Hedging

It is amusing to note that the academic literature essentially focuses on pricing
models, while what most practitioners need are effective hedging models. Many
people see model risk as the risk that a theoretical value produced by a model differs
from the value obtained in the market. But model risk is the potential loss that can
arise from the use of a particular model, and hedging is also a field that relies heavily
on mathematical models.
For hedging contingent claims, accurate forecasts of the volatility and correlation
parameters are the key issues. Such estimates are generally obtained by statistical
analysis of historical data or the use of implied volatility. In any case, the result is an
estimate or a confidence interval for the required parameter, and not its precise value.
The sampling variations of parameter estimates will induce sampling variation in the
estimated contingent-claims about their true values, and therefore, about the hedging
parameters. When there is uncertainty in the parameters, a model must be viewed as
a prediction model. Given the parameter’s estimates, it provides an answer.
Indeed, the effects of engaging in a hedging strategy using an incorrect value of the
volatility can be severe. As a simple example, the scatterplot of Figure 14.3 illustrates
what happens when a trader sells an at-the-money one-year call option on a stock
and engages in a daily hedging strategy to cover his position. The realized volatility
is 20%, while the hedger uses 10% in his own model. Each point corresponds to one
path out of the 1000 Monte Carlo simulations. It is very clear that the result of a
hedging strategy is a loss, whatever the terminal price of the stock.
Therefore, it may be tempting to resume the model risk in hedging to two numbers,
namely, the vega of the position and the expected volatility. For an option, we would
say, for instance, that the risk is larger for the short-term at-the-money option,
where the impact of volatility is very high. One has to remember that calculating the
vega of an option requires a pricing model, and that the calculation of volatility
requires another model – that is, two potential sources of model risk.
Very often, model risk is only examined in pricing, and hedging is considered as a
direct application of the pricing model. This is not necessarily true. Some decisions
may not affect pricing, but will have an impact on hedging. For instance, what
matters in the Black and Scholes model is in fact the ‘cumulative variance’. If we
carefully examine the pricing equation, we can see that everywhere it appears, the
variance (p2) is multiplied by the time to maturity (T-t). Therefore, as evidenced by
Merton (1973), if the volatility is not constant, but is a deterministic function of time
(póp(t)), we can still use the Black and Scholes formula to price a European option. 430