Mathematical and Statistical Methods for Actuarial Sciences and Finance

Exact and approximated option pricing 137

Using (8) and (9), the valueS(ti)givenS(ti− 1 )can be written as

S(ti)=S(ti− 1 )βiexp

{(

r−

σ^2 i

2

)

ti+σi

√

tiR

}

, (10)

whereR∼N( 0 , 1 ), henceS(ti)is a lognormal random variable.

5 Barrier options and their Greeks

To price barrier options, we choose to apply the conditional Monte Carlo (CMC)
technique, first used in finance in [16]. This method is applicable to path-dependent
derivatives whose prices have a closed-form solution in the B&S setting. It exploits
the following variance-reducing property of conditional expectation: for any random
variablesXandY,var[E[X|Y]]≤var[X], with strict inequality excepted in trivial
cases.
Now, we illustrate the CMC method for discrete barrier options. LetC(S( 0 ),
K,r,T,σ)denote the B&S price of a European call option with constant volatility
σ, maturityT, strikeK, written on an asset with initial priceS( 0 ). The discounted
payoff for a discrete knock-out option with barrierH>S( 0 )is given by

f(X)=e−rT(S(T)−K)+ (^1) {max 1 ≤i≤MS(ti)<H}, (11)
whereS(ti)is the asset price at timetifor a time partition 0=t 0 <t 1 <...<tM=
T. Using the law of iterated expectations, we obtain the following unconditional price
of the option
E

[

e−rT(S(T)−K)+ (^1) {max 1 ≤i≤MS(ti)<H}

]

=E

[

E

[

e−rT(S(T)−K)+ (^1) {max 1 ≤i≤MS(ti)<H}

∣

∣

∣

∫T

0

v(q)dq,

∫T

0

v(q)dW 2 (q),JS

]]

=E

[

C(S( 0 )βM,K,r,T,σM) (^1) {max 1 ≤i≤MS(ti)<H}

]

, (12)

whereσMandβMare defined in (8) and (9), respectively.
This approach can also be used to generate an unbiased estimator for delta, gamma
and rho, exploiting the likelihood ratio (LR) method.
Suppose thatp∈Rnis a vector of parameters with probability densitygp(X),
whereXis a random vector that determines the discounted payoff functionf(X)
defined in (11). The option price is given by

α(p)=E[f(X)], (13)

and we are interested in finding the derivativeα′(p). From (13), one gets

α′(p)=

d

dp

E[f(X)]=

∫

Rn

f(x)

d

dp

gp(x)dx=E

[

f(X)

g′p(x)

gp(x)

]

. (14)