STAT0013旨在介绍金融业使用的数学概念和工具，特别是用于金融建模和衍生品定价的随机模型和技术。该课程主要面向本科三年级和四年级学生，以及在统计科学系或与其他系联合开设的学位课程中注册的授课式研究生。

The Black-Scholes formula for the price of a European call option under the standard assumptions, with strike price $K$ and time to expiry $T$, is
$$S_0 N\left(d_1\right)-K e^{-r T} N\left(d_2\right),$$
where $N(\cdot)$ denotes the cumulative distribution function of a standard Normal, and
\begin{aligned} & d_1=\frac{\log \left(\frac{S_0}{K}\right)+\left(r+\frac{\sigma^2}{2}\right) T}{\sigma \sqrt{T}} \ & d_2=\frac{\log \left(\frac{S_0}{K}\right)+\left(r-\frac{\sigma^2}{2}\right) T}{\sigma \sqrt{T}}=d_1-\sigma \sqrt{T} \end{aligned}
The Black-Scholes formula for the price of a European put option with strike price $K$ and time to expiry $T$, is
$$K e^{-r T} N\left(-d_2\right)-S_0 N\left(-d_1\right),$$
where the notation is the same as above.

Problem 1.

A financial institution sold for $\$ 300,000$a European call option on 100,000 shares of a non-dividend paying stock. We assume that the stock price is$S_0=49$, the strike price is$K=50$, the risk-free interest rate is$r=5 \%$per annum, the stock price volatility is$\sigma=20 \%$per annum and the time to maturity is 20 weeks. Assume that the assumptions of the Black-Scholes model hold. (a) Find the value of the European call option on the 100,000 shares. (b) Find the value of a European put option with same strike price and expiration date on the 100,000 shares. (c) Verify that the put-call parity holds in this case. Problem 2. Assume that the current stock price is$£ 110$, the riskless interest rate is$3 \%$per annum and the volatility is$7 \%$per annum. Let$S_2$be the stock price after two years. What is the value of a financial product that pays$S_2-120$if$S_2>120,100-S_2$if$S_2<100$and zero otherwise? Assume that all Black-Scholes assumptions hold (including no dividends). Problem 3. For each of the following European-style derivatives with the given payoffs (in$£$), draw the payoff diagram at maturity and find the price of the derivative. In each case assume that the Black-Scholes assumptions hold, that all derivatives have maturity dates 1 year from now and that the underlying asset price process follows the stochastic differential equation $$d S_t=\mu S_t d t+\sigma S_t d B_t,$$ with parameters$\mu=0.1$and$\sigma=0.4$. Assume also that the underlying asset is currently priced at$£ 4$and that the risk-free rate is$0 \%$. (a) A payoff of 0 if$S_T \leq 4$, and a payoff of$S_T-4$if$S_T \geq 4$. (b) A payoff of 0.2 if$S_T \leq 4$, a payoff of$S_T-4$if$4 \leq S_T \leq 5$, and a payoff of 1 if$S_T \geq 5\$.