MTH 4500: Introductory Financial Mathematics
# Final (Practice 1)

**Problem 1**
**Problem 2**
**Problem 3**
**Problem 4**
**Problem 5**
**Problem 6**
**Problem 7**
**Problem 8**
**Problem 9**
**Problem 10**

Find the rate of continuous compounding equivalent to daily compounding of \(90\%\), if we assume that a year has \(365\) days.

The three year bond has face value USD \(100\), and pays USD \(5\) coupons annually, the last one at maturity. Assume that the continuously compounding rate is \(7\%\). Find the price of this bond.

Consider a two-period binomial model. Let the current stock price be \(60\) and the risk-free rate is assumed to be \(10\%\). In each period the stock can go either up by \(20\%\) or down by \(10\%\). The strike price of a European call option is \(62\), and the option expires at the end of the second period.

**(a)**Find the stock price tree.**(b)**Determine the call price tree.**(c)**Determine the hedge ratio tree.

Three risky securities have expected returns \(\mu_1=20\%\), \(\mu_2=50\%\), \(\mu_3=70\%\), standard deviations \(\sigma_1=0.1\), \(\sigma_2=0.3\), \(\sigma_3=0.25\), and correlations \(\rho_{12}=0.4\), \(\rho_{23}=-0.1\), and \(\rho_{31}=0.2\).

**(a)**Compute the expected return \(\mu_V\) and the risk \(\sigma_V\) of the portfolio with weights \(w_1=29\%\), \(w_2=-17\%\), \(w_3=88\%\).**(b)**Find the weights of the portfolio of the minimal risk that can be constructed using the given three securities. What is the expected return and risk of this minimal risk portfolio?

Assume that the stock price is governed by the binomial model. The initial price of the stock is \(S(0)=100\). The return on a risk-less security over one period of time is \(R=20\%\). The parameters \(u\) and \(d\) are not known.

**(a)**Which of the following two options is more expensive: A European put option with strike \(140\) and expiration \(1\) or a European call option with strike \(140\) and expiration \(1\)?**(b)**Which of the following two options is more expensive: A European put option with strike \(130\) and expiration \(2\) or a European call option with strike \(140\) and expiration \(2\)?

Suppose that the annual continuously compounded rate on a risk-less security is \(r=7\%\). Consider a \(120\)-day European call option with strike price \(X=60\) dollars written on a stock with current price \(S=50\). Assume that the stock volatility is \(\sigma=30\%\). Use the Black-Scholes formula to obtain the price of this option and the delta of the option. Assume that the year has 365 days.

Assume that \(S\) is a process given by the equation \(S(t)=S(0)e^{\mu t+ \sigma B(t)}\), where \(B(t)\) is a standard Brownian motion. Evaluate the expected value \(\mathbb E[S(t)]\) in terms of \(\mu\), \(\sigma\), \(t\), and \(S(0)\).

Assuming that the vega for European call is given by \(\displaystyle \mbox{vega }_{C_E}=\frac{S\sqrt T}{\sqrt{2\pi}}\cdot e^{-\frac{d_+^2}2}\), derive the formula for vega for European put on the same stock with the same strike price and the same expiration.

Given positive real numbers \(S(0)\), \(\mu\) and \(\sigma\), assume that the stock price is \(S(t)=S(0)e^{\mu t+\sigma W(t)}\), where \(W(t)\) is a standard Brownian motion. Assume that \(r\) is the rate of continuously compounded interest. For given \(X\) and \(T\), compute the price of the digital put option with expiration \(T\) that pays \(1\) dollar if the price of the stock at time \(T\) is below \(X\).

Given two positive numbers \(r\) and \(T\), and a positive differentiable function \(g\), consider the function \(u\) defined in the following way: \[u(t,x)=\frac{e^{-(T-t)r}}{\sqrt{2\pi(T-t)}}\int_{-\infty}^{+\infty} e^{-\frac{y^2}{2(T-t)}}g\left(xe^{\left(r-\frac{\sigma^2}2\right)(T-t)+\sigma y}\right)\,dy,\] for \(0\leq t\leq T\) and \(x\geq 0\). If \(u_t(x,t)\leq 0\) and \(u_x(x,t)\leq 0\) for all \(x\geq 0\) and \(t\in[0,T]\), prove that for each \(x,y\geq 0\), \(t\in(0,T)\), and \(\alpha\in(0,1)\) the following inequality holds \[u\left(t,\alpha x+(1-\alpha )y\right)\leq \alpha u(t,x)+(1-\alpha)u(t,y).\]