Consider the following model, where $r=0$, and a dividend of 1 unit of currency is paid at time 1.5. $$ \begin{array}{|c|c|c|c|} \hline & S(0,\omega) & S(1,\omega)^* & S(2,\omega)^* \\ \hline \omega_1 & 6& 9& 11\\ \hline \omega_2 & 6& 9& 7\\ \hline \omega_3 & 6& 4& 7\\ \hline \omega_4 & 6& 4& 1\\ \hline \end{array} $$ a) Calculate the risk-neutral probabilities at each node of the information tree.

b) Calculate the value at each node of an American call option with exercise price $K=5$

c) Construct a hedging strategy for the portfolio

My solution:

a) Risk neutral probabilities give expected value equal to the value. Make sure to use dividend in expected value.

$6=9\times p + 4 \times (1-p) \implies P(S(1)=9) = \frac{2}{5},\quad P(S(1)=4) = \frac{3}{5}$ $9=12\times p + 8 \times (1-p) \implies P(S(2)=11|S(1)=9) = \frac{1}{4}, \quad P(S(2)=7|S(1)=9) = \frac{3}{4}$ $4=8\times p + 2 \times (1-p) \implies P(S(2)=7|S(1)=4) = \frac{1}{3},\quad P(S(2)=1|S(1)=4) = \frac{2}{3}$

b) Work backwards from end, working out the max of value of exercising at that point or expected value of continuing. $$ \begin{array}{|c|c|c|c|} \hline & V_{amer}(0,\omega) & V_{amer}(1,\omega)^* & V_{amer}(2,\omega)^* \\ \hline \omega_1 & 2& 4 \text{ (stopping point)}& 6\\ \hline \omega_2 & 2& 4 \text{ (stopping point)}& 2\\ \hline \omega_3 & 2& \frac{2}{3}& 2\\ \hline \omega_4 & 2& \frac{2}{3}& 0\\ \hline \end{array} $$

c) ???

My question:

What is part C asking? What is the portfolio, is it the stock, is it the option, is it something else? Aren't there different ways to hedge something? Does it mean working out how many units of stock at each node to short to eliminate the risk at each step?

Also, are my answers correct for the other two parts? Is my notation correct?

  • $\begingroup$ I think you are right, they want to know how much stock to short against the option at each node. $\endgroup$ – dm63 Jan 13 '17 at 13:52

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