PERPETUAL OPTIONS .1 The perpetual American put

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Chapter 10

Numeric methods and single asset American

options

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If we substituteS¼exp (X) we then have:

dV dS¼dV

dX dX

dS¼expðXÞdV dX d²V

dS² ¼dX dS

d dX

dV

dXexpðXÞ

¼expð2XÞd²V dX²dV

dXexpð2XÞ Substituting the above results into Equation 10.1 we obtain:

²expð2XÞexpð2XÞ 2

d²V dX²dV

dX

þ ðrqÞexpðXÞexpðXÞdV

dXrV¼0 ²

2 d²V

dX² þ ðrqÞ ² 2

dV

dXrV¼0 So

d²V

dX² þ 2ðrqÞ ² 1

dV dX2r

²V ¼0 ð10:2Þ

Equation 10.2 is a homogeneous equation with constant coefficients, so we can look for solutions of the formV¼exp (mX). This gives:

m²þ 2ðrqÞ ² 1

m2r

²¼0 ð10:3Þ

which can be solved to yield:

m1¼1 2

2ðrqÞ ² þ1

þ1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðrqÞ

² 1

2

þ8r ² s

ð10:4Þ and

m2¼1 2

2ðrqÞ ² þ1Þ

1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðrqÞ

² 1

2

þ8r ² s

ð10:5Þ The general solution to Equation 10.2 is therefore:

VðXÞ ¼A1expðm1XÞ þA2expðm2XÞ ð10:6Þ

However, since we are solving Equation 10.1 we would like the solution in terms of the asset price S. So re-substituting S¼exp (X), and using the fact that exp (aX)¼exp (X)^a, we obtain:

A1expðm1XÞ ¼A1ðexpðXÞÞ^m1¼A1S^m1 and

A2expðm2XÞ ¼A2ðexpðXÞÞ^m2¼A2S^m2

Numeric methods and single asset American options 117

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The general solution of Equation 10.2 as a function ofSis therefore:

VðSÞ ¼A1S^m1þA2S^m2 ð10:7Þ

If we assume that (2(rD)=²)>1 then m1>0 and m2<0. (Note: When (2(rD)=²)<1,m₁<0 andm₂>0.)

For the perpetual American put asS! 1we haveP(S, E)!0. This means that the coefficientA₁in Equation 10.7 must be zero, andP(S,E)¼A₂S^m2. Suppose we decide that we will exercise the option whenSS , whereS is termed thecritical value of S, then the payoff (which is positive) atS¼S will be

PðS ;EÞ ¼ES ð10:8Þ

This gives

PðS ;EÞ ¼A2ðSÞ^m2¼ES ð10:9Þ Solving forA2 gives:

A2¼ES

ðS Þ^m2 ð10:10Þ

So we have:

PðS;EÞ ¼ ðES Þ S S

m2

ð10:11Þ We are now going to find the value ofS which maximizes the option value at any time before exercise. Differentiating Equation 10.11 and setting the value to zero we have:

@

@S ðESÞ S S

m2

¼ 1 S

S S

m2

S m₂ðES Þ

f g ¼0

and

S m2ðES Þ ¼0; so S ¼ E 11=m2

So substituting into Equation 10.10 results in:

A2¼1 m2

E 11=m2

1m2

When there are no dividends,q¼0, we have from Equation 10.5 that m2¼1

2 2r

² þ1

1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r

²1

2

þ8r ² s

ð10:12Þ but

2r ²1

2

þ8r

²¼ 1þ2r ²

2

118 Pricing Assets

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Therefore m₂¼1

2 2r

²þ12r ²1

and m₂¼2r

² ð10:13Þ

Substituting form2andA2in Equation 10.9 we thus obtain the value for a perpetual American put without dividends as:

PðS;EÞ ¼²S^2r=2 2r

E 1þ ð²=2rÞ

1þð2r=²Þ

ð10:14Þ see Merton (1973), Equation 52, p. 174.

10.2.2 The perpetual American call

Here we derive the value,C(S,E), for a perpetual American call with strike priceE on an asset of current value S. For the perpetual American call asS!0 we have C(S,E)!0. In the previous section we mentioned thatm2<0 which means that the A2S^m2! 1 as S!0. Thus if Equation 10.7 is to yield a finite solution for the perpetual American call we must setA2¼0 and look for solutions of the form:

CðS;EÞ ¼A1S^m1

The payoff for the call option is max (SE, 0), so whenS ¼Swe have:

CðS;EÞ ¼S E¼A₁ðS Þ^m1 ð10:15Þ and

A1¼ðS EÞ

ðS Þ^m1 ð10:16Þ

This gives

CðS;EÞ ¼ ðS EÞ S S

m₁

ð10:17Þ As in Section 10.2.1 we find the value S which maximizes the option value by differentiating Equation 10.17 w.r.t.S and setting the value to zero. This yields:

@

@S ðES Þ S S

m₁

¼ 1 S

S S

m₁

S m1ðS EÞ

f g ¼0

and

S m1ðS EÞ ¼0; so S ¼ E 11=m1

ð10:18Þ Numeric methods and single asset American options 119

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Now usingA1¼(S E)=(S )^m1we obtain A1¼ Ef1=ð11=m1Þ 1g

E^m1nð11=m1Þð11=m1Þ^m11o

¼ 1

E^m11ð11þ1=m₁Þð11=m₁Þ^m11 A1¼ 1

m₁

11=m1

E

m₁1

¼ 1 m₁

E 11=m₁

1m1

ð10:19Þ Therefore the value of the perpetual American call option is:

CðS;EÞ ¼ 1 m1

E 11=m1

1m1

S^m1 ð10:20Þ

When there are no dividends,q¼0, we have from Equation 10.4 that m1¼1

2 2r

² þ1

þ1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r

²1

2

þ8r ² s

ð10:21Þ but

2r ²1

2

þ 8r

²¼ 1þ2r ²

2

ð10:22Þ so substituting into Equation 10.21 we obtain

m1¼1 2 2r

²þ1þ2r ²þ1

¼1 ð10:23Þ

Settingm1¼1 in Equation 10.18 we thus find thatS ¼ 1. Therefore from Equa- tion 10.16:

A1¼ðS EÞ

ðS Þ^m1 ¼ðS EÞ

ðSÞ ¼1 ð10:24Þ

This means that the value of a perpetual American call with zero dividends is:

CðS;EÞ ¼A1S^m1¼1S¼S ð10:25Þ 10.2.3 Perpetual European options

We can easily derive expressions for perpetual European options by using the Black–

Scholes formulae given in Section 9.3.3. It can be seen that as the option maturity,, tends to infinityd1!1andd2!1. This means that for perpetual options we should use N1(d1)1 andN1(d2)0 in the Black–Scholes formulae. Therefore whenq>0, we have c(S,E)0 andp(S,E)0. Also whenq¼0 we havec(S,E)Sandp(S,E)0.

The value of a European call (whenq¼0) is therefore:

cðS;EÞ ¼CðS;EÞ ¼S ð10:26Þ

which means that, when there are no dividends, the perpetual American call and the perpetual European call options have the same value; the current asset priceS.

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10.2.4 Perpetual European down and out call

Here we find the value of a perpetual down and out European call barrier option, see Merton (1973).

Let the exercise price beEand the barrier be atBwhereB<E.

Since the Black–Scholes partial differential equation governs the price of the option we can, as before, look for solutions of the form:

cðS;EÞ_do¼A1S^m1þA2S^m2 ð10:27Þ subject to the boundary conditions: (i) cdo(B,E)¼0 and (ii)c(1,E)_do¼S, see the previous section.

From (i) we have:

cdoðB;EÞ ¼A1B^m1þA2B^m2¼0; so A1¼ A2B^m2m1 Therefore

cdoðS;EÞ ¼ A2B^m2m1S^m1þA2S^m2 From (ii), asS! 1:

cdoðS;EÞ ¼ A2B^m2m1S^m1þA2S^m2¼S

However, sincem2<0, we haveA2S^m2!0, asS! 1, giving cdoðS;EÞ ¼ A2B^m2m1S^m1¼S

So

A2¼ S^1m1

B^m2m1 and cdoðS;EÞ ¼S^1m1S^m1B^m2m1

B^m2m1 S^1m1S^m2 B^m2m1 which results in:

cdoðS;EÞ ¼SS^1þm2m1

B^m2m1 ð10:28Þ

When there are no dividends (q¼0) we have already shown in Sections 10.2.1 and 10.2.2 thatm1 ¼1 andm2¼ 2r=²so the value of a perpetual down and out call is (see Merton (1973)):

cdoðS;EÞ ¼S S^m2

B^m21¼SB S B

2r=²

ð10:29Þ

10.3 APPROXIMATIONS FOR VANILLA AMERICAN OPTIONS