MTH 4500: Introductory Financial Mathematics

# Final (Practice 2)

Problem 1

Assume an one period binomial model in which the initial stock price is $$S=150$$ and in each period the stock price can go either up by a factor of $$u=\frac{7}{5}$$ or down by a factor of $$d=\frac{2}{5}$$. Assume that the simple interest rate over one time period is $$r=\frac{1}{5}$$. Determine the price of the European call option with strike $$K=90$$.

Problem 2

Assume that the return $$K$$ of a risky security depend on the market scenario in the following way: $\begin{array}{|c|c|c|}\hline \mbox{Scenario} & \mbox{Probability}& \mbox{Return }K \newline \hline \omega_1& 0.6& -20\% \newline \hline \omega_2&0.4 & 80\% \newline \hline\end{array}$ Determine the expected return and the risk of the given security.

Problem 3

Assume that there are three risky securities whose expected returns and covariance matrix are given by $m=\left[\begin{array}{ccc} 10\%&30\%&40\% \end{array}\right]\quad\quad\quad \mbox{and}\quad\quad\quad C=\left[\begin{array}{ccc}5 & 3&0\newline 3&2&0\newline 0&0&1 \end{array}\right].$ Assume that the rate of risk free return is $$R=10\%$$. Determine the weights of the market portfolio.

Problem 4

Assume that the price of the stock follows the equation $$S(t)=S_0e^{7B_t}$$, where $$B_t$$ is the standard Brownian motion. Calculate the probability of the event $$\left\{S\left( 9\right)< 17 S\left(4\right)\right\}$$.

Problem 5

Assume that the stock price is given by $$S(t)=S_0e^{\mu t+\sigma B_t}$$, where $$S_0$$, $$\mu$$, and $$\sigma$$ are given real numbers that satisfy $$S_0 > 0$$ and $$\sigma > 0$$. Assume that the risk-free interest rate is $$r$$. Given positive real numbers $$T$$, $$K_1$$, and $$K_2$$ that satisfy $$K_2 > K_1$$, determine the price of the derived security that can be exercised only at time $$T$$ and that pays $$K_2-K_1$$, or $$S(T)-K_1$$, or nothing, depending on whether the stock price $$S(T)$$ is higher than $$K_2$$, between $$K_1$$ and $$K_2$$, or below $$K_1$$, respectively. In other words, the payoff function $$g$$ for this security is given by $g(x)=\left\{\begin{array}{ll} K_2-K_1, & \mbox{if } x > K_2,\newline x-K_1,& \mbox{if }x\in[K_1,K_2],\newline 0,&\mbox{if }x < K_1.\end{array}\right.$

Problem 6

Given $$n$$ risky securities, assume that the covariance matrix between their returns is $$C$$. For each vector of weights $$w=[\begin{array}{cccc}w_1&w_2&\dots&w_n\end{array}]$$ denote by $$\sigma(w)$$ the risk of the portfolio whose weights are $$w_1$$, $$\dots$$, $$w_n$$. Find the vector of weights $$w$$ that minimizes the function $$F(w)=\sigma^2(w) + 12\left(w_1^2+w_2^2+\cdots+w_n^2\right)$$. We are assuming that all securities are risky (i.e. that all variances are non-zero) and that their covariance matrix is invertible.

Problem 7

Assume two period binomial model for bond prices. Assume that we know the values of $$B(0,2)$$, $$B(0,1)$$, $$B^u(1,2)$$, $$B^u(1,3)$$, $$B^d(1,2)$$, $$B^d(1,3)$$, $$B^{uu}(2,3)$$, $$B^{ud}(2,3)$$, $$B^{du}(2,3)$$, and $$B^{dd}(2,3)$$. Determine the value of $$B(0,3)$$.

Problem 8

Assume that the risk free rate is $$r$$ and that the stock price follows the equation $$S(t)=S_0e^{\mu t+\sigma B_t}$$, where $$B_t$$ is a standard Brownian motion and $$\mu$$, $$S_0$$, $$\sigma$$, are constants ($$S_0 > 0$$, $$\sigma > 0$$). Assume that $$T_1$$, $$T_2$$, and $$K$$ are positive real numbers such that $$T_2 > T_1$$. Determine the formula for the price of the Bermudan put option with strike $$K$$ at time $$0$$ that can be exercised either at time $$T_1$$ or at time $$T_2$$.