numerical pde methods for exotic options - logowanieapalczew/cfp_lecture8.pdf · numerical pde...

Lecture 8

Numerical PDE methodsfor exotic options

Lecture Notesby Andrzej Palczewski

Computational Finance – p. 1

Barrier options

For barrier option part of the option contract is triggered ifthe asset price hits some barrier S = X, at some time priorto 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 andexpiry of the option;

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

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


Barrier options – cont.

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

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

down-and-in – the option expires worthless unlessthe barrier S = X is reached from above;

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

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


Barrier options – valuation

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

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

To solve the Black-Scholes equation for barrier options wehave to supplement the equation with proper boundary andterminal 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 expiresworthless when the price of the underlying asset is to theleft of the lower barrier X1 or to the right of the upper barrierX2.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

Down-and-in option.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),

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


Discretely monitored barrier

A boundary constraint which holds at a point in time canbe applied directly in an explicit manner. We compute thesolution 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 intime moments tα. At time level ν we compute V ν and thenapply the boundary constraint

V νi = V (si, tν) =

{0, if si ≤ δ(tν ,tα)X,

V νi , otherwise

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


Discretely monitored barrier – cont.

Boundary conditions on outer boundaries

For discretely monitored barriers we need boundary condi-tions for S → 0 and/or S → ∞. To impose these boundaryconditions 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 samefor in and out options.

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

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


Discretely monitored barrier – cont.

To impose proper conditions on the lower boundary S = 0let 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 asimilar manner.


Lookback options

For the basic lookback contracts, the payoff comes in twovarieties: the fixed strike and the floating strike. Theseoptions have payoffs that are the same as vanilla optionsexcept that in the floating strike option the vanilla exerciseprice is replaced by the maximum or minimum. In the fixedstrike option it is the asset value in the vanilla option that isreplaced by the maximum or minimum.

H(ST , JT ) =

MT − ST , floating put,

ST −mT , floating call

(MT −K)+, fixed call,

(K −mT )+, fixed put,

where JT denotes MT or mT and

MT = max0≤t≤T St, mT = min0≤t≤T St.Computational Finance – p. 12

Lookback options – cont.

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

continuous monitoring – security price is monitoredcontinuously between the issue and expiry of theoption;

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

The pair (St, Jt) is a Markov process, and the price at timet of the lookback option is equal to V (t, St, Jt) where thefunction V is defined for t ∈ [0, T ] and (S, J) in an appropriateset depending whether J = M or J = m

(S, J) ∈

{{(S,M) ∈ R

2+ : 0 ≤ S ≤ M}, for J = M,

{(S,m) ∈ R2+ : 0 ≤ m ≤ S}, for J = m.


Lookback options – valuation

The value of a lookback option can be computed using anextended version of the Black-Scholes equation. We derivethis equation assuming J = M .Let

Mn(t) =(∫ t

0

(Sτ

)n)1/n.

We have

limn→∞

Mn(t) = max0≤τ≤t

Sτ = Mt.

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


Black-Scholes equation

Using the Feynman-Kac theorem and the differential

dMn(t) =1

n

Snt(

Mn(t))n−1

dt

we obtain

∂V (S,Mn, t)

∂t+ rS

∂V (S,Mn, t)

∂S+

1

n

Sn

Mn−1n

∂V (S,Mn, t)

∂Mn+

+1

2σ2S2∂

2V (S,Mn, t)

∂S2− rV (S,Mn, t) = 0.


Black-Scholes equation – cont.

Since St ≤ max0≤τ≤t Sτ = Mt then for S < M

limn→∞

1

n

Sn

(Mn

)n−1= 0

and we obtain the Black-Scholes equation

∂V (S,M, t)

∂t+rS

∂V (S,M, t)

∂S+1

2σ2S2∂

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 variableM plays the role of a parameter. This parameter entershowever into boundary and terminal conditions.



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 St close to current maximum.Since the probability that the current maximum is still themaximum 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.



Additional boundary conditions depend on the type of theoption.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 fora 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 = SM . When the payoff can

be written as (this is the case of the floating put and call)

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

then we search for a solution of the Black-Scholes equationassuming

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

The equation for W has the form

∂W (x, t)

∂t+ rx

∂W (x, t)

∂x+

1

2σ2x2

∂2W (x, t)

∂x2− rW (x, t) = 0.


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

∂x= W at x = 1.


Asian options

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

(S −K)+, average price call,

(K − S)+, average price put,

(ST − S)+, average strike call,

(S − ST )+, average strike put,

where S denotes the average of asset prices.Computational Finance – p. 21

Asian options – cont.

There are several ways the average of past values of St canbe formated:

arithmetic average continuously monitored

S = 1T

∫ T0 Stdt,

arithmetic average discretely monitored

S = 1n

∑ni=1 Sti ,

geometric average continuously monitored

S = exp(

1T

∫ T0 log(St)dt

),

geometric average discretely monitored S =(∏n

i=1 Sti

)1/n,


Asian options – cont.

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

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

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


Asian options – valuation

The value of an arithmetic average continuously monitoredAsian option can be computed using an extended version ofthe Black-Scholes equation.Let us consider a generalized arithmetic average

At =

∫ t

0

f(Sτ , τ)dτ,

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

The pair (St, At) is a Markov process, and the price at time tof an arithmetic average Asian option is equal to V (t, St, At)where the function V is defined for t ∈ [0, T ], St > 0 andAt > 0.


Black-Scholes equation

Using the Feynman-Kac theorem and the differential

dAt = f(St, t)dt

we obtain

∂V (S,A, t)

∂t+ rS

∂V (S,A, t)

∂S+ f(S, t)

∂V (S,A, t)

∂A+

+1

2σ2S2∂

2V (S,A, t)

∂S2− rV (S,A, t) = 0.


Similarity reduction

We consider an arithmetic average strike call with payoff

(ST −

1

TAT

)+= ST

(1−

1

TST

∫ T

0

Sτdτ

)+

.

This form of the payoff suggest introduction of an auxiliaryvariable

Rt =1

St

∫ t

0

Sτdτ.

Then the payoff is

ST

(1−

1

TRT

)+

,

and we can look for the factorization

V (S,A, t) = SH(R, t). Computational Finance – p. 26


The equation for H has the form

∂H(R, t)

∂t+ (1− rR)

∂H(R, t)

∂R+

1

2σ2R2∂

2H(R, t)

∂R2= 0.

The terminal condition follows from the payoff

H(RT , T ) =(1−

1

TRT

)+.

Since St is integrable, then, from the definition of Rt, forR → ∞ we have S → 0. Remembering that for S → 0 calloption is not exercised, we obtain

H(R, t) = 0 for R → ∞.



To obtain the boundary condition on the boundary R = 0 wereturn to the PDE which H fulfills. For R → 0 this equationgives

∂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

2σ2R2∂

2H

∂R2= 0,

H = 0 for R → ∞,

∂H

∂t+

∂H

∂R= 0 for R → 0,

H(RT , T ) =(1−

1

TRT

)+.


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 becausethe boundary conditions can be easily implemented.The equation reads

∂u(x, t)

∂t+ α(x)

∂u(x, t)

∂x+

1

2σ2x2

∂2u(x, t)

∂x2− ρu(x, t) = 0,

with appropriate terminal and boundary conditions.


Finite difference approximation

Let vi,ν be an approximation of u(xi, tν) and ηi approximation

of2α(xi)

xi+1−xi−1. Then

vi,ν−1 − vi,νδt

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

+ (1− θ)Aivi−1,ν + (1− θ)Bivi,ν + (1− θ)Civi+1,ν ,

(2)

where

Ai =1

2(σ2i2 − ηi), Bi = −(σ2i2 + ρ), Ci =

1

2(σ2i2 + ηi).

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 toavoid instabilities, can be a subject of spurious oscillations.


Spurious oscillations – cont.

To understand the phenomenon of spurious oscillations letus 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 finitedifferences

wj,ν+1 − wjν

∆t+ a

wj+1,ν − wj−1,ν

2∆x= b

wj+1,ν − 2wjν + wj−1,ν

(∆x)2.



Let us express the approximations wjν of the ν-th time level

by a sum of Fourier modes

wjν =∑

k

c(ν)k eikj∆x,

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

c(ν+1)k = Gkc

(ν)k ,

where Gk is the growth factor. For |Gk| ≤ 1 it is guaranteed

that the modes eik∆x are not amplified.



For the model equation we arrive at

Gk = 1− 2λ+ 2λ cos k∆x− iβλ sin k∆x

and

|Gk|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 |Gk|2 reveals

that |Gk| ≤ 1 for

0 ≤ λ ≤1

2, and, in addition, for β > 2 we need λβ2 ≤ 2.



The inequality 0 ≤ λ ≤ 1/2 brings back the stability criterionof 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∂

2u∂x2 in

the model equation.

It is clear that controlling oscillations means that β must besmall.


Spurious oscillations for BS equation

Theorem In order to prevent the formation of spuriousoscillations in the numerical scheme (2), the followingconditions must be satisfied

(xi − xi−1)2

x2i<

σ2

ηi

and

1

(1− θ)δt> σ2

x2i(xi+1 − xi)(xi − xi−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

wj,ν+1 − wjν

∆t+ a

wj,ν − wj−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 approximatethe derivative

∂u

∂x≈

wj,ν − wj−1,ν

∆x,

i.e. the information flows from downstream to upstreamnodes.The stability analysis for this scheme gives

Gk = 1− γ + γe−ik∆x,

where γ = a∆t∆x . Then |Gk| ≤ 1 for γ ≤ 1.


Upwind scheme – cont.

For a < 0, the upwind scheme is

wj,ν+1 − wjν

∆t+ a

wj+1,ν − wj,ν

∆x= 0.

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


numerical pde methods for exotic options - logowanieapalczew/cfp_lecture8.pdf · numerical pde...

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