MTH 4500: Introductory Financial Mathematics

# Final (Practice 5)

Problem 1

The initial price of the stock is $$S=140$$. The price is assumed to follow one-period binomial model. At time one the price can be either $$Su=224$$ or $$Sd=112$$. Assume that the simple interest rate over one time period is $$r=\frac{2}{5}$$.

• (a) What is the price of the European call option with strike $$K=140$$?

• (b) What is the price of the European put option with strike $$K=140$$?

Problem 2

Assume that the risk free rate is $$0$$ and that the stock price is given by the equation $S(t)=10e^{2 t + 2B(t)},$ where $$B(t)$$ is the standard Brownian motion. Determine the price at time $$0$$ of the European put option with strike $$K=10e^{18}$$ and expiration $$9$$.

Problem 3

There are only three stocks in the stock market called $$X$$, $$Y$$, and $$Z$$. Their current prices are $$10$$, $$20$$, and $$30$$, respectively. The covariance matrix between the returns on the stocks $$X$$, $$Y$$, and $$Z$$ is $C=\left[\begin{array}{ccc}1 & 1&0\newline 1&2&0\newline 0&0&1 \end{array}\right].$ What is the risk of the portfolio that consists of $$2$$ shares of $$X$$, $$3$$ shares of $$Y$$, and $$4$$ shares of $$Z$$?

Problem 4

The prices $$B(0,9)$$, $$B(0,5)$$, and $$B(5,9)$$ of the zero coupon bonds with face value $$1$$ are known at time $$0$$ and are: $B(0,9)=0.22, \; B(0,5)=0.3,\; \mbox{and}\; B(5,9)=0.7.$ Prove that there is an arbitrage opportunity and explain how to achieve this arbitrage.

Problem 5

Assume that the price of a security follows Black-Scholes model and that its price at time $$0$$ is $$20$$. Consider a portfolio $$M$$ that is long three European put options with strike $$25$$ and expiration $$1$$ and short three European call options with strike $$25$$ and expiration $$1$$ written on the given security. Sketch the graph of the price of the portfolio $$M$$ at time $$0$$ as the function of the interest rate $$r$$.

Problem 6

A $$5\times 5$$ matrix $$C$$ is the covariance matrix between the returns of $$5$$ risky securities. It is known that $\left[9,3,9,5,6\right] \cdot C=\left[8,8,8,8,8\right].$ Determine $$\overrightarrow u C^{-1}\overrightarrow u^T$$, where $$\overrightarrow u\in\mathbb R^5$$ is the vector whose all components are $$1$$.

Problem 7

Assume that $$X$$ is a standard normal random variable and that $$Y$$ is independent from $$X$$ and satisfies $$\mathbb P(Y=1)=\mathbb P(Y=-1)=\frac12$$. Assume that the return on the security $$A$$ is $$K_A=e^X-1$$ and the return on the security $$B$$ is $$K_B=e^{XY}-1$$. Determine the covariance matrix between the returns on portfolios $$A$$ and $$B$$ and the minimal possible variance for the return on a portfolio that consists entirely of shares of $$A$$ and $$B$$?

Problem 8

Assume that the numbers $$\mu$$, $$r$$, $$\sigma_1$$, $$\sigma_2$$, $$T_1$$, $$T_2$$, $$K$$, and $$S_0$$ are given and that $$0 < T_1 < T_2$$. The function $$\sigma(t)$$ is defined as \begin{eqnarray*} \sigma(t)&=&\left\{\begin{array}{ll}\sigma_1, & \mbox{ if } t\in[0,T_1),\newline \sigma_2,&\mbox{ if }t\in[T_1,T_2].\end{array}\right. \end{eqnarray*} What is the price of European call option with strike $$K$$ whose underlying security has the price that satisfies $$S(t)=S_0e^{\mu t+\sigma(t)B(t)}$$, where $$B(t)$$ is a standard Brownian motion?