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FM320/FM322
LT 2022 Summative Assignment
Instructions to candidates
This assignment contains two questions, each worth 25 marks.
Answer both questions.
1. Consider a European call option on a non-dividend-paying underlying with maturity date T ,
for which the strike price is equal to the forward price of the underlying. We say that the
option is at-the-money-forward, and denote its price (at t = 0) by CAMF .
(a) [12 marks]
Show that, under Black-Scholes assumptions,
CAMF = S [2Φ(σˆ/2)− 1] ,
where S is the price of the underlying at t = 0, Φ(·) is the standard normal cdf, and
σˆ := σ
√
T .
(b) [7 marks]
Use a first-order approximation of the formula for CAMF in part (a) around σˆ = 0 to
obtain the ratio CAMF/S as a linear function of σˆ.
Hint: Recall that the first-order Taylor series approximation of a function f around x0
is given by f(x) ≃ f(x0) + f ′(x0)(x− x0).
(c) [6 marks]
An at-the-money-forward European call expiring in 3 months is worth 4% of the under-
lying. Using the approximation derived above, calculate its implied (annual) volatility.
2. (a) [15 marks]
In this question, the instantaneous riskfree rate r follows an arbitrary stochastic process.
Let P (t, t′) denote the time t price of a zero-coupon bond with maturity at time t′ > t.
All bonds have face value £1. Show that
P (t, T ) = EQt
[
e−
∫
S
t
rudu
P (T, S)
]
,
for all t < T < S, where Q is the risk-neutral measure.
(b) [10 marks]
Consider a coupon bond that matures in T years and has face value £1. The coupons
are paid annually, and the coupon at time t, t = 1, . . . , T , is given by
ct =
1
P (t− 1, t) − 1,
where P (t− 1, t) is the price at t− 1 of a one-year zero with face value £1. What is the
price of the coupon bond at t = 0? Prove all your assertions.