Sample Derivative Problems

Question 1
Calculate the risk neutral probability of an upward move in a binomial problem wherein the underlying asset starting at 100 may increase to 130 under favorable conditions or decrease to 70 under unfavorable conditions if the risk-free interest rate is 10%?

(1)
\begin{align} \pi=\frac{1+r_f-d}{u-d} \end{align}
(2)
\begin{align} \pi=\frac{1+0.1\frac{70}{100}}{\frac{130}{100}-\frac{70}{100}} \end{align}
(3)
\begin{align} \pi=\frac{0.4}{0.6}=0.667 \end{align}

Question 2
If the futures or forward price for an asset can appreciate by 20% or decrease by 20% over one period, calculate the risk-neutral probabilities in both the up and down states.

(4)
\begin{align} \pi=\frac{F-F_d}{F_u-F_d} \end{align}
(5)
\begin{align} \pi=\frac{1-0.8}{1.2-0.8}=0.5 \end{align}
(6)
\begin{align} 1-\pi=1-0.5=0.5 \end{align}

Question 3
If a put option on a stock with an exercise price of $90 per share is about to expire when the underlying stock price is $96 per share, what is the entire put contract worth?

Since there are 100 shares represented by one contract, the value would be 100 x (96 - 90) = $600

Question 4

Find the forward rate two years in the future for British pounds if the spot rate is 1.90 $ per pound and the dollar interest rate is 4% per year and the pound interest rate is 6% per year.

(7)
\begin{align} F=S\frac{(1+r_\$)^2}{(1+r_p_o_u_n_d)^2} \end{align}
(8)
\begin{align} F=1.90\frac{(1.04^2)}{(1.06^2)}=1.83\ \$ {\ per\ pound} \end{align}

An underlying asset has future payoffs as illustrated by this diagram.