The risk-neutral measure

1. Suppose Xt
   is the price of a risky asset under the Black-Scholes model so that it satisfies
   the equation
   dXt = rXt dt + σXt dwˆt
   ,
   under the risk-neutral measure Pˆ, where X0 > 0, σ > 0, and ˆwt
   is a standard Pˆ-Wiener
   processes. [35 marks]
   (a) By applying Itˆo’s lemma to ln
   X
   −α
   t

and solving the resulting equation, show that
X−α
u = X
−α
t
e
α(
1
2
σ
2
(α+1)−r)X
−α
t
dt−σα X−α
t
dwˆt
for any 0 ≤ t ≤ u.. [5 marks]
(b) Let α1 > 0, α2 > 0, and suppose t1 and t2 are given such that t < t1 < t2. Show that
we may write
X
−α1
t1 X
−α2
t2 = X
−α1−α2
t
e
−µ−ςξ
,
where ξ is a standard normal random variable under the risk-neutral measure, and
µ and ς are functions of r, σ, t, t1 and t2. Determine µ and ς explicitly and provide
the details of your calculations. [8 marks]
(c) Show that the price, pt
, at time t of a European derivative that provides the payoff
hT =

K − X
−α1
t1 X
−α2
t2

+
at time T, where t < t1 < t2 < T, is given by [1 mark]
pt = e
−r(T −t)Eˆ
t
h
K − X
−α1
t1 X
−α2
t2

+
i
.
(d) Show that the price of the derivative in (c) can be written as
pt = e
−r(T −t)KEˆ
t

1
n
K>X−α1
t1
X
−α2
t2
o

− e
−r(T −t)Eˆ
t

X
−α1
t1 X
−α2
t2
1
n
K>X−α1
t1
X
−α2
t2
o

,
where 1
n
K>X−α1
t1
X
−α2
t2
o denotes the indicator function. [3 marks]
(e) Using the result from part (b), show that
e
−r(T −t)KEˆ
t

1
n
K>X−α1
t1
X
−α2
t2
o

= e
−r(T −t)KΦ (d+),
for some d+ and determine d+ explicitly in terms of the parameters µ and ς introduced
in part (b). You must provide the details of your calculations. [6 marks]
(f) Using the result from part (b) once again, show that we can write
e
−r(T −t)Eˆ
t

X
−α1
t1 X
−α2
t2
1
n
K>X−α1
t1
X
−α2
t2
o

= X
−α1−α2
t
e
−µ−r(T −t)
Z ∞
−d+
e
−ςξφ(ξ) dξ,
where d+ is as introduced in part (e) and φ(ξ) denotes the standard normal density
function. Using the definition,
φ(ξ) = 1
√
2π
e
− 1
2
ξ
2
,
of the standard normal density function show that we can write
e
−r(T −t)Eˆ
t

X
−α1
t1 X
−α2
t2
1
n
K>X−α1
t1
X
−α2
t2
o

= X
−(α1+α2)
t
e
−µ−r(T −t)+ 1
2
ς
2
Φ (d−),
for some d− and determine d− explicitly. Hence, or otherwise, determine the price,
pt
, of the derivative introduced in part (c). [12 marks]

2. Let T1 < T2 < · · · < Tn be a sequence of times and let P(t, T) denote the price of a
   T-maturity zero coupon bond at time t. Let 1 < k < n be fixed and let L(t, Tk, Tk+1) be
   defined by
   L(t, Tk, Tk+1) = 1
   τ

P(t, Tk)
P(t, Tk+1)
− 1

.
Assume that you are given a 1-dimensional interest rate model where L(t, Tk, Tk+1) satisfies the equation
dL(t, Tk, Tk+1) = σk(t)L(t, Tk, Tk+1) dwk+1
t
under the Tk+1-forward measure P
k+1, where σk(t) is a deterministic function and w
k+1
t
is a standard P
k+1-Wiener process. Show that the dynamics of L(t, Tk, Tk+1) under the
Tk-forward measure is given by
dL(t, Tk, Tk+1) = τσ2
k
(t)L(t, Tk, Tk+1)
2
1 + τL(t, Tk, Tk+1)
dt + σk(t)L(t, Tk, Tk+1) dwk
t
,
where w
k
t
is a standard P
k
-Wiener process. [10 marks]
2

Sample Solution

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