MTH 4500: Introductory Financial Mathematics

# Final (Practice 3)

Problem 1

Assume an one period binomial model in which the initial stock price is $$S=80$$ and in each period the stock price can go either up by a factor of $$u=\frac{7}{4}$$ or down by a factor of $$d=\frac{3}{4}$$. Assume that the simple interest rate over one time period is $$r=\frac{1}{4}$$. Determine the price of the European call option with strike $$K=80$$.

Problem 2

Assume that there are only three stocks in the stock market called $$P$$, $$Q$$, and $$R$$ and that their prices at time $$0$$ are $$10$$, $$10$$, and $$20$$, respectively. The covariance matrix between the returns on the stocks $$P$$, $$Q$$, and $$R$$ is given by $C=\left[\begin{array}{ccc}2 & 1&0\newline 1&1&0\newline 0&0&1 \end{array}\right].$ Calculate the risk of the portfolio that consists of $$13$$ shares of $$P$$, $$3$$ shares of $$Q$$, and $$2$$ shares of $$R$$.

Problem 3

The bond with face value $$F$$ and maturity $$T=26$$ years has two coupon payments in the amount $$C$$ at the end of years $$7$$ and $$8$$. Determine the formula for the price of this bond if the continuous compounding rate is $$r$$?

Problem 4

Assume that the risk free rate is $$0$$ and that the stock price is given by the equation $$S(t)=6e^{2 t + 2B(t)}$$, where $$B(t)$$ is the standard Brownian motion. Determine the price at time $$0$$ of the European call option with strike $$K=6e^{18}$$ and expiration $$9$$.

Problem 5

Assume that $$B(t)$$ is a Brownian motions and that $$S(t)$$ is a random variable defined as $$S(t)=7 \cdot 2^{B(t)}$$. Determine $$\mathbb P\left(S(11)> 4S(5)\right)$$.

Problem 6

Assume that the covariance matrix between the returns of $$100$$ given securities is $$C$$. For each vector of weights $$w=[\begin{array}{cccc}w_1&w_2&\dots&w_{100}\end{array}]$$ denote by $$\sigma(w)$$ the risk of the portfolio whose weights are $$w_1$$, $$\dots$$, $$w_{100}$$. Find the vector of weights $$w$$ that minimizes the function $F(w)=\sigma^2(w) + 9\left(w_1^2+w_2^2+\cdots+w_{50}^2\right)+ 2\left(w_{51}^2+w_{52}^2+\cdots+w_{100}^2\right).$ 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 that bond prices are following a binomial model and assume that we know the values of the zero-coupon bond with maturity $$4$$ in all possible scenarios, i.e. assume that the values $$B(0,4)$$, $$B^u(1,4)$$, $$B^d(1,4)$$, $$B^{uu}(2,4)$$, $$B^{ud}(2,4)$$, $$B^{du}(2,4)$$ , $$B^{dd}(2,4)$$, $$B^{uuu}(3,4)$$, $$\dots$$, $$B^{ddu}(3,4)$$, and $$B^{ddd}(3,4)$$ are known. Assume also that the short rates $$r(0)$$, $$r^u(1)$$, and $$r^d(1)$$ are known. Determine the price of the zero-coupon bond with maturity $$2$$.

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 $$K$$ is a positive real number. Determine the formula for the price of the derived security whose payoff at time $$T$$ depends on the stock price $$S(T)$$ in the following way $g(S(T))=\left(\mbox{max }\left\{S(T)-K, 0\right\}\right)^2.$