Let's say trader is long a one year ATM call option on an index at 1,000 points (assume that the current spot for the index is also at 1,000 points). The strike prices of the index (which is listed and traded) has a spacing of 100 points. The one year (implied) volatility at strike 1,000 is 15% and at strike 1,100 is 17%. Thus there is a volatility smile.

The trader happily uses a Black-Scholes delta hedging for his position and ignores the smile (or is unaware of how to factor it into his hedging). He finds out that even though he did delta hedge his position according to Black-Scholes model he lost money every time the spot moved by 10 points. Why?

The Black-Scholes delta is given by:

............(1)

In the above, the differential is with respect to the spot with the assumption - and this is the big assumption in BS model - that all the parameters, except for the spot (asset/underlying) within the parenthesis, including volatility, is constant. Now if there is smile (skew) in the market, as we see in the above example, then it means that different strikes will have different implied volatilities. And hence the value of sigma inside the parenthesis will not be constant. Thus the differential will become a partial differential with the delta being the total differential as shown below:

..........(2)

The above equation simply follows from the rules of calculus when a function that is being differentiated with respect to a variable has two underlying variables in its argument. However, the above can also be intuitively explained by the dynamics of the smile. The smile implies that the implied volatility will vary with respect to strike (which can be approximated by the underlying) and due to this the value of the call itself will vary with the volatility; hence the product of two derivatives. The second term in the above equation (2) is called the hedging error; the quantity that will modify the theoretical value of the Black-Scholes delta due to varying volatility (with strike) in the market.

Anyway, coming back to the example mentioned above let's see how the trader will lose out if he ignores the hedging error.

Calculation of the hedging error in our example (assuming low interest rates) using equation (2) above:

Therefore, the product of the two derivatives above gives us the value of the hedging error in index points:

points

Thus for a 10 point move ( ) in the index the trader will lose (due to imperfect hedge) 0.797 points. And how much does he make from a perfect (Black-Scholes) hedge?

Now, the profit that the trader will earn from perfect (Black-Scholes) delta hedg when the index moves by 10 points will be his gamma profit and will be given by:

points

Therefore, you can see that the profit he will earn due to perfect (Black-Scholes) delta hedge will be completely eaten up by the losses from the hedging error that will be there in his trade due to a volatility smile.

Even today, after almost two decades since the phenomenon of smile was first observed and with so much research, talk, training, practice and guidance from the top many junior traders fall victim to this problem when they first start trading. If anything the problem has gotten much more complex.

*Reference: **For more details on the above see Emanuel Derman's paper. *

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