Estimating Delta-Hedge Cost Variability by Simulation

Summary

The document explains how to interpret a table reporting the risk of a dynamically delta-hedged option position. It describes simulating stock-price paths at the stated weekly frequency, computing the option delta at each step, and adjusting the share position as delta changes. The present value of the resulting hedge costs is calculated for each path.

The procedure is repeated across the stated simulations, and the standard deviation of hedge cost is divided by the option’s initial price to produce the table’s measure. The quantity is intended to capture hedge-cost volatility, rather than average cost. Under Black–Scholes assumptions such as constant volatility, the average present value of hedge costs is expected to match the initial option price. That conclusion depends on the model assumptions; the document does not discuss transaction costs, discrete hedging error beyond the simulated schedule, or departures from the assumed price process.

Key ideas

• Simulate stock prices at the rebalancing frequency over the option’s life.
• Recompute option delta at each time step and trade shares to maintain the hedge.
• Discount hedge costs to present value for each simulated path.
• The reported measure is the standard deviation of hedge costs divided by the initial option price.
• The expected hedge cost matches the option price only under the stated Black–Scholes assumptions.

Tags

# How is the performance measure computed here?


# How is the performance measure computed here?












The image is from John C Hull Textbook titled Options, Futures and Other Derivatives ( page 407 - Ninth Edition). The table above was obtained after computing the delta of stock price, shares purchased and interest cost.

## Answer by Magic is in the chain (score 1, accepted)

https://quant.stackexchange.com/a/42446

The labelling is indeed a bit confusing- this guy is normally very smooth!

Let’s focus on the weekly rebalancing column, here are the detailed steps.

1) Simulate the path of the stock price as per weekly frequency (20 time steps here as the option maturity is 20 weeks).

2) Calculate the delta of the option at each time step.

3) At the initial time, buy or sell shares, to hedge the delta of the option at time zero.

4) Readjust the delta hedge at each time step, buying or selling shares, depending on the change in delta from the previous step.

5) Generate the present value of the cost of this dynamic delta hedge.

Repeat the above precedure 1000 times, and calculate its standard deviation. Dividing this by the option price at time zero is the ratio you see in the table.

He is only after risk or volatility of the delta hedge. Average is not a problem as the average of the present value of the delta hedge would match the option price at time zero, if the black scholes assumptions are assumed to hold (e.g. constant volatility).

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.