Hedging strategy for American Option

Question

Good day,

I was asked to devise a hedging strategy for an American Option given the following claims.

Note, $r=0$ and the underlying stock pays a dividend of $1$ at time $t=1.5$

\begin{array}{|c|c|c|c|} \hline & S(t=0,\omega) & S(t=1,\omega)^* & S(t=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}

I found this question which seems identical except this person did not use dynamic programming to find the value of the option at each node.

I found the same risk neutral probabilities but I found the values at the nodes to be as follows

$$V_{amer}(0)=\dfrac{8}{5}, V_{amer}(1,\{ \omega_1,\omega_2\})=3, V_{amer}(1,\{ \omega_3,\omega_4\})=\dfrac{2}{3}$$ $$V_{amer}(1,\{ \omega_1\})=6, V_{amer}(1,\{ \omega_2\})=2, V_{amer}(1,\{ \omega_3\})=2, V_{amer}(1,\{ \omega_4\})=0$$

The comment on this (Constructing a hedging strategy for an American option) post says its how much stock we need to short against the option at each node but Im not quite sure how that makes a hedging strategy.

Answer 1

The hedging strategy the commenter describes is called a “delta” hedge to neutralize the option position to the change in the underlying. This “delta” hedging strategy is often used by traders that are attempting to isolate their position to the volatility of the underlying. Through “gamma scalping” they are trying to capitalize on higher realized volatility than the implied volatility they paid for the option. Of course the short side has the exact opposite position and is taking the position of realized volatility being lower than the implied volatility they sold by having less “delta” hedging costs over the life of the option. If you search “gamma scalping” on this forum you will find an example of how this “gamma scalping” works.