The Mathematics of Arbitrage

3.3 The Binomial and the Trinomial Model 47

X̂ 1 (x)=−V′

(

ŷ(x)

dQ

dP

)

=−V′(U′(x))c

α^1 − 1

U

(

dQ

dP

)α−^11

,

=xc−V^1

(

dQ

dP

)α^1 − 1

,

wherewehaveused−V′=(U′)−^1 andV′(y)=−y
α^1 − 1

. Hence

X̂ 1 (x)=

⎧

⎪⎨

⎪⎩

xc−V^1

(

− 2 d ̃

̃u−d ̃

)α^1 − 1

=xc−V^1 (2q)

α^1 − 1

, forω=g,

xc−V^1

(

2 ̃u

̃u−d ̃

)α (^1) − 1
=xc−V^1 (2(1−q))
1
α− (^1) ,forω=b.
Let us explicitly verify thatX̂ 1 (x) is indeed of the form
X̂ 1 (x)=x+̂h∆S 11 , (3.36)
for somêh∈R, or, equivalently, thatEQ[X 1 (x)] =x. Indeed:
EQ

[

X̂ 1 (x)

]

=x

(

1

2

(

(2q)

αα− 1

+(2(1−q))

αα− 1 )

)− 1

·

[

q(2q)

α^1 − 1

+(1−q)(2(1−q))

α^1 − 1 ]

=x

To calculatêhexplicitly we may apply (3.36), e.g., forω=gto obtain

x+̂hu ̃=xc−V^1 (2q)

α−^11

which yields
̂h=x

[

c−V^1 (2q)

1

α− (^1) − 1

]

̃u−^1. (3.37)

In the special case ofα =^12 (so thatβ = αα− 1 =−1andβ−1=

1

α− 1 = −2) the constants become somewhat nicer: (2q)

α^1 − 1

=^14

(

u ̃−d ̃

d ̃

) 2

,

cV=^12

(

̃u−d ̃

− 2 d ̃+

u ̃−d ̃

2 u ̃

)

=−( ̃u−

d ̃)^2

4 ̃ud ̃ so that

̂h=x

[

− 4 ̃ud ̃

(u ̃−d ̃)^2

·

1

4

(u ̃−d ̃)^2

d ̃^2

− 1

]

̃u−^1

=x

̃u+d ̃

| ̃ud ̃|

.