306 G. Villani
The most relevant models that value exchange options are given in Margrabe [7],
McDonald and Siegel [8], Carr [5, 6] and Armada et al. [1]. Margrabe [7] values an
European exchange option that gives the right to realise such an exchange only at
expiration. McDonald and Siegel [8] value an European exchange option considering
that the assets distribute dividends and Carr [5] values a compound European exchange
option in which the underlying asset is another exchange option. However, when the
assets pay sufficient large dividends, there is a positive probability that an American
exchange option will be exercised strictly prior to expiration. This positive probability
induced additional value for an American exchange option as given in Carr [5, 6] and
Armada et al. [1].
The paper is organised as follows. Section 2 presents the estimation of a Simple
European Exchange option, Section 3 introduces the Monte Carlo valuation of a
Compound European Exchange option and Section 4 gives us the estimation of a
Pseudo American Exchange option. In Section 5, we apply new techniques that allow
reduction of the variance concerning the above option pricing and we also present
some numerical studies. Finally, Section 6 concludes.
2 The price of a Simple European Exchange Option (SEEO)
We begin our discussion by focusing on a SEEO to exchange assetDfor assetVat
timeT. Denoting bys(V,D,T−t)the value of SEEO at timet, the final payoff at
the option’s maturity dateTiss(V,D, 0 )=max( 0 ,VT−DT),whereVTandDT
are the underlying assets’ terminal prices. So, assuming that the dynamics of assets
VandDare given by:
dV
V=(μv−δv)dt+σvdZv, (1)dD
D=(μd−δd)dt+σddZd, (2)Cov(
dV
V,
dD
D)
=ρvdσvσddt, (3)whereμvandμdare the expected rates of return on the two assets,δvandδdare the
corresponding dividend yields,σv^2 andσd^2 are the respective variance rates andZvand
Zdare two Brownian standard motions with correlation coefficientρvd, Margrabe [7]
and McDonald and Siegel [8] show that the value of a SEEO on dividend-paying
assets, when the valuation date ist=0, is given by:
s(V,D,T)=Ve−δvTN(d 1 (P,T))−De−δdTN(d 2 (P,T)), (4)where:
- P=VD; σ=
√
σv^2 − 2 ρvdσvσd+σd^2 ; δ=δv−δd;- d 1 (P,T)=
logP+(σ 2
2 −δ)
T
σ
√
T ; d^2 (P,T)=d^1 (P,T)−σ