Dynamic Hedging in a Multi-Period Binomial Option Model

Summary

The document addresses how to hedge an option across multiple periods in a binomial stock-price model. Its central point is that the hedge must be recalculated at each node of the tree, since the appropriate holdings depend on the price path taken so far.

The response describes maintaining separate holdings in the underlying and the risk-free asset as the tree evolves. After the first period, those holdings vary with the history of up and down moves. It points readers to a textbook chapter for a fuller treatment, but provides no derivation, numerical example, or detailed arbitrage construction. The note therefore gives a useful orientation to the method while leaving the calculations and assumptions to other sources.

Key ideas

• Recalculate the hedge ratio at each node as the binomial tree evolves.
• Holdings after the initial period depend on the sequence of up and down moves.
• A multi-period hedge requires tracking state-contingent positions in the underlying and risk-free asset.
• The response points to a textbook for a fuller derivation but does not work through an example.

Tags

# Hedging/Arbitrage with multiple period binomial tree


# Hedging/Arbitrage with multiple period binomial tree












Plenty of material is written on how to hedge/arbitrage option price in one period binomial model, but I cannot find anything about hedging in multiple periods. If one to use multiple periods binomial model, how would one try to hedge? It seems like hedge ration always depends on the binary outcome of stock prices. Could somebody point me in the right direction?

## Answer by Bob Jansen (score 1)

https://quant.stackexchange.com/a/54321

Baxter and Rennie treat the multi-period model in chapter 2 and should be easy to follow.

The trick is to adjust the hedge ratio at every node. So you'll have holdings at time $t=1$ of $x_1$ and $y_1$ and different holdings on $x_2$ and $y_2$ and so on for all the steps. The holdings, except for the first one, depend on the history of up or down moves.

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.