MATH20912
Two hours
A formula sheet is attached.
THE UNIVERSITY OF MANCHESTER
INTRODUCTION TO FINANCIAL MATHEMATICS
xxxx
xxxx
Answer THREE of the FOUR questions.
If more than THREE questions are attempted, then credit will be given for the best THREE
answers.
No prepared notes of any kind or mathematical tables are to be brought into the examination room.
Electronic calculators may be used, provided that they cannot store text.
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P.T.O.
MATH20912
1.
(a) The current share price is S(0) = £261 and you predict that the share price S(T ) in one year
will be £251 with probability 0.25, £283 with probability 0.4, and £316 with probability 0.35.
(i) Find the expected profit (or loss) at maturity for the holder of a portfolio who is long two
calls with the exercise price E = £280, and long two puts with exercise price E = £300,
with all options to be exercised in one year; the total cost of the portfolio is £60 and the
risk-free interest rate is 4% per annum.
(ii) You must decide whether to invest in the portfolio or an equivalent amount of money in
shares, which do you choose and why?
[6 marks]
(b)
(i) Find the value of a zero coupon bond, B(t, T ), with face value F and maturity time T
when the risk-free interest rate is
1
r(t) =
a+t
and a is a constant to be determined from the market.
(ii) If a bond with face value F = £100 expiring at T = 5 is trading at time t = 2 for £88,
find the value of a.
(iii) Plot the term structure of interest rate Y (0, T ) for T ∈ [0, 25] and the value of a given by
part (ii).
[7 marks]
(c) Payoff Diagrams:
(i) Sketch the graph of the payoff diagram for the following portfolio: long one bond with
face value F = 1000, long two European put options with exercise price £400, and short
three European call options all with exercise price £500. All options are written on the
same underlying and all financial contracts have the same expiry (or maturity) date.
(ii) A trader currently holds a portfolio Π compromising of short one share and long two call
options with exercise price E = 100. You are asked to buy (or sell) one more call option
with the same expiry date so that the final payoff from the portfolio satisfies
ΠT ≤ 200.
State the final portfolio that you should choose (stating whether you are long or short
and the exercise price of the additional call option) and then sketch the resulting payoff
diagram.
[7 marks]
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P.T.O.
MATH20912
2.
(a) Let C(S, t; E, T ) denote the value of a European call option expiring at time T with exercise
price E.
(i) Show using arbitrage arguments that
C(S, t; E1 , T ) ≥ C(S, t; E2 , T )
if
E1 ≤ E2 .
(ii) Show using arbitrage arguments that
C(S, t; E1 , T ) ≥ 2C(S, t; E2 , T ) − C(S, t; E3 , T )
if the exercise price
E2 = E1 + x
and
E3 = E1 + 2x
for any x > 0.
(iii) Consider the case where the current share price is S0 = £297, a call option with E1 = £280
is priced at £83, a call with E2 = £300 is priced at £74, and a call with E3 = £320 is
priced at £56, all of the options have one year left until expiry. The current risk free
interest rate is a constant 4% per annum. Construct a portfolio (using part (ii)) to exploit
the arbitrage opportunity and calculate the profit made.
[12 marks]
(b) Assume that the value of the portfolio
Π(S, t) = 2P (S, t) + C(S, t)
can be found from the standard Black-Scholes formula. Here, C(S, t) is the value of a European
call option with exercise price E and expiry date T , and P (S, t) is the value of a European put
option, also with exercise price E and expiry date T .
(i) Use the Black-Scholes formula along with put-call parity to derive a formula for the portfolio Π in terms of S, t, E, r, N (d1 ) and N (d2 ).
(ii) Find
lim Π(S, t).
r→∞
(iii) Can you give a financial interpretation of this result?
[8 marks]
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P.T.O.
MATH20912
3.
(a) A share price is currently 78, and over each of the next two 3 month periods it is expected
to go up by 12.5% or down by 7.5%. The risk free interest rate is 2% per annum. There is a
contract V trading on the market which has the payoff

0
if
S ≤ 80

S − 80 if 80 < S ≤ 90
VT =

10
if
S > 90
What is the value of the contract V (it can be exercised only on the expiry date) in 6 months
time? (Hint: use a two-step binomial tree.)
[8 marks]
(b) The value of a European call option with a continuous dividend rate D0 is given by
C(S, t) = Se−D0 (T −t) N (d10 ) − Ee−r(T −t) N (d20 )
where
1
N (x) = √
2π
Z
x
2
− y2
e
−∞
dy,
d10
ln(S/E) + (r − D0 + σ 2 /2)(T − t)
√
,
=
σ T −t
√
d20 = d10 − σ T − t.
and E is the exercise price, σ is the volatility, r is the interest rate, T is the expiry date.
(i) Find ∆ =
∂C
.
∂S
e−D0 (T −t) N 0 (d10 )
√
(ii) Substitute your result for ∆ and the result Γ =
into the modified BlackSσ T − t
∂C
Scholes equation with continuous dividends to find a formula for
.
∂t
(iii) Show that for large S the value of a call option is an increasing function of time.
[12 marks]
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P.T.O.
MATH20912
4.
(a) Assume that a share is paying out a continuous constant dividend yield D0 Sdt at each instant
in time, and that the holder of a share will use these dividends to buy new shares. Then in a
one-step binomial model, a portfolio holding a share at the initial price S0 will either go up to
S0 eD0 T u or down to S0 eD0 T d at t = T .
Set up the portfolio Π = ∆S − P where ∆ is a number of shares, then use hedging arguments
to derive a formula for the European put option P0 at time t = 0 in a one-step binomial tree
model with dividends. Write down the risk-neutral pricing formula and calculate the value of
p when shares pay out a continuous constant dividend yield.
[11 marks]
(b) The equation
dB
= r(t)B − K(t)
dt
describes the bond price B(t, T ), where r(t) is the risk-free interest rate at time t and K(t) is
the coupon payment rate, T is the maturity date, and the final condition is B(T, T ) = F at
t = T.
(i) With r(t) = r0 , assume now that the coupon payment rate function is given by
0
if t < t0
K(t) =
k F if t ≥ t0
where k and t0 are positive constants meaning that coupons only start paying after t0
years. Find a formula for the bond price at t = 0.
(ii) For the parameters F = 100, r0 = 0.1, t0 = 1, and T = 2, find the value of the parameter
k which satisfies B(0, 2) = F .
(iii) For the parameters in (ii) including your calculated value of k, sketch the value of B(0, T )
for different maturity dates in the range T ∈ [0, 2].
[9 marks]
END OF EXAMINATION PAPER
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