Mathematical and Statistical Methods for Actuarial Sciences and Finance

A Monte Carlo approach to value exchange options using a single stochastic factor 309

a SEEOs(V,D,τ)whose time to maturity isτ=T−t 1 witht 1 <T, the exercise
price is a ratioqof assetDat timet 1 and the expiration date ist 1. So, considering
that the valuation date ist=0, the final payoff of CEEO at maturity datet 1 is:

c(s(V,D,τ),qD, 0 )=max[0,s(V,D,τ)−qD].

Assuming that the evolutions of assetsVandDare given by Equations (1) and (2),
under certain assumptions, Carr [5] shows that the CEEO price at evaluation date
t=0is:

c(s(V,D,τ),qD,t 1 )=Ve−δvTN 2

(

d 1

(

P

P 1 ∗

,t 1

)

,d 1 (P,T);ρ

)

−De−δdTN 2

(

d 2

(

P

P 1 ∗

,t 1

)

,d 2 (P,T);ρ

)

−qDe−δdt^1 N

(

d 2

(

P

P 1 ∗

,t 1

))

, (19)

whereN 2 (x 1 ,x 2 ;ρ)is the standard bivariate normal distribution evaluated atx 1 and

x 2 with correlationρ=

√

t 1

TandP

∗
1 is the critical price ratio that makes the underlying
asset and the exercise price equal and solves the following equation:

P 1 ∗e−δvτN(d 1 (P 1 ∗,τ))−e−δdτN(d 2 (P 1 ∗,τ))=q. (20)

It is obvious that the CEEO will be exercised at timet 1 ifPt 1 ≥P 1 ∗.WepricetheCEEO
as the expectation value of discounted cash-flows under the risk-neutral probability
Qand, after some manipulations and usingDt 1 as numeraire, we obtain:

c(s,qD,t 1 )=e−rt^1 EQ[max(s(Vt 1 ,Dt 1 ,τ)−qDt 1 , 0 )]

=D 0 e−δdt^1 E∼

Q

[gc(Pt 1 )], (21)

where

gc(Pt 1 )=max[Pt 1 e−δvτN(d 1 (Pt 1 ,τ))−e−δdτN(d 2 (Pt 1 ,τ)−q,0]. (22)

Using a Monte Carlo simulation, it is possible to approximate the value of CEEO as:

c(s,qD,t 1 )≈D 0 e−δdt^1

(∑n

i= 1 g

i

c(Pˆ

i

t 1 )

n

)

, (23)

wherenis the number of simulated paths andgci(Pˆti 1 )are thensimulated payoffs of
CEEO using a single stochastic factor.

4 The price of a Pseudo American Exchange Option (PAEO)

Lett=0 be the evaluation date andTbe the maturity date of the exchange option. Let
S 2 (V,D,T)be the value of a PAEO that can be exercised at timeT 2 orT. Following