Performance of Stop Loss Strategy

In section 19.3 of Hull (9th), it defines the performance of stop-loss strategy as "the performance measure is the ratio of the standard deviation of the cost of writing the option and hedging it to the theoretical price of the option."

I don't really understand how this definition work. What does it mean by "hedging it to the theoretical price of the option"? Can anyone help outline the steps to calculate this performance measure, please?

So after reading the above again, I think I understand what it tries to say. But it would be great if anyone can help confirm the following.

• Step 1 is to simulate N (a large round number) stock price path.
• Step 2 is to calculate the hedging cost for each path.
• Step 3 is to calculate the standard deviation of the hedging cost from step 2.
• Step 4 is to calculate the BSM price of the option (a single number).
• Step 5 is to take the ratio of step 3 over step 4.

Also what the point of such measure? Small standard deviation only says that the hedging cost is stable. But it can be a very large constant. Right? So this measure doesn't necessarily give us a good metric to measure the performance of the hedging strategy (at least not on its own). Right? What other common performance measures for hedging is used in practice?

 

Your steps look right.

The point of hedging is to reduce risk, and standard deviation of the strategy is one way to measure the risk involved. So here we're saying if the standard deviation is small it's a good hedge because the risk is small.

You can use other risk measures if you'd like: VaR, TVaR, ES, variance, semi-variance, etc. Standard deviation is commonly used.

You divide by the option price at t0 in order to standardize it. This means you can compare hedging performance more easily across different strikes and maturities. There are other ways of standardizing here, for example dividing by implied vol, vega, gamma or delta. Dividing by price is pretty standard.