, ) : H 1 (R) H 1 (R) R the bilinear form associated to the operator A BS a BS (ϕ, φ) := 1 2 σ2 (ϕ, φ ) + (σ 2 /2 r)(ϕ, φ) + r(ϕ, φ). (8) The variational formulation of the Black-Scholes equation (7) reads: Find u L 2 (J; H 1 (R)) H 1 (J; L 2 (R)) such that ( t u, v) + a BS (u, v) = 0, v H 1 (R), a.e. in J, (9) u(0) = u 0, where u 0 (x) := g(e x ).

13 If u 0 L 2 (R), then problem (9) admits a unqiue solution, since a BS (, ) is continuous and satisfies a Gårding inequality on H 1 (R). Proposition There exist constants C i = C i (σ, r) > 0, i = 1, 2, 3, such that there holds for all ϕ, φ H 1 (R) a BS (ϕ, φ) C 1 ϕ H 1 φ H 1, a BS (ϕ, ϕ) C 2 ϕ 2 H C 1 3 ϕ 2 L. 2 Proof. a BS (ϕ, φ) 1 2 σ2 ϕ L 2 φ L 2 + σ 2 /2 r ϕ φ L 2 +r ϕ L 2 φ L 2 C 1 (σ, r) ϕ H 1 φ H 1. a BS (ϕ, ϕ) = 1 2 σ2 ϕ 2 L 2 + r ϕ 2 L 2 = 1 2 σ2 ϕ 2 H 1 + (r σ 2 /2) ϕ 2 L σ2 ϕ 2 H 1 r σ 2 /2 ϕ 2 L 2.

14 Outline Black-Scholes equation

15 u 0 (x) = g(e x ) L 2 (R) implies an unrealistic growth condition on the payoff g. For example, the payoff of both the put g(e x ) = max{0, K e x } and the call g(e x ) = max{0, e x K} are not in L 2 (R). We can weaken this assumption by reformulating the problem on a bounded domain (which we have to do anyway for discretizing the problem) The unbounded domain R of the log price x = log s is truncated to a bounded domain G. In terms of financial modeling, this corresponds to approximating the option price by a knock-out barrier option. Let G = ( R, R), R > 0, be an open subset and let τ G = inf{t 0 X t G c } be the first hitting time of the complement set G c = R\G by X.

16 The price of a knock-out barrier option in log-price with payoff g(e x ) is given by [ ] v R (t, x) = E e r(t t) g(e X T )1 {T <τg } X t = x. (10) We show that the barrier option price v R converges to the option price [ ] v(t, x) = E e r(t t) g(e X T ) X t = x, exponentially fast in R. Theorem Suppose there exist C > 0, q 1 such that the payoff function g : R + R satisfies g(s) C(s + 1) q for all s R +. Then, there exist C(T, σ), γ 1, γ 2 > 0, such that v(t, x) v R (t, x) C(T, σ)e γ 1R+γ 2 x.

17 We see from this Theorem that v R v exponentially for a fixed x as R. The artificial zero Dirichlet barrier type conditions at x = ±R are not describing correctly the asymptotic behavior of the price v(t, x) for large x. Since the barrier option price v R is a good approximation to v for x R, R should be selected substantially larger than the values of x of interest. The barrier option price v R can again be computed as the solution of a PDE provided some smoothness assumptions.

18 Theorem Let v R (t, x) C 1,2 (J R) C 0 (J R) be a solution of t v R + A BS v R rv R = 0, (11) on [0, T ) G where the terminal and boundary condition given by v R (T, x) = g(e x ), x G, v R (t, x) = 0, on (0, T ) G c. Then, v R (t, x) can also be represented as in (10). Now, we can restate the problem (9) on the bounded domain: Find u R L 2 (J; H 1 0 (G)) H1 (J; L 2 (G)) such that ( t u R, v) + a BS (u R, v) = 0, v H 1 0(G), a.e. in J, (12) u R (0) = u 0 G.

19 Outline Black-Scholes equation

20 Finite difference discretization We discretize the PDE (11) directly using finite differences on bounded domain with homogeneous Dirichlet boundary conditions. Using xx v R (t m, x i ) h 2 (u m i+1 2um i u m i 1 ), x v R (t m, x i ) (2h) 1 (u m i+1 um i 1 ) we obtain the matrix problem: Find u m+1 R N such that for m = 0,..., M 1, ( I + θkg BS ) u m+1 = ( I (1 θ)kg BS) u m, u 0 = u 0, where G BS = σ 2 /2 R + ( σ 2 /2 r ) C + ri, is given explicitly with R := h 2 tridiag ( 1, 2, 1 ), C := (2h) 1 tridiag ( 1, 0, 1 ).

21 Finite element discretization We discretize (12) using the θ-scheme and the finite element space V N = S 1 T H1 0 (G). Setting again u 0(x) := g(e x ) and assuming uniform mesh width h and constant time step k, we obtain the matrix problem: Find u m+1 R N such that for m = 0,..., M 1 (M + kθa BS )u m+1 = (M k(1 θ)a BS )u m, u 0 N = u 0, where M ij = (b j, b i ) L 2 (G) and A BS ij = a BS (b j, b i ). Using the methods we have developed in chapter 3, we find A BS = σ 2 /2 S + ( σ 2 /2 r ) B + rm, explicitly with S := h 1 tridiag ( 1, 2, 1 ), B := 2 1 tridiag ( 1, 0, 1 ), M := 6 1 h tridiag ( 1, 4, 1 ).

22 Outline Black-Scholes equation

23 Graded mesh Know: achievable convergence rate of linear continuous FEM + θ-scheme is u u N L 2 (J;L 2 (G)) = O(h 2 + k r ) provided the initial data u 0 (x) = g(e x ) (the payoff) satisfies u 0 H 2 (G). Here, r = 1 for θ [0, 1] \ {1/2} and r = 2 for θ = 1/2 (constant time step k is used). However, for put and call contracts (u 0 H 3/2 ε (G)) or for digital options (u 0 H 1/2 ε (G)), this regularity assumption is not satisfied, and we expect a reduction of the rate of convergence w.r. to time, i.e. we expect a lower r. To recover the optimal convergence rate for u 0 H s (G), 0 < s < 2 we need to use graded meshes in time.

24 Set T = 1. Let λ : [0, 1] [0, 1] be a grading function which is strictly increasing and satisfies λ C 0 ([0, 1]) C 1 ((0, 1)), λ(0) = 0, λ(1) = 1. We define for M N the graded mesh by the time points, t m = λ ( m ), m = 0, 1,..., M. M It can be shown that we obtain again the optimal convergence rate if λ(t) = O(t β ) where β depends on r and s, β = β(r, s). We give an example for a European call and digital (or binary) option.

25 Example We set strike K = 100, volatility σ = 0.3, interest rate r = 0.01, maturity T = 1. For the discretization we use M = O(N), θ = 1/2, R = 3 and apply the L 2 -projection for u 0. We measure the discrete L 2 (J; L 2 (G))-error defined by M m=1 k ih ɛ m 2 l 2 where ɛ m 2 l 2 := N u(t m, x i ) u N (t m, x i ) 2. i=1 both with constant time steps and with graded time steps. We use the grading factor β = 3 for the call option g(s) = max{0, s K} and β = 25 for the digital option g(s) = χ {s>k}.

26 10 0 Call option Digital option 10 0 Call option Digital option L 2 -error 10 2 L 2 -error ments 10 3 s = 2.0 PSfrag replacements 10 4 s = N N