Mathematical Finance (20443/20444)
ASSESSED PROBLEM SHEET 2
The submission deadline for Problem Sheet 2 is 17:00 on 5 November
2021
(a) Consider the Black-Scholes equation for the option price V (S, t) if the
underlying does not pay out dividends.
(i) Show that V (S, t) = AS, where S is the underlying price, t is time,
and A is a constant, is a solution to the Black-Scholes equation.
(ii) For the solution V (S, t) = AS, define the value ∆ used for hedging.
(b) Today is time t = 0, the current value of the underlying is S =£10.00,
and the risk-free interest rate is r = 0.05 per annum. Today it is possi-
ble to buy a European vanilla call option with exercise price E =£8.00
and expiry date T = 6 months for C = 275 pence. What is the fair
value in pence for the corresponding European vanilla put?
(c) The Black-Scholes equation for the value of an option V (S, t), where S
is the value of the underlying asset and t is time, can be transformed
into the diffusion equation for a new variable u(x, τ) using S = Eex,
t = T − 2τ/σ2, k = 2r/σ2 and
V = Eu (x, τ) exp
(
−(k − 1)x
2
− (k + 1)
2 τ
4
)
,
where r is the risk-free interest rate, σ is the volatility, E is the strike
price, and T is the expiry date. The diffusion equation has the solution
u(x, τ) =
1
2
√
τpi
∫ ∞
−∞
u0 (s) exp
(
− 1
4τ
(x− s)2
)
ds.
Determine, showing all steps, the function u0(s) for a European asset-
or-nothing call with the pay-off Λ(S) = SH(S − E).
(d) Let a discrete dividend with the dividend yield dy be paid out on the
underlying S at time td.
(i) Formulate the jump condition for the underlying price S at time td
and explain carefully the meaning of all parameters and variables
you use in the jump condition.
(ii) Consider the dividend yield dy = 0.2. The value of S immediately
after the dividend payment is S = £10.00. What is the value of
S immediately before the dividend payment?
(e) Explain why the value of an American option is always at least the
value of its payoff function.























































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