- In this question, we consider the lognormal analytical approximation for the Asian put. A random
variable Z is said to be lognormal if ln Z is normally distributed. Let Z be a lognormal random variable
so that ln Z
P∼ N (µ, ς2
), where µ ∈ R, ς ∈ R+, and P is a probability measure. Then it is known that
the probability density of Z under P is given by
f(z) = 1
ςz√
2π
e
−
(ln z−µ)
2
2ς2
, (4)
for z > 0. We are interested in the quantity
p(T, K) = e
−rT E
P
[(K − Z)+] = e
−rT Z ∞
0
(K − z)+ f(z) dz,
2
which is similar to the price of a put option with strike K and expiry T. [50 marks]
(a) Since ln Z
P∼ N (µ, ς2
), note that we can write ln Z = µ+ςξ where ξ
P∼ N (0, 1). Show by substituting
z = e
µ+ςξ in the expression for c(K, T) that [5 marks]
p(T, K) = e
−rT
√
2π
Z ∞
−∞
K − e
µ+ςξ
+
e
− 1
2
ξ
2
dξ. (5)
(b) Show that the expression for p(T, K) in part (a) can be written [4 marks]
p(T, K) = Ke−rT
√
2π
Z −
(µ−ln K)
ς
−∞
e
− 1
2
ξ
2
dξ −
e
−rT
√
2π
Z −
(µ−ln K)
ς
−∞
e
− 1
2
ξ
2+ςξ+µ
dξ.
(c) Show that the first term in the expression for p(T, K) in part (b) is given by
Ke−rT
√
2π
Z −
(µ−ln K)
ς
−∞
e
− 1
2
ξ
2
dξ = Ke−rT Φ
−
(µ − ln K)
ς
,
where Φ[·] denotes the cumulative normal distribution function. [2 marks]
(d) By making the change of variable ζ = ξ−ς, show that the second term in the expression for p(T, K)
in part (b) is given by [5 marks]
e
−rT
√
2π
Z −
(µ−ln K)
ς
−∞
e
− 1
2
ξ
2+ςξ+µ
dξ = e
µ+ 1
2
ς
2−rT Φ
−
(µ + ς
2 − ln K)
ς
(e) Conclude that if Z is lognormally distributed such that ln Z
Pˆ
∼ N (µ, ς), where Pˆ is the risk-neutral
measure, then the price, p0(T, K) of a European option with expiry T and payoff (K − Z)+ at
time t = 0 in the Black-Scholes model is given by [3 marks]
p0(T, K) = Ke−rT Φ
−
(µ − ln K)
ς
− e
µ+ 1
2
ς
2−rT Φ
−
(µ + ς
2 − ln K)
ς
. (6)
Note: This is a generalization of the Black-Scholes formula for the price of European put options, and (6)
allows us to value a European derivative with expiry T and strike K on any underlying that is lognormally
distributed under the risk-neutral measure Pˆ. If a given underlying is not lognormally distributed, we
can approximate it as a lognormal random variable by matching the moments and use (6) to obtain an
approximate price.
(f) Using (1) show that for the arithmetic average An we have [4 marks]
Eˆ
0[An] = X0
n
Xn
i=1
e