Mathematical and Statistical Methods for Actuarial Sciences and Finance

308 G. Villani

are two Brownian motions under the risk-neutral probability measure

∼

QandZ′is

a Brownian motion under

∼
Qindependent ofZˆd. By the Brownian motions defined
in Equations (11) and (12), we can rewrite Equation (7) for the assetPunder the

risk-neutral probability

∼

Q. So it results that:

dP

P

=−δdt+σvdZˆv−σddZˆd. (13)

Using Equation (12), it results that:

σvdZˆv−σddZˆd=(σvρvd−σd)dZˆd+σv

(√

1 −ρ^2 vd

)

dZ′, (14)

whereZˆv andZ′are independent under

∼
Q. Therefore, as(σvdZˆv−σddZˆd)∼
N( 0 ,σ

√

dt), we can rewrite Equation (13):

dP

P

=−δdt+σdZp, (15)

whereσ=

√

σv^2 +σd^2 − 2 σvσdρvdandZpis a Brownian motion under

∼

Q.Usingthe

log-transformation, we obtain the equation for the risk-neutral price simulationP:

Pt=P 0 exp

{(

−δ−

σ^2

2

)

t+σZp(t)

}

. (16)

So, using the assetDTas numeraire given by Equation (10), we price a SEEO as the
expectation value of discounted cash-flows under the risk-neutral probability measure:

s(V,D,T)=e−rTEQ[max( 0 ,VT−DT)]

=D 0 e−δdTE∼

Q

[gs(PT)], (17)

wheregs(PT)=max(PT− 1 , 0 ). Finally, it is possible to implement the Monte Carlo
simulation to approximate:

E∼

Q

[gs(PT)]≈

1

n

∑n

i= 1

gsi(PˆTi), (18)

wherenis the number of simulated paths effected,PˆTifori= 1 , 2 ...nare the simulated

values andgsi(PˆTi)=max( 0 ,PˆTi− 1 )are thensimulated payoffs of SEEO using a
single stochastic factor.

3 The price of a Compound European Exchange Option (CEEO)

The CEEO is a derivative in which the underlying asset is another exchange option.
Carr [5] develops a model to value the CEEO assuming that the underlying asset is