Discrete Delta-Hedging Error and Black–Scholes Call Replication
Summary
This note describes a question about reconciling a discretely rebalanced Black–Scholes hedge with the payoff of a written call. The questioner tracks a stock-and-bond portfolio, observes that its terminal value matches the option payoff in a simulated path, and wonders why separately delivering the share at the strike price seems to leave cash over after repaying borrowing. The apparent discrepancy concerns how the hedge portfolio, option settlement, and stock delivery are accounted for.
The response focuses on the practical diagnostic: as rebalancing becomes more frequent, the standard deviation of hedging error should approach zero; the mean error should fluctuate around zero with shrinking deviations. It mentions plots from simulated sample paths as evidence, but gives no plots or detailed derivation in the supplied text. The specific accounting puzzle is not directly resolved here, and the questioner's code is absent, so the response cannot identify an implementation error or quantify transaction costs and other real-world frictions.
Key ideas
- A discrete-time delta hedge can have a nonzero hedging error before the continuous-rebalancing limit.
- The response says hedging-error dispersion should decline as rebalancing frequency increases.
- The mean hedging error is described as fluctuating around zero with decreasing deviations.
- The supplied response does not explain the specific terminal stock-delivery accounting puzzle.
- The simulation discussion does not address transaction costs or other market frictions.
Tags
Full text
# Basic practical question about Delta hedging # Basic practical question about Delta hedging I am trying to understand a simple thing about Delta hedging in the Black-Scholes world. I know I'm doing something blatantly wrong, I just can't see it now. Let's say I write a call option and sell it to someone. With that money, and whatever else I need to borrow, I can set-up a self-financing portfolio of stock and bond, where the amount of stock I hold at all times is Delta. At maturity, the value of this portfolio is exactly the payoff of the option. So if the buyer exercises the option, I can cover the pay exactly with this portfolio. However, there's something bugging me: the final value of the portfolio is obviously calculated using the final value of the stock. Suppose that $S_T>K$, so the buyer exercises. In this case, I can just give him the stock I've been holding (will most likely be one unit), and receive $K$. For concreteness, I've run a simulation of daily Delta-hedging, with $S_0=49$, $K=50$ and maturity 3 months. In this particular sample path, the final price of the stock is $S_T=50.277668$. I also have at maturity Delta=1, so I hold one unit of stock, and I am short $49.37889$ bonds, so that's what I owe in dollars. This is consistent with the above: the total value of the portfolio is $(S_T-K)_+$. The option buyer thus ends up with a position of $0.2776678$. This is the value of my portfolio, so I can just give it to him and we're even. However I can also sell the stock to him for $50$, so he gets his stock at his price, I get $50$, pay back the $49.37889$ I owe, and I end up with some money. Where did this come from? ## Answer by LocalVolatility (score 4, accepted) https://quant.stackexchange.com/a/32966 As has been remarked in the comments already, the standard deviation of your hedging error should approach zero as your re-hedging frequency (the number of time steps) increases. Here is a sample plot of how it should behave like. It was generated using $T = 1$, $K = 100$, $S_0 = 100$, $r = 5\%$, $\sigma = 20\%$ and 1,000 sample paths. Just as a sanity check - here is the same plot but with the mean hedging error, which fluctuates around zero with decreasing deviations. As you didn't provide your code, I created a simple Jupyter notebook for this that you can clone from GitHub. This should hopefully help you debug your own program.
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