Mathematical and Statistical Methods for Actuarial Sciences and Finance

310 G. Villani

Carr [5,6], the payoff of PAEO can be replicated by a portfolio containing two SEEOs
and one CEEO. Hence, the value of PAEO is:

S 2 (V,D,T)=Ve−δvTN 2

(

−d 1

(

P

P 2 ∗

,

T

2

)

,d 1 (P,T);−ρ

)

+Ve−δv

T 2

N

(

d 1

(

P

P 2 ∗

,

T

2

))

−De−δdTN 2

(

−d 2

(

P

P 2 ∗

,

T

2

)

,d 2 (P,T);−ρ

)

−De−δd

T

(^2) N

(

d 2

(

P

P 2 ∗

,

T

2

))

, (24)

whereρ=

√

T/ 2

T =

√

0 .5andP 2 ∗is the unique value that makes the PAEO exercise

indifferent or note at timeT 2 and solves the following equation:

P 2 ∗e−δv

T 2

N

(

d 1

(

P 2 ∗,

T

2

))

−e−δd

T 2

N

(

d 2

(

P 2 ∗,

T

2

))

=P 2 ∗− 1.

The PAEO will be exercised at mid-life timeT 2 if the cash flows(VT/ 2 −DT/ 2 )exceed
the opportunity cost of exercise, i.e., the value of the options(V,D,T/ 2 ):

VT/ 2 −DT/ 2 ≥s(V,D,T/ 2 ). (25)

It is clear that if the PAEO is not exercised at timeT 2 , then it’s just the value of a
SEEO with maturityT 2 , as given by Equation (4). However, the exercise condition
can be re-expressed in terms of just one random variable by taking the delivery asset
as numeraire. Dividing by the delivery asset priceDT/ 2 , it results in:

PT/ 2 − 1 ≥PT/ 2 e−δv

T

(^2) N(d 1 (PT/ 2 ,T/ 2 ))−e−δd
T
(^2) N(d 2 (PT/ 2 ,T/ 2 )). (26)
So, if the condition (26) takes place, namely, if the value ofPis higher thanP 2 ∗at
momentT 2 , the PAEO will be exercised at timeT 2 and the payoff will be(VT/ 2 −DT/ 2 );
otherwise the PAEO will be exercised at timeTand the payoff will be max[VT−
DT,0].So, using the Monte Carlo approach, we can value the PAEO as the expectation
value of discounted cash flows under the risk-neutral probability measure:
S 2 (V,D,T)=e−r
T
(^2) EQ[(VT/ 2 −DT/ 2 ) (^1) (PT/ 2 ≥P∗
2 )]
+e−rTEQ[max( 0 ,VT−DT) (^1) (PT/ 2 <P 2 ∗)]. (27)
Using assetsDT/ 2 andDTas numeraires, after some manipulations, we can write
that:
S 2 (V,D,T)=D 0

(

e−δd

T

(^2) E∼
Q
[gs(PT/ 2 )]+e−δdTE∼
Q
[gs(PT)]

)

, (28)