Numerical PDE methods for exotic options

Lecture 8

Numerical PDE methods

for exotic options

Lecture Notes

by Andrzej Palczewski

Barrier options

For barrier option part of the option contract is triggered if the asset price hits some barrier S = X, at some time prior to expiry.

Depending on the way the hitting time is monitored we dis-tinguish following options:

continuous monitoring (called American barrier) – hitting moment can be any time between issue and expiry of the option;

discrete monitoring – security price is monitored only in selected moments of time (say daily or weekly);

discrete monitoring called European barrier – security price is monitored only at expiry.

Barrier options – cont.

Depending on the conditions under which the option gets or losses value we have following type of barrier options:

up-and-in – the option expires worthless unless the barrier S = X is reached from below;

down-and-in – the option expires worthless unless

the barrier S = X is reached from above;

up-and-out – the option expires worthless if the barrier S = X is reached from below;

down-and-out – the option expires worthless if the barrier S = X is reached from above.

Barrier options – valuation

The value of a barrier option can be computed using the Black-Scholes equation.

For European options with barriers of European or Ameri-can type there are explicit analytic formulas (although some of them are quite complicated). But American options or op-tion with the barrier monitored discretely have to be valued numerically.

To solve the Black-Scholes equation for barrier options we have to supplement the equation with proper boundary and terminal conditions.

Knock-out options

Terminal conditions are as for vanila options modified by lim-itations coming from boundary conditions.

Boundary conditions:

Down-and-out option

V (S, t) = 0, for S = X,

V (S, t) = boundary value for vanila option, for S > X. Up-and-out option

V (S, t) = boundary value for vanila option, for S < X, V (S, t) = 0, for S = X.

Double knock-out options

A double knock-out option is a barrier option which expires worthless when the price of the underlying asset is to the left of the lower barrier X₁ or to the right of the upper barrier

X₂.

Terminal conditions are as for vanila options modified by lim-itations coming from boundary conditions.

Boundary conditions:

V (S, t) = 0, for S = X₁, V (S, t) = 0, for S = X₂.

Knock-in options

For S ∈ [0, X] this is a plain vanila option. We show condi-tions for S > X. Terminal condition V (S, T ) = 0 for S > X. Boundary condition V (S, t) → 0 as S → ∞, V (X, t) = C(X, t),

where C(S, t) is the price of corresponding vanila option.

Up-and-in option.

For S ∈ [X, ∞) this is a plain vanila option. We show condi-tions for S < X. Terminal condition V (S, T ) = 0 for S < X. Boundary condition V (S, t) → 0 as S → 0, V (X, t) = C(X, t),

Discretely monitored barrier

A boundary constraint which holds at a point in time can be applied directly in an explicit manner. We compute the solution for a given time level and apply the constraint if nec-essary. Then move to the next time level.

Consider a down-and-out option with barrier X monitored in time moments t_α. At time level ν we compute V ν and then apply the boundary constraint

V_iν = V (s_i, t_ν) =

(

0, if s_i ≤ δ_(tν,tα)X, V_iν, otherwise

where δ_(tν,tα) is the Kronecker delta.

Discretely monitored barrier – cont.

For discretely monitored barriers we need boundary condi-tions for S → 0 and/or S → ∞. To impose these boundary conditions correctly we can use the relation

knock-in option + knock-out option = vanilla option (1) which holds as well for continuously and discretely moni-tored barriers when the monitoring moments are the same for in and out options.

Consider again a down-and-out option with barrier moni-tored discretely. For the upper boundary we have

Discretely monitored barrier – cont.

To impose proper conditions on the lower boundary S = 0

let us observe that for the down-and-in option we have

V (S, t) = boundary value for vanila option, for S → 0.

Using relation (1) we obtain for a down-and-out option

V (S, t) = 0, for S → 0.

Boundary conditions for other options can be obtained in a similar manner.

Lookback options

For the basic lookback contracts, the payoff comes in two varieties: the fixed strike and the floating strike. These options have payoffs that are the same as vanilla options except that in the floating strike option the vanilla exercise price is replaced by the maximum or minimum. In the fixed strike option it is the asset value in the vanilla option that is replaced by the maximum or minimum.

H(S_T , J_T) =          M_T − S_T, floating put, S_T − m_T, floating call (M_T − K)+, fixed call, (K − m_T)+, fixed put,

where J_T denotes M_T or m_T and

Lookback options – cont.

Depending on the way the maximum or minimum is moni-tored we distinguish:

continuous monitoring – security price is monitored continuously between the issue and expiry of the option;

discrete monitoring – security price is monitored only in selected moments of time (say weekly or monthly).

The pair (S_t, J_t) is a Markov process, and the price at time

t of the lookback option is equal to V (t, S_t, J_t) where the function V is defined for t ∈ [0, T ] and (S, J) in an appropriate set depending whether J = M or J = m

(S, J) ∈

(

{(S, M ) ∈ _R2₊ : 0 ≤ S ≤ M }, for J = M ,

Lookback options – valuation

The value of a lookback option can be computed using an extended version of the Black-Scholes equation. We derive this equation assuming J = M.

Let M_n(t) =  Z t 0 S_τ
n1/n. We have lim n→∞ Mn(t) = max0≤τ ≤t Sτ = Mt.

Next we consider the function V (S, M_n, t), which can be think of as a price of an instrument depending on the vari-able M_n.

Black-Scholes equation

Using the Feynman-Kac theorem and the differential

dM_n(t) = 1 n S_tn M_n(t)
n−1dt we obtain ∂V (S, M_n, t) ∂t + rS ∂V (S, M_n, t) ∂S + 1 n Sn M_nn−1 ∂V (S, M_n, t) ∂M_n + + 1 2σ 2_S2 ∂2V (S, Mn, t) ∂S2 − rV (S, Mn, t) = 0. Computational Finance – p. 15

Black-Scholes equation – cont.

Since S_t ≤ max_{0≤τ ≤t} S_τ = M_t then for S < M lim n→∞ 1 n Sn M_n
n−1 = 0

and we obtain the Black-Scholes equation

∂V (S, M, t) ∂t +rS ∂V (S, M, t) ∂S + 1 2σ 2_S2∂2V (S, M, t) ∂S2 −rV (S, M, t) = 0.

Observe that the above equation is the standard one-dimensional Black-Scholes equation in which the variable

M plays the role of a parameter. This parameter enters however into boundary and terminal conditions.

Black-Scholes equation – cont.

At expiry we have

V (S, M, T ) = H(S, M ).

We require a boundary condition on the line S = M. Con-sider the stochastic process S_t close to current maximum. Since the probability that the current maximum is still the maximum at expiry is zero, the option price must be insen-sitive to small changes of M when S is close to M. Hence

∂V (S, M, t)

∂M = 0 for S = M.

Black-Scholes equation – cont.

Additional boundary conditions depend on the type of the option.

For a floating put option we have

V (0, M, t) = Me−r(T −t).

For a floating call option we have boundary conditions as for a vanila option

V (S, m, t) ≈ S for S very large.

Observe that in all these cases we have to solve a 3-dimensional PDE.

Similarity reduction

For floating strike lookback options the 3-dimensional Black-Scholes equation can be reduced to 2 dimensions. We per-form all calculations for a floating put option.

We introduce the new variable x = _MS . When the payoff can be written as (this is the case of the floating put and call)

H(S, M ) = M h(x),

then we search for a solution of the Black-Scholes equation assuming

V (S, M, t) = M W (x, t).

The equation for W has the form

∂W (x, t) ∂t + rx ∂W (x, t) ∂x + 1 2σ 2_x2∂2W (x, t) ∂x2 − rW (x, t) = 0. Computational Finance – p. 19

Similarity reduction – cont.

The terminal and boundary conditions are as follows:

W (x, T ) = max(1 − x, 0), W (0, t) = e−r(T −t),

∂W

Asian options

For the basic Asian contracts, the payoff comes in two va-rieties: the average price and the average strike. These options have payoffs that are the same as vanilla options except that in the average strike option the vanilla exercise price is replaced by the average of asset prices. In the av-erage price option it is the asset value in the vanilla option that is replaced by the average of asset prices.

(S − K)b +, average price call, (K − S)b +, average price put, (S_T − S)b +, average strike call, (S − Sb _T )+, average strike put,

where bS denotes the average of asset prices.

Asian options – cont.

There are several ways the average of past values of S_t can be formated:

arithmetic average continuously monitored b

S = _T1 R₀T S_tdt,

arithmetic average discretely monitored b

S = _n1 Pn_i=1 S_ti,

geometric average continuously monitored b

S = exp_T1 R₀T log(S_t)dt,

geometric average discretely monitored Sb =

Qn

i=1 Sti

1/n

Asian options – cont.

Since for geometric average Asian options we have analytic expressions for the price, we are only interested in calcula-tion of arithmetic average Asian opcalcula-tion.

In what follows, we shall describe the PDE method for the calculation of an arithmetic average continuously monitored Asian option. This problem leads to a 3-dimensional Black-Scholes equation. For an arithmetic average continuously monitored Asian option this equation can be reduced to two dimensions and solved numerical by finite difference algo-rithms.

For an arithmetic average discretely monitored Asian op-tion we can use straightforwardly the known 2-dimensional Black-Scholes equation appropriately modifying initial con-ditions at the time of monitoring.

Asian options – valuation

The value of an arithmetic average continuously monitored Asian option can be computed using an extended version of the Black-Scholes equation.

Let us consider a generalized arithmetic average

A_t =

Z t 0

f (S_τ, τ )dτ,

where f (x, t) depends on the type of averaging (for a simple arithmetic average f (x, t) = x up to a constant factor).

The pair (S_t, A_t) is a Markov process, and the price at time t

of an arithmetic average Asian option is equal to V (t, S_t, A_t)

where the function V is defined for t ∈ [0, T ], S_t > 0 and

Black-Scholes equation

Using the Feynman-Kac theorem and the differential

dA_t = f (S_t, t)dt we obtain ∂V (S, A, t) ∂t + rS ∂V (S, A, t) ∂S + f (S, t) ∂V (S, A, t) ∂A + + 1 2σ 2_S2 ∂2V (S, A, t) ∂S2 − rV (S, A, t) = 0. Computational Finance – p. 25

Similarity reduction

We consider an arithmetic average strike call with payoff  S_T − 1 T AT + = S_T  1 − 1 T S_T Z T 0 S_τdτ + .

This form of the payoff suggest introduction of an auxiliary variable R_t = 1 S_t Z t 0 S_τdτ.

Then the payoff is

S_T  1 − 1 T RT + ,

and we can look for the factorization

Similarity reduction – cont.

The equation for H has the form

∂H(R, t) ∂t + (1 − rR) ∂H(R, t) ∂R + 1 2σ 2_R2∂2H(R, t) ∂R2 = 0.

The terminal condition follows from the payoff

H(R_T, T ) = 1 − 1

T RT

+ .

Since S_t is integrable, then, from the definition of R_t, for

R → ∞ we have S → 0. Remembering that for S → 0 call option is not exercised, we obtain

H(R, t) = 0 for R → ∞.

Similarity reduction – cont.

To obtain the boundary condition on the boundary R = 0 we return to the PDE which H fulfills. For R → 0 this equation gives

∂H(R, t) ∂t +

∂H(R, t)

∂R = 0 for R → 0.

Finally we obtain the following boundary value problem

∂H ∂t + (1 − rR) ∂H ∂R + 1 2σ 2_R2 ∂2H ∂R2 = 0, H = 0 for R → ∞, ∂H ∂t + ∂H ∂R = 0 for R → 0, H(R_T, T ) = 1 − 1 T RT + .

Numerical scheme

We solve the 2-dimensional Black-Scholes PDE of the pre-vious slides (barrier, lookback and options) without transfor-mation of variables. This approach is very usefull because the boundary conditions can be easily implemented.

The equation reads

∂u(x, t) ∂t + α(x) ∂u(x, t) ∂x + 1 2σ 2_x2 ∂2u(x, t) ∂x2 − ρu(x, t) = 0,

with appropriate terminal and boundary conditions.

Finite difference approximation

Let v_i,ν be an approximation of u(x_i, t_ν) and η_i approximation of 2α(xi)

xi+1−xi −1

. Then

v_i,ν−1 − v_i,ν

δt = θAivi−1,ν−1 + θBivi,ν−1 + θCivi+1,ν−1+

+ (1 − θ)A_iv_i−1,ν + (1 − θ)B_iv_i,ν + (1 − θ)C_iv_i+1,ν,

(2) where A_i = 1 2(σ 2_i2 _{− η} i), Bi = −(σ2i2 + ρ), Ci = 1 2(σ 2_i2 _{+ η} i).

The choice of θ gives the explicit (θ = 0), the implicit (θ = 1) and the Crank-Nicolson (θ = 1₂) scheme.

Spurious oscillations

Numerical scheme (2), even if the grid is chosen properly to avoid instabilities, can be a subject of spurious oscillations.

Spurious oscillations – cont.

To understand the phenomenon of spurious oscillations let us consider a model equation

∂u ∂t + a ∂u ∂x = b ∂2u ∂x2.

The source of oscillations is the first order term a∂u_∂x.

To see this let us perform the von Neumann stability analy-sis. To this end we approximate the above equation by finite differences w_j,ν+1 − w_jν ∆t + a w_j+1,ν − w_j−1,ν 2∆x = b w_j+1,ν − 2w_jν + w_j−1,ν (∆x)2 .

Spurious oscillations – cont.

Let us express the approximations w_jν of the ν-th time level by a sum of Fourier modes

w_jν = X k

c(ν)_k eikj∆x,

The linearity of the numerical scheme allows to find a rela-tion

c(ν+1)_k = G_kc(ν)_k ,

where G_k is the growth factor. For |G_k| ≤ 1 it is guaranteed that the modes eik∆x are not amplified.

Spurious oscillations – cont.

For the model equation we arrive at

G_k = 1 − 2λ + 2λ cos k∆x − iβλ sin k∆x

and |G_k|2 = (1 − 4λs2)2 + 4β2λ2s2(1 − s2), where λ = b∆t (∆x)2, β = a∆x b , s = sin k∆x 2 .

A straightforward analysis of the polynomial |G_k|2 reveals that |G_k| ≤ 1 for

0 ≤ λ ≤ 1

2, and, in addition, for β > 2 we need λβ

Spurious oscillations – cont.

The inequality 0 ≤ λ ≤ 1/2 brings back the stability criterion of the heat equation.

β is called a mesh Péclet number and corresponds to the ratio of the convection term a∂u_∂x to the diffusion term b∂_∂x2u2 in

the model equation.

It is clear that controlling oscillations means that β must be small.

Spurious oscillations for BS equation

Theorem In order to prevent the formation of spurious

oscillations in the numerical scheme (2), the following conditions must be satisfied

(x_i − x_i−1)2 x2_i < σ2 η_i and 1 (1 − θ)δt > σ 2 x2i (x_i+1 − x_i)(x_i − x_i−1) + ρ.

Upwind scheme

The source of oscillations is the first order term a∂u_∂x. Let us consider the model equation

∂u

∂t + a ∂u

∂x = 0.

The solution of this equation is F (x−at), where F (y) = u₀(y)

is the initial condition for our equation u(x, 0) = u₀(x).

For a > 0, we approximate the above equation by finite dif-ferences

w_j,ν+1 − w_jν

∆t + a

w_j,ν − w_j−1,ν

∆x = 0

This scheme is called upwind scheme.

Upwind scheme – cont.

To explain the name we return to the solution F (x − at).

As t increases the profile F (y) drifts in positive x-directions. We say that ”the wind blows to the right”.

In agreement with that ”blowing direction” we approximate the derivative

∂u ∂x ≈

w_j,ν − w_j−1,ν

∆x ,

i.e. the information flows from downstream to upstream nodes.

The stability analysis for this scheme gives

G_k = 1 − γ + γe−ik∆x,

Upwind scheme – cont.

For a < 0, the upwind scheme is

w_j,ν+1 − w_jν

∆t + a

w_j+1,ν − w_j,ν

∆x = 0.

The stability analysis for this scheme gives |G_k| ≤ 1 for |γ| ≤ 1.