Pricing of American Options

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University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe

Numerical Methods for Option Pricing in Finance

Chapter 6: Pricing of American Options

6.1 The Black-Scholes Inequality

6.1.1 American put option

We know from Chapter 1 that the price PA of an American put option satisfies

() PA(S, t) (K S)+

, 0 t T .

Since PA(0, t) = K = (K S)+|S=0 and PA(K, t) = 0, due to the continuity and monotonicity

of PA there exists 0 SF(t) < K such that the payoff is reached, i.e.,

PA(SF(t), t) = K SF(t) .

Consequently, we have

PA(S, t)> (K S)

+

, S > SF(t)= K S , 0 S SF(t)

.


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Stopping and Continuation Region

S

T

t

S

F

Continuation region

region

Stopping

(t)FS

(T)

In the (S, t)-plane, the set

{(SF(t), t) | 0 t T}

defines a curve which partitions the semi-infinite strip

[0,) [0, T] into two parts:

The stopping region

S := {(S, t) | S SF(t)}

and the continuation region

C := {(S, t) | S > SF(t)}.

The economical meaning of the continuation and the stopping region is as follows:

In the stopping region we have PA = K S and the put option should be exercised, since = K S + S = K can be invested at the risk-free interest rate r > 0.In the continuation region we have PA > (K S)

+ and an early exercise of the option does not

make sense.


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Free boundary and stopping time

FS

T

t

S

(T)

Free boundary

(t)FS

Continuation region

region

Stopping

Since the curve (SF(t), t) | 0 t T} is an unknown

of the problem, it is referred to as a free boundary.

The time instanttS := min {0 t T | S = SF(t)}

is called the stopping time.

At tS, the put option should be exercised, since holding it longer would reduce the profit of the

risk-free investment.


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The determination of the free boundary requires an additional boundary condition besides

PA(SF(t), t) = K SF(t). For that purpose we consider the slopePAS (SF(t), t) where PA(S, t) touches

the straight line K S which obviously has the slope 1. We can rule out the case PAS

(SF(t), t) 1 can be excluded as well. In other words,

we obtain PAS

(SF(t), t) = 1 .

Hence, in case of an American put option with proportional dividend D0 > 0, the price PA(S, t) of

the put option satisfies the free boundary problem

S SF(t) : PA(S, t) = K S

S > SF(t) :PAt

+1

2

2 S2

2PAS2

+ (rD0) SPAS

rPA = 0

with final condition PA(S,T) = (K S)+ and boundary conditions

PA(0, t) = K , limS

PA(S, t) = 0 , PA(SF(t), t) = K SF(t) , (PA/S)(SF(t), t) = 1 .


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6.1.2 American call option

As shown in Chapter 1, the price CA of an American call option is the same as the price CE ofa European call option, if no dividends are paid. However, in case of a proportional dividend

D0 > 0, for t < T and sufficiently large S we have

CA(S, t) (SK)+ = SK > S exp(D0(T t)) (d1) K exp(r(T t)) (d2) = CE(S, t) ,

where d1/2 = ln(SK) + (rD0

2

2 )(T t)

T t .

Since CA(K, t) = 0 and limS CA(S, t) = SK > 0, due to the continuity and monotonicity ofCAthere exists SF(t) > K such that the payoff is reached, i.e.,

CA(SF(t), t) = SF(t) K ,whence

CA(S, t)

= SK , S SF(t)> (SK)+ , 0 S < SF(t) .


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University of Houston/Department of Mathematics

Dr. Ronald H.W. HoppeNumerical Methods for Option Pricing in Finance

Stopping and Continuation Region

F (T)

T

t

S

S

Continuation region

(t)FS

region

Stopping

As in case of the put, the free boundary

{(SF(t), t) | 0 t T}defines a continuation region

C := {(S, t) | S SF(t)} ,

where it is advantageous to keep the option, and a

stopping region

S := {(S, t) | S > SF(t)} ,where an early exercise makes sense.

The price CA of the call option satisfies the free boundary problem

S SF(t) : CA(S, t) = SK

S < SF(t) :CAt

+1

2

2 S22CAS2

+ (rD0) SCAS

rCA = 0

with final condition CA(S,T) = (SK)+ and boundary conditions

CA(0, t) = 0 , limS

CA(S, t) = O(S) , CA(SF(t), t) = SF(t)K , (CA/S)(SF(t), t) = 1 .


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6.1.3 The Black-Scholes inequality for American optionsThe free boundary problem for the price V = PA of an American put can be formulated as a

Linear Complementarity Problem (LCP) which does not explicitly contain the free boundary.

As in the derivation of the Black-Scholes equation, we assume a risk-free, self-financing portfolio

()1 Y = c1 B + c2 S V .

As in Chapter 3, we deduce c2 = VS and

()2 dY = (c1 r B V

t

1

2

2 S2

2V

S2) dt .

The owner of the portfolio has sold the put. If the buyer does not exercise it optimally, the

seller can realize a higher profit than for a risk-free investment, i.e., dY rYdt. Inserting ()1

and ()2 into this inequality results in the so-called Black-Scholes inequality

()V

t+

1

2

2 S2

2V

S2+ (rD0) S

V

S r V 0 .


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When does equality and when does strict inequality hold true in ()?

Case I (S > SF(t)) : We already know from 6.1.1 that in this case the equality sign applies.

Case I (S SF(t)) : We claim that in this case strict inequality holds true.

The proof is easy for D0 = 0: Inserting PA = K S into the Black-Scholes equation yields

PA

t

+1

2

2S2

2PA

S2

+ (rD0)SPA

S

rPA = (rD0)S r(K S) = rK < 0 .

The proof for D0 > 0 is much more involved. It can be shown that SF(t) min(K, rK/D0)

which implies for PA = K S

PAt

+1

2

2S2

2PAS2

+ (rD0)SPAS

rPA = D0S rK < D0SF(t) rK min(D0K, rK) rK 0 .

A similar reasoning applies in case of an American call option:

Equality holds true for 0 S < SF(t), whereas strict inequality applies for S SF(t). In the

latter case, one has to use SF(t) max(K, rK/D0).


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Theorem 6.1 (LCP for American options)

The price of an American option satisfies the LCP

(V

t+

1

2

2 S2

2V

S2+ (rD0) S

V

S r V) 0 ,

V (S) 0 ,

(

V

t +

1

2 2

S2

2V

S2 + (rD0) S

V

S r V) (V(S)) = 0,

where(S) =

(K S)+ , put option(SK)+ , call option

.

The final condition is V(S, T) = (S) and the boundary conditions are

V(0, t) = K , limS

V(S, t) = 0 for a put option ,

V(0, t) = 0 , limS

V(S, t) = O(S) for a call option .


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6.2 Finite Difference Approximation of the Black-Scholes Inequality

6.2.1 Transformation of the Linear Complementarity ProblemAs for the Black-Scholes equation, we perform the transformations

x = ln(S

K) , =

1

2

2 (T t) ,

u(x, ) =1

Kexp(

1

2(k0 1)x +

1

4(k0 1)

2+ k) V(S, t) ,

where k := 2r2

and k0 =2(rD0)2

, and obtain the transformed linear complementarity problem

u

2u

x2 0 , u f 0 , (

u

2u

x2)(u f) = 0 ,

where f(x, ) := exp(12(k0 1)x +14(k0 1)

2+ k)(Kexp(x))K . The initial condition is u(x, 0) = f(x, 0),

and the boundary conditions are given by

limx

(u(x, ) f(x, )) = 0 , limx+

u(x, ) = 0 for a put option ,

limx

u(x, ) = 0 , limx+

(u(x, ) f(x, )) = 0 for a call option .


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6.2.2 Finite Difference Approximation

Discretizing the transformed linear complementarity by finite differences in time and space inexactly the same way as in case of the transformed Black-Scholes equation (cf. Chapter 5),

we obtain the finite-dimensional linear complementarity problem: Find uj+1h lRN1 such that

Ah()uj+1h b

jh 0 , u

j+1h f

j+1h 0 , (Ah()u

j+1h b

jh)(u

j+1h f

j+1h ) = 0 ,

where bjh := Bh()ujh + d

jh , f

j+1h := (f(x1, j+1), ..., f(xN1, j+1))

T, and

djh :=

(1 ) kh2 u(a, j) + k

h2 u(a, j+1)

0

0(1 ) kh2 u(+a, j) +

kh2 u(+a, j+1)

.

The matrices Ah(), Bh() are given as in Chapter 5 (with = 1).


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6.2.3 The discrete LCP and constrained Quadratic Programming (QP)

Dropping the indices h,j as well as the dependence on , the discrete LCP reads as follows:

Find u lRN1 such that

() Au b 0 , u f 0 , (Au b)(u f) = 0 .

Introducing the quadratic objective functional

J(v) := 12

vTAv bTv , v lRN1 ,

and the closed, convex setK := { v lRN1 | vi fi , 1 i N 1 } ,

the discrete LCP () represents the first order necessary optimality conditions (Karush-Kuhn-

Tucker conditions) of the constrained quadratic programming problem

() J(u) = min J(v) .v K

Since A is symmetric, positive definite, the LCP () is also sufficient for a minimizer of ().


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6.3 Numerical Solution of the Constrained QP

6.3.1 Method of Projected Successive Over-Relaxation (PSOR)Using the min-function, the LCP () can be equivalently written as the problem

() min (Au b,u f) = 0 .

As we know from Numerische Mathematik I, the SOR method is based on a triangular decom-

position of the spd matrix A lRN1N1 according to A = D LU, where D := diag(A) and

L resp. U represent the lower resp. upper triangular part ofA.

Then, () is equivalent to

min (uD1(Lu + Uu + b),u f) = 0 u = max (D1(Lu + Uu + b), f) .

This equivalence motivates to solve the LCP by the following projected SOR method:

Given u(0) lRN1 and 1 < < 2, compute u(+1), 0, according to

z()i = a

1ii (bi

i1

j=1aij u

(+1)j

N1

j=i+1aij u

()j ) , 1 i N 1 ,

u(+1)i = max (u

()i + (z

()i u

()i ), fi) , 1 i N 1 .


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6.3.2 Active Set Strategy

For w lR

N1

we denote byIAC(w) := {1 i N 1 | wi = fi} , IIN(w) := {1, ..., N 1} \ IAC(w)

the active set and the inactive set, respectively.

Then, the active set strategy for the solution of the constrained QP () proceeds as follows:

Step 1 (Initialization): Choose u(0) lRN1 and determine IAC

(u(0)) and IIN

(u(0)).

Step 2 (Iteration Loop): For 0 compute:

Step 2.1: Set u(+1)i = fi, i IAC(u

()), and compute u(+1)i , i IIN(u

()), as the solution of the

reduced linear system

jIIN(u()

)

aij u(+1)

j = fi

jIAC(u()

)

aij fj , i IIN(u()) .

Step 2.2: Determine IV(u(+1)) := {i IAC(u

(+1)) |N1

j=1 aij u(+1)

j < fi} and define

IAC(u(+1)) := IAC(u

(+1)) \ IV(u(+1)) , IIN(u

(+1)) := {1, ..., N 1} \ IAC(u(+1)) .

Step 2.3: IfIAC(u(+1)) = IAC(u

()), then STOP, else set := + 1 and go to Step 2.1.

Pricing of American Options

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