MATH39032 MATHEMATICAL MODELLING IN FINANCE

1.通过不使用套利参数，得出欧洲看跌期权P和

[6分]
2.绘制以下每个投资组合的到期收益图：
（i）做空两股，做多三份看涨期权，行使价为X。
（ii）做空两股，做多三手看跌期权，行使价为X1，做多四次看涨期权，行使价为X2。

[12分]
3.资产S的价格变化满足随机差分方程
dS = 0.5Sdt +（1+ t）

SDW，

[12分]
4.考虑Black-Scholes方程的股票价格S的期权价格（期权V）

∂V
∂t
+1
2个
σ2
S2 2

∂S2
+ rS
∂V
∂S
− rV = 0，

（i）代入证明
V（S，t）= Aert +βln S，

（ii）确定常数β的值。
（iii）找到满足V（Z，t）的偏微分方程，其中Z = −lnS。
[20分]

1. By using no arbitrage argument, derive put-call parity for the European put option, P,andthe
European call option, C, with the same exercise price X and expiry date T. Assume the constant
risk-free interest rate r.
[6 marks]
2. Draw the expiry payoﬀ diagrams for each of the following portfolios:
(i) Short two shares, long three calls with an exercise price X.
(ii) Short two shares, long three puts with exercise price X1, long four calls with exercise price X2.
Consider only the case X1 <X2.
[12 marks]
3. The change in price of an asset S satisﬁes the stochastic diﬀerential equation
dS =0.5Sdt +(1+ t)

SdW,
where W(t) is a Wiener process. Using Ito’s Lemma along with the delta-hedge argument, derive
the partial diﬀerential equation satisﬁed by the price V (S, t) of an option on the underlying asset.
[12 marks]
4. Consider the Black-Scholes equation for the price for an option, V of an option, on a stock S that
pays no dividends, namely
∂V
∂t
+ 1
2
σ2
S2 ∂2
V
∂S2
+ rS
∂V
∂S
− rV =0,
where r and σ are both constants.

(i) Show by substitution that
V (S, t)= Aert+β ln S,
where A and β are constants, is a solution of the Black-Scholes equation.
(ii) Determine the values of the constant β.
(iii) Find the partial diﬀerential equation satisﬁed by V (Z, t), where Z = −ln S.
[20 marks]