In a stochastic volatility world where the volatility is bound to stay below a certain level, Nicole El Karoui1 has shown that it is possible to delta hedge a convex position, such as a call option, with the Black Scholes model and make sure that the profit and loss from this strategy is almost surely positive. This gives a non-competitive upper replication price.
In reality, practitioners capture the fair price of options by not only delta hedging but also vega hedging. In this setting we derive estimation for the fair price of a delta/vega hedging strategy basing our approach on the foundations of pricing and hedging derivatives.
The New PDE
We make all the usual relevant assumptions about the financial market--no restrictions on short sales, no commissions, no taxes and frictions, etc--and we shall attempt to hedge a plain-vanilla option of expiry T.
Since the volatility is strongly linked to the movements of the spot as it is shown in statistical studies, we shall follow a Shadow Delta Hedging strategy, which consists of adjusting the Black and Scholes delta while the volatility is changing with the spot.
The main goal of the trader is to construct a strategy that will realise an expected profit and loss equal to zero. This will constitute the fair price. The option price has then to satisfy the following equation:
Where loc is the local volatility developed by Bruno Dupire2, which represents the coefficient of the local diffusion of the spot.
We not only recognise the usual Greeks for the Black Scholes equation but also less usual risk ratios that are equally important.
For the usual Greeks, we have the and the :
* A positive gamma position, >0, allows the holder to buy
the stock after downticks and to sell after up-ticks.
* A negative theta position, <0 means the holder is losing
time value.
For the other Greeks, Vanna and Volga, they represent the change of the usual important greeks delta and vega when the volatility is changing:
* A long Volga position, >0, allows the holder to buy
volatility after downticks and to sell after up-ticks.
* A short Vanna position, <0, allows the holder to sell
stock after downticks in volatility--and sell vol after
downticks in spot.
The equations behind and ß are derived from writing the expected P&L equal to zero under the historical probability. Therefore they represent risk measures that have to be taken into account in the practical use of the model.
* measures the correlation of the spot and the volatility
and therefore induces the skew observed in the market.
* ß measures the volatility of the volatility and therefore has
an impact on the wings.
Implying The Black Scholes Volatility
The previous price differential equation gives the price of a call/put that is consistent with the hedging strategy--delta and vega--and the statistical movements of the volatility when spot moves.
What we finally care about is to write this result in light with the Black Scholes model. Solving numerically the previous equation produces the new volatility and by the Black Scholes model we have the value of the price of the vanilla options.
It is however very interesting to derive approximations for this implied volatility in a simplified framework to have a better understanding of the method.
A First Approximation Of The Fair Volatility
In this section, we derive an approximation of the implied volatility in the previous context with another assumption of small parameters and ß .
These assumptions are quite natural if we think in a perturbative way, moving slightly from the log normal distribution to a skewed one.
A small expansion of the implied volatility gives the following result:
Where * and ß* are proportional respectively to and ß.1
This result is very intuitive and shows how the price of calls is being calibrated to take into account the skew and the convexity.
Equation 2 is remarkable because it gives a sound theoretical basis for some models of volatility that have already been used by traders in practice, for example in the foreign exchange market. Also it gives the limitations of this kind of formulae.
Conclusion
Never compute a price without having a clear idea of your hedging strategy. This approach reminds us of the importance of the hedging strategy in the computation of a price. It involves both the liquid observable as the current at-the-money spot--or ATM forward in the fx market--volatilities but also a statistical measure of risk from and ß that will constitute the weights for the Vanna and the Volga.
This methodology shown in the plain-vanilla market has to be extended to exotics where the risks are more difficult to understand, estimate and manage.
Recalling the different steps we have followed:
* choice of a hedging strategy and the set of liquid instruments;
* fair pricing condition and the derivation of statistical risk measures;
* reading the result under simpler model as the Black Scholes in the vanilla case.
We would like to stress the fact that this method extends to several underlyings, where the volatility now is represented by a matrix of variance covariance.
Last but not least, is the impact on optimal trading levels of volatility given the position and hedged position.
This week's Learning Curve was written by Adil Reghaï, co-head of the equity derivatives quantitative team, and Frederic Hatt, exotic trader, at Dresdner Kleinwort Wasserstein in London.
References
[1] N. El Karoui (1998) Robustness of the Black and Scholes
model, Mathematical Finance.
[2] B. Dupire (1994) Pricing with a smile, Risk.
The equations are available from the authors at readers' request.
