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EdisonCruise
Topic Author
Posts: 117
Joined: September 15th, 2012, 4:22 am

### What can we do during in delta-hedging when volatility increases?

What can we do during in delta-hedging when volatility increases?Suppose that we made a forecast in the coming one year, the realized volatility is sigma_0. Then we sold an option at sigma_0. After a few days we may make a new forecast, in the coming year, the realized volatility should be sigma_1, which is much bigger than sigma_0.Then what can we do to reduce loss when this happens?Shall we still delta-hedge with sigma_0? Or Hedge with sigma_1? Or is there any method to calculate an optimal sigma to hedge?I hope to figure out a way to reduce standard deviation of PnL in such a case.Thank you.I have read some papers about this issue, but I am still not sure what to doWhich Free Lunch Would You Like Today, Sir?: Delta Hedging, Volatility Arbitrage and Optimal PortfoliosPARAMETER RISK IN THE BLACK AND SCHOLES Model

acastaldo
Posts: 1416
Joined: October 11th, 2002, 11:24 pm

### What can we do during in delta-hedging when volatility increases?

What do you think will happen to volatility in the future?(a) It is reliably announced that vol will be constant and equal to sigma_1 in the future. Then we are back in the BS case and we must all hedge with sigma_1(b) Volatility will continue to fluctuate. Then we are in the Stochastic Volatility case and we need a model of these fluctuations.

DevonFangs
Posts: 3004
Joined: November 9th, 2009, 1:49 pm

### What can we do during in delta-hedging when volatility increases?

QuoteOriginally posted by: EdisonCruiseShall we still delta-hedge with sigma_0? Or Hedge with sigma_1?In both cases you lose the same money on average, the difference is that if you hedge at sigma_1 the daily pnl is stochastic whereas if you hedge at sigma_0 the daily pnl is not (but the total pnl is). How about, if it goes OTM, i.e. good for you, you hedge at sigma_0 and lock your loss (because it's low gamma) and it if goes ITM, i.e. bad for you, you hedge at sigma_1 so you let the pnl stochastic hoping it moves back to OTM a little?
Last edited by DevonFangs on October 22nd, 2014, 10:00 pm, edited 1 time in total.

EdisonCruise
Topic Author
Posts: 117
Joined: September 15th, 2012, 4:22 am

### What can we do during in delta-hedging when volatility increases?

Yes, you are right. We must loss if the option is sold at volatility lower than the realized one. However, is there any trading strategy to reduce the loss or the standard deviation of final PnL if this happens? It seems there is a stop-loss/start-gain strategy in such a case, given by Peter Carr in Delta Hedging under stochastic Volatility. I am not sure how good this method is? Is there any better anlternative?

EdisonCruise
Topic Author
Posts: 117
Joined: September 15th, 2012, 4:22 am

### What can we do during in delta-hedging when volatility increases?

I know there is a relationship as below:$PnL=(Option price sold-Estimated Option value)+ $\int_0 ^T (sigma_h(t)^2-sigma_r(t)^2) S^2*Gamma_h*dt$$given in PARAMETER RISK IN THE BLACK AND SCHOLES Model where sigma_h is the sigma used to calculate delta, sigma_r is the real volatility. Assume we sold the option at sigma_h, and then use it to hedge. If sigma_r is bigger than sigma_h, the integration term is negative, meaning a loss. However, if we can use a bigger sigma_h2 than sigma_r, the integration term can be zero or positive. In such a case, Estimated Option value should be calculated with sigma_h2, otherwise the first term is negative. In particular, when option Gamma and underlying price is big, it is tempting to use a sigma_h close to sigma_r so the second term can be small. So I wonder if there is an optimal sigma_h to minimize the PnL after we sell the option at a low volatility?I have tried to use historical data to do simulation and find that the mean PnL and its standard deviation is quite dependent on the path. There is no obvious advantage of using a sigma_h close to sigma_r or just keep a constant sigma_h.

Engy
Posts: 34
Joined: February 21st, 2005, 9:19 pm

### What can we do during in delta-hedging when volatility increases?

>> have tried to use historical data to do simulation and find that the mean PnL and its standard deviation is quite dependent on the pathThe optimal sigma_h to hedge is the expected realized volatility over your hedging period. Also note that when you use discrete hedging with total number of re-hedging N, the variance of your delta-hedging P&L is proportional to 1/N. Your expected P&L is just the gamma P&L not the spread between option price sold and estimated option value.

EdisonCruise
Topic Author
Posts: 117
Joined: September 15th, 2012, 4:22 am

### What can we do during in delta-hedging when volatility increases?

The optimal sigma_h to hedge is the expected realized volatility over your hedging period. Could you please explain further why "The optimal sigma_h to hedge is the expected realized volatility over your hedging period"? The "expected realized volatility" changes as time passes, so we need to keep changing the sigma_h in the whole hedging period. This means our estimated option value changes because of volatility. It makes difficulty to make our profolio mark-to-market. Moreover, it is also based on the assumption that we are confident on "expected realized volatility", but actually not.

Engy
Posts: 34
Joined: February 21st, 2005, 9:19 pm

### What can we do during in delta-hedging when volatility increases?

>> assumption that we are confident on "expected realized volatility", but actually not.If you are not confident about the expected realized volatility, then don't trade.You can think of realized volatility of returns in terms of the GARCH process V(t). The expected realized volatility is an integral over the hedging period, sigma_h=sqrt{ int E[ V(t)] dt} so it's not very sensitive to the shock in instantaneous variance V(0) because it will be relaxed by the mean-reversion. >> This means our estimated option value changes because of volatility. It makes difficulty to make our profolio mark-to-market.You mark-to-market your positions at market prices not at your estimated option values. Typically you would also hedge at implied vols. You would have more discretion to choose re-hedging periods and skew-delta adjustments.