Volatility Models: Local volatility
In the local volatility models the security prices followed a nonlinear diffusion process with drift and volatility depending on the price level
and time
(3.1)
The model is complete but the conjecture of continuous trading is essentially for the local volatility models in view of the fact that the
corresponding discrete-time model is incomplete.
For a given maturity  and current stock price  , the call option price  is related to the
risk-neutral probability density function (“pdf”)  of the final spot price  through the rela-tionship:
(3.2)
By taking twice the derivative with respect to  , we obtain the Breeden – Litzenberger
formula for calculating  :
(3.3)
In the Black-Scholes framework, (3.3) can be calculated analytically and leads to the lognormal den-sity. In general, based on call option
prices, it is possible to derive an implied density by using (3.3) directly.
The pdf  evolves according to the Fokker-Planck equation:
(3.4)
Application of Itô’s lemma to the call option price and the self-financing assumption gives rise to a partial differential equation
(PDE) for which is a straightforward generalization of the famous Black-Scholes PDE
(3.5)
Rearranging this we find that:
(3.6)
As the right-hand side of (3.6) can be computed if we assume a continuum of call option prices, this implies that under this assumption,
the local volatility is given uniquely by the analytical form
(3.6). This formula was derived by Dupire.
We can view (3.6) as a definition of the local volatility function regardless of what kind of process (stochastic volatility for example)
actually governs the evolution of volatility. But if volatility is a de-terministic function of stock price and time, then the model is
complete: there is only one source of risk, the stock price itself, which can be dynamically hedged the same way as in the
Black-Scholes model. Market completeness and uniqueness of local volatility surface are the two crucial implications of this model.
This calculation of the volatility surface from option prices worked because of the particular form of the payoff, the call payoff, which
allowed us to derive the very simple relationship between derivatives of the option price and the transition probability density function.
In practice, however, there are only a finite number of liquid European call options in the market, and determining the local volatility
surface can be regarded as a function approximation problem from a finite data set with a nonlinear observation function. This is a
well-known ill-posed problem: there are typically an infinite number of solutions to the problem. So local volatility models are a
theoretical dream and a numerical nightmare.
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