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

Computational Methods for Quant. Finance II

Finite difference and finite element methods

Lecture 10

Sensitivities and Greeks

I Key task in financial engineering: fast and accurate

calculation of sensitivitiesof market models with respect to model parameters.

I Necessary in modelcalibration, risk analysis and in the pricing andhedging of certain derivative contracts.

I Example: variations of option prices with respect to the spot price or with respect to time-to-maturity, the so-called

“Greeks”.

Outline

Option pricing

Sensitivity analysis

Numerical examples

I 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.

I For notational simplicity assume that r = 0.

I As shown in many instances of this course ¹, the fair price of a Europeanstyle contingent claim with payoff g and

underlying X, i.e.,

v(t, x) = E

g(X_T) | X_t = x , is the solution of

∂_tv+ Av = 0 in J × R^d, v(T, x) = g(x) in R^d. (1)

1provided some smoothness assumptions

I In (1), we denote by A theinfinitesimal generator of X and assume that it has the following form

(Af )(x) = 1

2tr[Q(x)D²f(x)] + b(x)^>∇f (x) + c(x)f (x) +

Z

R^d

f(x + z) − f (x) − z^>∇f (x)
ν(dz),

(2) where Q : R^d → R^d×d, b : R^d → R^d, c : R^d→ R and ν a d-dimensional L´evy measure satisfying

R

R^dmin{1, |z|²}ν(dz) < ∞ andR

|z|>1|z_i|ν(dz) < ∞, i= 1, . . . , d.

I We calculate the sensitivitiesof the solution v of (1) with respect to parametersin the infinitesimal generator A and with respect tosolution arguments x and t.

I Let S_η be the set of admissible parameter.

I Write A(η₀) for a fixed parameter η₀ ∈ S_η to emphasize the dependence of A onη₀ and change to time-to-maturity t→ T − t.

I In (1), admit a non-trivial right hand side f (will become important later on).

I Thus, we consider, instead of (1), the equation

∂_tu− A(η₀)u = f in J × R^d, u(0, x) = u₀ in R^d, (3) with u₀= g.

I For the numerical implementation we truncate (3) to a bounded domain G ⊂ R^d and impose boundary conditions on

∂G. Typically, G =Q_d

k=1(a_k, b_k), a_k, b_k∈ R, bk> a_k, k = 1, . . . , d.

I Bilinear form a(η₀; ·, ·) : V × V → R associated with infinitesimal generator A(η₀), η₀∈ S_η,

a(η₀; u, v) := −hA(η₀)u, vi_V∗,V, u, v ∈ V,

where we consider the abstract setting as given in Chapter 3 with Hilbert spaces V ⊂ H ≡ H^∗⊂ V^∗.

I Assume that a(η₀; ·, ·) is continuous and satisfies a G˚arding inequality for all η₀ ∈ S_η.

I For f ∈ L²(J; V^∗) and u₀ ∈ H the weak formulationof problem (3) is given by:

Find u ∈ L²(J; V) ∩ H¹(J; H) such that (∂_tu, v)_H+ a η₀; u, v

= hf, vi_V^∗_,V, ∀v ∈ V , (4) u(0, ·) = u₀.

Outline

Option pricing

Sensitivity analysis

Numerical examples

For a market model X we distinguish two classes of sensitivities.

1. The sensitivity of the solution u to a variation S_η 3 η_s:= η₀+ sδη, s >0,

of an input parameter η₀ ∈ Sη. Typical examples are the Greeks Vega (∂_σu), Rho (∂_ru) and Vomma (∂_σσu). 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 (∂_tu), Delta (∂_xu) and Gamma (∂_xxu).

Today, we focus on the first class!

Sensitivity with respect to model parameters

I Let C ⊇ S_η be a Banach space over a domain G ⊂ R^d. (C is the space of parameters or coefficients in A)

I Denote by u(η₀) the unique solution to (4). Introduce the derivative of u(η₀) with respect to η₀ ∈ S_η as the mapping D_η0u(η₀) : C → V

e

u(δη) := D_η0u(η₀)(δη) := lim

s→0⁺

1

s u(η₀+ sδη) − u(η₀)

, δη∈ C.

I Also introduce the derivative of A(η0) with respect to η0 ∈ Sη

A(δη)ϕ := De _η0A(η₀)(δη)ϕ := lim

s→0⁺

1

s A(η₀+ sδη)ϕ − A(η₀)ϕ

, ϕ∈ V, δη ∈ C.

I Assume that eA(δη) ∈ L( eV, eV^∗) with eV a real and separable Hilbert space satisfying

V ⊆ V ⊂ H ∼e = H^∗⊂ V^∗⊆ eV^∗.

I Further assume that there exists a real and separable Hilbert space V ⊆ eV such that eAv ∈ V^∗, ∀v ∈ V.

Lemma

Let eA(δη) ∈ L( eV, eV^∗), ∀δη ∈ C and u(η₀) : J → V, η₀ ∈ S_η be the unique solution to

∂_tu(η₀) − A(η₀)u(η₀) = 0 in J × R^d,

u(η₀)(0, ·) = g(x) in R^d. (5) Then, eu(δη) solves

∂_tu(δη) − A(ηe ₀)eu(δη) = A(δη)u(ηe ₀) in J× R^d,

u(δη)(0, ·) = 0e in R^d. (6)

I Associate to the operator eA(δη) the bilinear form ea(δη; ·, ·) : eV × eV → R given by

ea(δη; u, v) = −h eA(δη)u, vi_{Ve∗, eV}.

I The variational formulationto (6) reads:

Find eu(δη) ∈ L²(J; V) ∩ H¹(J; H) such that ∀v ∈ V (∂_teu(δη), v)_H+ a η₀; eu(δη), v

= −ea δη; u(η0), v

, (7) eu(δη)(0) = 0 .

I Problem (7) admits a unique solution eu(δη) ∈ V due to the assumptions on a(η₀; ·, ·), eA and u(η₀) ∈ V.

Example: BS model

I One-dimensional diffusion process X with inf. generator (r = 0)

A^BSf = 1

2σ²∂_xxf −1 2σ²∂_xf.

I Sensitivity of the price with respect to the volatility σ. The set of admissible parameters S_η is S_η = R+ with η = σ.

I Have

A(δσ)f = δσσe ₀∂_xxf− δσσ₀∂_xf ∈ L(V, V^∗), with δσ ∈ R = C.

I Bilinear form ea(δσ; ·, ·) appearing in the weak formulation (7) of eu(δσ) is given by

ea(δσ; ϕ, φ) = δσσ0(∂_xϕ, ∂_xφ) + δσσ₀(∂_xϕ, φ).

I In this example, eV = V = H₀¹(G).

Example: Tempered stable model

I 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^Jf = Z

R

(f (x + z) − f (x) − zf⁰(x))k(z)dz.

I Sensitivity of the price with respect to the jump intensity parameter α of X. Have S_η = (0, 2) with η = α and A(δα)f = δαe

Z

R

(f (x + z) − f (x) − zf⁰(x))ek(z)dz ∈ L( eV, eV^∗), where the kernel ek is given by

ek(z) := − ln |z|k(z).

I In this example, eV = eH^α/2+ε(G) ⊂ eH^α/2(G) = V, ε > 0.

I Consider the finite element discretization of (6), (7). We obtain the matrix form

Find eu^m+1∈ R^N such that for m = 0, . . . , M − 1, (M + θ∆tA)eu^m+1 = (M − (1 − θ)∆tA

e u^m

−∆t eA(θu^m+1+ (1 − θ)u^m), e

u⁰ = 0 .

I Here, eA is matrix of the bilinear form ea(δη; ·, ·) in the basis of V_N, eAij = ea(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).

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 η₀ ∈ S_η, δη ∈ C.

Calculate the matrices M, A and eA.

Let u⁰ be the coefficient vector of u⁰_N in the basis of V_N. Set eu⁰ = 0.

For j = 0, 1, . . . , M − 1,

u¹ ← solve M + θ∆tA, (M − (1 − θ)∆tA)u⁰
Set f := eA(θu¹+ (1 − θ)u⁰).

e

u¹ ← solve M + θ∆tA, M − (1 − θ)∆tA)eu⁰− ∆tf)
Set u⁰:= u¹, eu⁰ := eu¹.

Next j

Theorem

Assume u,ue∈ C¹(J; V) ∩ C³(J ; V^∗). Then, there holds the error bound

keu^M − eu^M_Nk²+ ∆t

M −1X

m=0

keu^m+θ− eu^m+θ_N k²_V

≤ C X

v∈{u,eu}

( (∆t)²R_T

0 k¨v(τ )k²_∗dτ θ∈ [0, 1]

(∆t)⁴R_T

0 k...

v(τ )k²_∗dτ θ= ¹₂ + Ch^2(s−r) X

v∈{u,eu}

Z _T

0

k ˙v(τ )k²_e

H^s−rdτ + Ch^2(s−r) max

0≤t≤Tku(t)k²_e

H^s.

Hence: the approximated sensitivities converge with the same, optimal rate as the approximated option price!

Outline

Option pricing

Sensitivity analysis

Numerical examples

One-dimensional models

I BS and variance gamma model. European put with K = 100, T = 1.0 and r = 0.01.

I We calculate the Greeks Delta, ∆ = ∂_sV, and Gamma, Γ = ∂_ssV. For BS, we additionally compute the Vega, V = ∂_σV.

I Choose σ = 0.3 and for the variance gamma model ν = 0.04, θ= −0.2.

I As predicted in Theorem 2, all Greeks convergence with the optimal rate as the price V itself.

10¹ 10² 10³ 10⁴ 10⁻⁷

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

s = −2.0

N

Error

Price Delta Gamma Vega

10¹ 10² 10³ 10⁴

10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰ 10²

s = −2.0

N

Error

Price Delta Gamma

Convergence rates of Greeks for a European put in the Black-Scholes (left) and variance gamma model (right).

Multi-variate models

I Heston stochastic volatility model.

I Calculate the sensitivity eu(δρ) with respect to correlation ρ of the Brownian motions that drive the underlying and the volatility.

I The derivative with respect to ρ of A^H_κ is Ae^H_κ(δρ) = 1

2δρβ(y∂_xy+ κy²∂_x)

I Corresponding stiffness matrix Ae^H_κ = −1

2βB¹⊗ (B^x2 + κM^x22).

I European call with K = 100 and T = 0.5. Model parameters:

α= 2.5, β = 0.5, m = 0.025 and ρ₀ = −0.4.

10² 10³ 10⁴ 10⁵ 10⁻³

10⁻² 10⁻¹ 10⁰ 10¹

s = −1.0

N

Error

Price Sensitivity

Convergence rate of sensitivity eu(δρ) in the Heston stochastic volatility model.