68 142 What data to use When it comes to estimation there is a radical

68 142 what data to use when it comes to estimation

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14.2 What data to use When it comes to estimation, there is a radical departure from the spirit of the Black-Scholes model both in practice and in the research literature. It has long been common practice in the industry to use implied volatility as a proxy for volatility. In other words, options are priced with the Black-Scholes formula, but using implied volatilities instead of historical volatilities. Market makers quote implied volatilities which may be generated from their own (non-Black-Scholes ) model, or via some projection from the previous day’s implied volatility surface. Other than the current stock price, no historical stock price data is used at all. Today’s options are priced using information from yesterday’s options data. The main reason for this procedure, which is of course mathematically inconsistent since volatility is assumed constant in the derivation of the Black-Scholes formula, is the belief that implied volatilities are better predictors of future realized volatility. Empirical studies disagree as to whether this is borne out by the data. (The other major use of the Black- Scholes formulas is for hedging large books of derivatives. Different securities are combined to reduce local sensitivities, measured by the Black-Scholes Greeks, to sudden changes in the stock price, volatility and so on). In tests of new models in the academic literature, calibration of parameters is also usually from derivative data (traded call option prices, for example), ignoring the history of the stock price. This is usually because the latter is easier to do compared to econometric methods for time series, especially when there is a formula for call option prices to fit to. Once the parameters have been estimated, the model can be used to price other derivatives in a consistent (no-arbitrage) manner, for example by simulation. Now some derivative data, most often around-the-money call option prices, are part of the basic data. The market in around-the-money calls (and puts) is liquid enough for this to be a valid procedure (we can trust the data). 14.3 Local Volatility Surface One popular way to modify the lognormal model is to suppose that volatility is a function of time and stock price: σ = σ ( t, S t ), called the local volatility surface. The SDE modeling the stock price is dS t = μS t dt + σ ( t, S t ) S t dW t , and the function C ( t, S ) giving the no-arbitrage price of a European derivative security at time t when the asset price S t = S then satisfies the generalized Black-Scholes PDE ∂C ∂t + 1 2 σ 2 ( t, S ) S 2 2 C ∂S 2 + r parenleftbigg S ∂C ∂S C parenrightbigg = 0 , the derivation being identical to the constant σ case, with σ ( t, S ) appearing instead of constant σ when we use Itˆo’s formula. The terminal condition is the payoff function. The hedging ratio is given by the delta of the solution to this PDE problem, ∂C/∂S , and a perfect hedge is achieved by holding this amount of stock. This is because the randomness of the volatility was introduced as a function of the existing randomness of the lognormal 69
model. This is not the case if the volatility is modeled to have a random component of its

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• Fall '11
• COULON
• Stochastic volatility, Smile Curve

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