Lecture 10. Finite difference and finite element methods. Option pricing Sensitivity analysis Numerical examples

Transcription

1 Finite difference and finite element methods Lecture 10

2 Sensitivities and Greeks Key task in financial engineering: fast and accurate calculation of sensitivities of market models with respect to model parameters. Necessary in model calibration, risk analysis and in the pricing and hedging of certain derivative contracts. Example: variations of option prices with respect to the spot price or with respect to time-to-maturity, the so-called Greeks.

3 Outline Option pricing

4 Consider the process X to model the dynamics of a single underlying, a basket or a underlying and its background volatility drivers in case of stochastic volatility models. For notational simplicity assume that r = 0. As shown in many instances of this course 1, the fair price of a European style contingent claim with payoff g and underlying X, i.e., is the solution of v(t, x) = E [ g(x T ) X t = x ], t v + Av = 0 in J R d, v(t, x) = g(x) in R d. (1) 1 provided some smoothness assumptions

5 In (1), we denote by A the infinitesimal generator of X and assume that it has the following form (Af)(x) = 1 2 tr[q(x)d2 f(x)] + b(x) f(x) + c(x)f(x) ( + f(x + z) f(x) z f(x) ) ν(dz), R d where Q : R d R d d, b : R d R d, c : R d R and ν a d-dimensional Lévy measure satisfying R d min{1, z 2 }ν(dz) < and z >1 z i ν(dz) <, i = 1,..., d. We calculate the sensitivities of the solution v of (1) with respect to parameters in the infinitesimal generator A and with respect to solution arguments x and t. (2)

6 Let S η be the set of admissible parameter. Write A(η 0 ) for a fixed parameter η 0 S η to emphasize the dependence of A on η 0 and change to time-to-maturity t T t. In (1), admit a non-trivial right hand side f (will become important later on). Thus, we consider, instead of (1), the equation t u A(η 0 )u = f in J R d, u(0, x) = u 0 in R d, (3) with u 0 = g. For the numerical implementation we truncate (3) to a bounded domain G R d and impose boundary conditions on G. Typically, G = d k=1 (a k, b k ), a k, b k R, b k > a k, k = 1,..., d.

7 Bilinear form a(η 0 ;, ) : V V R associated with infinitesimal generator A(η 0 ), η 0 S η, a(η 0 ; u, v) := A(η 0 )u, v V,V, u, v V, where we consider the abstract setting as given in Chapter 3 with Hilbert spaces V H H V. Assume that a(η 0 ;, ) is continuous and satisfies a Gårding inequality for all η 0 S η. For f L 2 (J; V ) and u 0 H the weak formulation of problem (3) is given by: Find u L 2 (J; V) H 1 (J; H) such that ( t u, v) H + a ( η 0 ; u, v ) = f, v V,V, v V, (4) u(0, ) = u 0.

8 Outline Option pricing

9 For a market model X we distinguish two classes of sensitivities. 1. The sensitivity of the solution u to a variation S η η s := η 0 + sδη, s > 0, of an input parameter η 0 S η. Typical examples are the Greeks Vega ( σ u), Rho ( r u) and Vomma ( σσ u). Other sensitivities which are not so commonly used in the financial community are the sensitivity of the price with respect to the jump intensity or the order of the process that models the underlying. 2. The sensitivity of the solution u to a variation of arguments t, x. Typical examples are the Greeks Theta ( t u), Delta ( x u) and Gamma ( xx u). Today, we focus on the first class!

10 Sensitivity with respect to model parameters Let C S η be a Banach space over a domain G R d. (C is the space of parameters or coefficients in A) Denote by u(η 0 ) the unique solution to (4). Introduce the derivative of u(η 0 ) with respect to η 0 S η as the mapping D η0 u(η 0 ) : C V ũ(δη) := D η0 u(η 0 )(δη) 1( := lim u(η0 + sδη) u(η 0 ) ), δη C. s 0 + s Also introduce the derivative of A(η 0 ) with respect to η 0 S η Ã(δη)ϕ := D η0 A(η 0 )(δη)ϕ := 1( lim A(η0 + sδη)ϕ A(η 0 )ϕ ), ϕ V, δη C. s 0 + s

11 Assume that Ã(δη) L(Ṽ, Ṽ ) with Ṽ a real and separable Hilbert space satisfying Ṽ V H = H V Ṽ. Further assume that there exists a real and separable Hilbert space V Ṽ such that Ãv V, v V. Lemma Let Ã(δη) L(Ṽ, Ṽ ), δη C and u(η 0 ) : J V, η 0 S η be the unique solution to t u(η 0 ) A(η 0 )u(η 0 ) = 0 in J R d, u(η 0 )(0, ) = g(x) in R d. (5) Then, ũ(δη) solves t ũ(δη) A(η 0 )ũ(δη) = Ã(δη)u(η 0) in J R d, ũ(δη)(0, ) = 0 in R d. (6)

12 Associate to the operator Ã(δη) the bilinear form ã(δη;, ) : Ṽ Ṽ R given by ã(δη; u, v) = Ã(δη)u, v e V, e V. The variational formulation to (6) reads: Find ũ(δη) L 2 (J; V) H 1 (J; H) such that v V ( t ũ(δη), v) H + a ( η 0 ; ũ(δη), v ) = ã ( δη; u(η 0 ), v ), (7) ũ(δη)(0) = 0. Problem (7) admits a unique solution ũ(δη) V due to the assumptions on a(η 0 ;, ), Ã and u(η 0 ) V.

13 Example: BS model One-dimensional diffusion process X with inf. generator (r = 0) A BS f = 1 2 σ2 xx f 1 2 σ2 x f. Sensitivity of the price with respect to the volatility σ. The set of admissible parameters S η is S η = R + with η = σ. Have Ã(δσ)f = δσσ 0 xx f δσσ 0 x f L(V, V ), with δσ R = C. Bilinear form ã(δσ;, ) appearing in the weak formulation (7) of ũ(δσ) is given by ã(δσ; ϕ, φ) = δσσ 0 ( x ϕ, x φ) + δσσ 0 ( x ϕ, φ). In this example, Ṽ = V = H1 0 (G).

14 Example: Tempered stable model One-dimensional pure jump process X with tempered stable density k(z) = z 1 α (c + e β + z 1 {z>0} + c e β z 1 {z>0} ) and infinitesimal generator A J f = (f(x + z) f(x) zf (x))k(z)dz. R Sensitivity of the price with respect to the jump intensity parameter α of X. Have S η = (0, 2) with η = α and Ã(δα)f = δα (f(x+z) f(x) zf (x)) k(z)dz L(Ṽ, Ṽ ), R where the kernel k is given by k(z) := ln z k(z). In this example, Ṽ = H α/2+ε (G) H α/2 (G) = V, ε > 0.

15 Consider the finite element discretization of (6), (7). We obtain the matrix form Find ũ m+1 R N such that for m = 0,..., M 1, (M + θ ta)ũ m+1 = (M (1 θ) ta ) ũ m ũ 0 = 0. tã(θum+1 + (1 θ)u m ), Here, Ã is matrix of the bilinear form ã(δη;, ) in the basis of V N, Ãij = ã(δη; b j, b i ) for 1 i, j N. Furthermore, u m+1, m = 0,..., M 1, is the coefficient vector of the finite element solution u N (t m+1, x) V N to (4).

16 Denote by y solve(b, x) the output of a generic solver for a linear system Bx = y. Then, we have following algorithm to compute sensitivities with respect to model parameters. Choose η 0 S η, δη C. Calculate the matrices M, A and Ã. Let u 0 be the coefficient vector of u 0 N in the basis of V N. Set ũ 0 = 0. For j = 0, 1,..., M 1, u 1 solve ( M + θ ta, (M (1 θ) ta)u 0) Next j Set f := Ã(θu1 + (1 θ)u 0 ). ũ 1 solve ( M + θ ta, M (1 θ) ta)ũ 0 tf) ) Set u 0 := u 1, ũ 0 := ũ 1.

17 Theorem Assume u, ũ C 1 (J; V) C 3 (J; V ). Then, there holds the error bound M 1 ũ M ũ M N 2 + t ũ m+θ ũ m+θ N 2 V C v {u,eu} { m=0 + Ch 2(s r) ( t) 2 T 0 v(τ) 2 dτ θ [0, 1] ( t) 4 T 0... v (τ) 2 dτ θ = 1 2 v {u,eu} T + Ch 2(s r) max 0 t T u(t) 2 e H s. 0 v(τ) 2 eh s rdτ Hence: the approximated sensitivities converge with the same, optimal rate as the approximated option price!

18 Outline Option pricing

19 One-dimensional models BS and variance gamma model. European put with K = 100, T = 1.0 and r = We calculate the Greeks Delta, = s V, and Gamma, Γ = ss V. For BS, we additionally compute the Vega, V = σ V. Choose σ = 0.3 and for the variance gamma model ν = 0.04, θ = 0.2. As predicted in Theorem 2, all Greeks convergence with the optimal rate as the price V itself.

20 Price Delta Gamma Vega Price Delta Gamma Error 10 3 Error s = s = N N Convergence rates of Greeks for a European put in the Black-Scholes (left) and variance gamma model (right).

21 Multi-variate models Heston stochastic volatility model. Calculate the sensitivity ũ(δρ) with respect to correlation ρ of the Brownian motions that drive the underlying and the volatility. The derivative with respect to ρ of A H κ is Corresponding stiffness matrix à H κ (δρ) = 1 2 δρβ(y xy + κy 2 x ) à H κ = 1 2 βb1 (B x 2 + κm x2 2 ). European call with K = 100 and T = 0.5. Model parameters: α = 2.5, β = 0.5, m = and ρ 0 = 0.4.

22 10 1 Price Sensitivity 10 0 Error s = N Convergence rate of sensitivity ũ(δρ) in the Heston stochastic volatility model.