Risky asset

  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


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]

  1. 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

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