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MATH39032
1.
A forward contract with delivery price F such that F < SerT (where S is the current value of an asset
S that does not pay out a dividend, r is the risk-free rate and T is the time to expiry) is trading on the
market with zero cost to enter a short or long position. Show that there exists an arbitrage opportunity.
2. Consider a financial contract depending on two assets, S1 and S2, that has the following payoff at
maturity
V (S1, S2, t = T ) = max(S1 −X1, S2 −X2).
Here X2 > X1 are constants, S1 ∈ [0,∞), S2 ∈ [0,∞), and t ∈ [0, T ].
(a) Provide a sketch of the payoff V against S2 when S1 = X2.
(b) Assuming that asset prices follow a geometric Brownian motion, interest rates are constant, and that
neither asset pays a dividend, state appropriate boundary conditions for the value of the financial
contract V (S1, S1, t) in the following two cases:
(i) S1 → 0 and S2 → 0;
(ii) S1 → 0 and S2 →∞.
3. Suppose that the interest rate process r is governed by the general stochastic differential equation
dr = κ(θ − r)dt+ σrdX,
where X is a Brownian motion. V1(r, t) is the value of a zero-coupon bond with maturity T1, and V2(r, t)
is the value of a zero-coupon bond with maturity T2.
Consider a portfolio comprising long one V1 bond and short ∆ of V2 bonds. Using Itoˆ’s Lemma (which
may be used without proof), show that the choice
∆ =
∂V1
∂r
/
∂V2
∂r
eliminates the random component of the portfolio. Hence show that the value of both V1 and V2 must
satisfy the equation
∂V
∂t
+
1
2
σ2r2
∂2V
∂r2
+ κ(θ − r − λσr)∂V
∂r
− rV = 0,
where λ(r, t) is the market price of risk.
Page 2 of 4 P.T.O.
MATH39032
4.
(a) Let F (S, t) = KeλtSα, where K, λ and α are constants. Determine λ (in terms of α, r and σ) so
that F satisfies the Black-Scholes PDE,
∂F
∂t
+
1
2
σ2S2
∂2F
∂S2
+ rS
∂F
∂S
− rF = 0,
where the volatility σ and interest rate r are constants.
(b) Now consider a particular financial instrument with price f(S, t) that satisfies the above PDE and
provides a single payoff at time t = T > 0 amounting to Sα; determine f(S, t < T ) using your
answer to part (a).
(c) Describe the monotonic behaviour of f(S, t) with respect to time for α < 1, α = 1 and α > 1.
5.
Consider the Black-Scholes equation for a put option, P , on a stock S which pays no dividends, namely
∂P
∂t
+
1
2
σ2S2
∂2P
∂S2
+ rS
∂P
∂S
− rP = 0
in the usual notation, where r and σ are both constants.
(a) By neglecting the time derivative in the above equation, seek solutions of the form
P (S, t) = P (S) = A1S
α1 + A2S
α2
where A1 and A2 are constants, and α1 > α2. Determine the values of α1 and α2.
(b) Consider a perpetual American put option P (S), i.e. an option with no expiry date but where it
is possible at any point in time to exercise, which has a (non-standard) exercise value of Ke−S,
where K is a constant. This can be valued using the time independent solutions in (i) above. The
exercise boundary is denoted by Sf . State and justify the two conditions at S = Sf and the other
appropriate boundary condition.
(c) Determine Sf and the values of A1 and A2.
(d) Describe the behaviour of P (S) and Sf as r → 0.
Page 3 of 4 P.T.O.
MATH39032
6. You may assume that the Black-Scholes equation (in the usual notation) for a put option on a non-
dividend paying asset is
∂P
∂t
+
1
2
σ2S2
∂2P
∂S2
+ rS
∂P
∂S
− rP = 0
where the interest rate r and the volatility σ are both constant.
(a) A binary put option Pb(S, t) has the payoff
Pb(S, T ) =
{
0 if S > X
K if S < X
.
Show that its value is given by
Pb(S, t) = K
∂P
∂X
where P (S, t;X) is the value of a vanilla put option with strike X.
(b) Given that the value of a European put option is
P (S, t;X) = Xe−r(T−t)N(−d2)− SN(−d1),
where N(x) is the cumulative distribution function of the standard normal distribution, namely
N(x) =
1√
2pi
∫ x
−∞
e−
y2
2 dy
and
d1 =
ln(S/X) + (r + σ2/2)(T − t)
σ
√
T − t , d2 = d1 − σ
√
T − t,
show that
Pb(S, t;X) = Ke
−r(T−t)N(−d2).
(c) Consider a stock whose price (6 months from the expiration of a binary put option) today is £10.50.
The option pays £0.5 if the stock value at expiry is less than £11, and nothing if the stock value
at expiry is greater than 11. The risk-free interest rate is 4% per annum (fixed) and the volatility
(constant) is 15% per (annum)
1
2 . Using the formula above for Pb(S, t;X), determine its value
today. The value of N(x) may be determined by a numerical (online) calculator.
END OF EXAMINATION PAPER