Delta Hedging an American Option Across a Binomial Tree
Summary
The document asks how node values for an American option on a small stock-price tree translate into a hedge. The accepted reply identifies the commenter’s suggested stock position as delta hedging: adjusting exposure to the underlying so that small price moves have less effect on the option position. In a discrete tree, node-by-node hedge ratios can describe the shares to hold or short against the option at each point, while the option’s exercise choices and dividend affect its value and hedge through time.
The reply connects dynamic hedging with volatility trading. A trader who owns an option may rebalance shares as the underlying moves; gamma scalping describes how this rebalancing can generate gains when realized volatility exceeds the volatility priced into the option, before costs. The short option position has the opposing exposure. This is a high-level explanation, not a worked hedge: it does not derive hedge ratios from the listed node values, specify rebalancing trades, or address transaction costs, model risk, and early exercise details. Its volatility interpretation is therefore conditional rather than a guaranteed outcome.
Key ideas
- Delta hedging uses stock exposure to offset an option’s sensitivity to underlying price changes.
- A hedge can be adjusted at successive nodes as the underlying price and option value change.
- An option holder may use gamma scalping to seek gains from realized volatility above the volatility paid for.
- The opposing option position has the reverse volatility and hedging exposure.
- The reply gives an intuition but does not calculate a complete hedge for the supplied tree.
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Full text
# Hedging strategy for American Option
# Hedging strategy for American Option
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 by AlRacoon (score 1, accepted)
https://quant.stackexchange.com/a/44375
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.