How Realized Volatility Paths Affect Black-Scholes Delta-Hedge P&L
Summary
The document asks whether the profit or loss of a Black-Scholes delta-hedged option depends only on realized volatility at expiry or also on its path. It references a hedge-error expression involving gamma exposure and the difference between implied variance and realized variance. The author describes a proposed test using historical S&P 500 returns, an EGARCH forecast of next-day realized volatility, and a 63-day hedging horizon, with option positions chosen according to the forecast relative to implied volatility.
The motivating example highlights the distinction between an average or terminal volatility comparison and daily mark-to-market results: volatility could rise above implied levels early and later fall below them. The document supplies no plotted results or resolution, so it is a question rather than evidence that the proposed forecasting strategy earns a profit. Its setup also leaves implementation and evaluation details unspecified, including how hedge rebalancing, option valuation, and transaction costs affect realized P&L.
Key ideas
- Delta-hedging error is linked to gamma exposure and the gap between implied and realized variance.
- The author proposes using EGARCH forecasts to choose long or short options and hedge dynamically.
- Daily mark-to-market P&L may be sensitive to the volatility path during the hedge horizon.
- The document poses the path-versus-expiry question but provides no empirical answer or performance evidence.
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# Empircal data analysis delta hedge error of Black-Scholes by Mark Davis
# Empircal data analysis delta hedge error of Black-Scholes by Mark Davis
Regarding Mark Davis derivation of the delta-hedging error occuring in the black-scholes as a result of difference in realized volatility and implied volatily. The formula reads as follows: $$ Z_t = \int_{0}^T e^{r(T-s)}\frac{1}{2} S_t^2 \Gamma_t (\hat{\sigma}- \beta_t^2)dt$$ I want to test the hypothesis, that e.g. if we can predict the realised volatility (I know it's only determinable at an option expiry), we could make volatility arbitrage. However, my question is regarding whether the determination of a negative or positive, final Profit and loss of a portfolio, is due to the path of the realised volatilty, or solely based upon the realised volatility at expiry. I have set up an hedge expirement, valuating previous log-returns of 126 days of the S&P-500 index, and then forecast the realised volatility 1-day ahead using EGARCH(1,1), in where we either buy the option if ($\sigma_{forecast}>\sigma_i$), or otherwise we short (The hedge horizon is 63 days). \ I have read in a paper that: "Our $\Delta$-hedge strategy only makes us a profit if realised volatility ends up "on the right side" of initial implied volatilty". However, regarding this, my profit and loss is based upon daily rebalancing of the portfolio. Thus I was wondering if this statement is true. Say my forecast is wrong, and I have forecasted $\sigma_i>\sigma_a$, I'm then shorting the option, and going long in the stock. However, if the realized volatilty rapidly growth above the initial implied volatilty in the first 50 days, and then in the last 13 days settles below the initial implied volatility. The statement would say I would earn a positive profit and loss, however the mean of the realised volatility is above the initial implied volatility. I'm then wondering if I would still earn money, based upon the daily mark-to-market profits. Can anybody elaborate on this? In example, is there a way to connect these two graphs plotted from an initial ATM implied volatility of 11.1%, then the first is the profit and loss path: and this volatilty graph from the hedging period: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.