# Implementing a replicating strategy from the order book

So I have futures data in an order book (one screenshot every day at 12 p.m. for one month) for various futures products (i.e. various delivery periods such as the next day, the day after and so on) and I want to apply a replication to estimate the option premium. My question is however what volatility i need to use to calculate the deltas for the hedging strategy with daily rebalancing and what would be the most feasible way to estimate it? Could I sort of estimate it for a product from the daily prices over a month?

• It is not entirely clear from the way you phrased the question what exactly you're trying to achieve. The way I understand your question is that you'd like to back-test delta hedging of an option, and you'd like to back-test on historical daily futures prices? Obviously, to delta hedge an option, you need to delta-hedge it with the underlying, so I assume the option itself is on the futures contract? What type of futures? Stocks, Rates, commodities...? Please specify your question a bit more if you'd someone to answer. – Jan Stuller Nov 21 '20 at 12:28
• @JanStuller it would be on power futures – Question Anxiety Nov 23 '20 at 13:35
• If you want to use the daily futures to estimate historical volatility, and use that as the volatility in your option pricing formula, you can do it, but the historical volatility is not a good estimator of the Implied Vol used in actual option pricing: you'd be better off taking an Implied Vol from a quoted option on these energy futures and use that Implied Vol in your $N(d_1)$ formula, which would be your delta. If you cannot get hold of the Implied Vol from quoted options on the energy futures, then you can just compute the Standard Deviation of the historical returns: – Jan Stuller Nov 23 '20 at 19:32
• $$\hat{\sigma}^2=\sum_{i=1}^{i=n}\frac{(x_i-\hat{\mu})^2}{n-1}, \hat{\mu}=\sum_{i=1}^{i=n}\frac{x_i}{n}$$ Above, $\hat{\sigma}$ would be your volatility estimate from the sample (so you'd need to take a square root of $\hat{\sigma}^2$). If you have 1 month worth of observations, I assume your $n$ will be about 20. Anyway, you can then take the computed $\hat{\sigma}$ and plug it into the Black-76 formula for $N(d_1)$, where: $$d_1=\frac{ln\left(\frac{S_0}{K}\right)+0.5\hat{\sigma}^2t}{\hat{\sigma}\sqrt{t}}$$ – Jan Stuller Nov 23 '20 at 19:42
• Choose an option that expires in one month (so $t=\frac{1}{12}$). Your energy future price on the first day in your 1-month sample can be selected as $S_0$. You can choose any strike you want. Your delta on your first day will be $N(d_1)$ as indicated above. On your second day, replace $S_0$ from the first day with a new $S_0$ from your second day. Also adjust $t$ as maturity shortens. Then recompute your $N(d_1)$ and that will be your new delta. Do this every day as you move forward in your one-month sample, and that way, you could "back-test" the delta-hedging process. Does this help? – Jan Stuller Nov 23 '20 at 19:45