Local Volatility

Given the prices of call or put options across all strikes and maturities, we may deduce the volatility which produces those prices via the full Black-Scholes equation (Dupire, 1994 and Derman and Kani, 1994).

• This function has come to be known as local volatility.

• Unlike the naive volatility produced by applying the Black-Scholes formulae to market prices, the local volatility is the volatility implied by the market prices and the one factor Black-Scholes.

In 1994, Dupire [3] showed that if the spot price follows a risk-neutral random walk of the form:

and if no-arbitrage market prices for European vanilla options are available for all strikes K and expiries T, then σL(K, T ) can be extracted analytically from these option prices.

Dupire Formulae

If C(S, t, K, T ) denotes the price of a European call with strike K and expiry T, we obtain Dupire’s famous equation:

If we rearrange this equation, we obtain the direct expression to calculate the local volatility (Dupire formulae):

(11)

Problems using Dupire formulae

One potential problem of using the Dupire formulae {11} is that, for some fi- nancial instruments, the option prices of different strikes and maturities are not available or are not enough to calculate the right local volatility. Another problem is that, for strikes far in- or out-the-money, the numerator and denominator of this equation may become very small, which could lead to numerical inaccuracies. Elder in [4] shows that if CBS(S, t, K, T, σI ) is the Black-Scholes value for a European call with strike K, expiry T and implied volatility σI, making this substitution in {11} gives us an alternative expression for local volatility in terms of the derivative of the implied volatility:

where:

If we construct a huge matrix (MxN) = (300x200) with option prices using the Black-Scholes formulae {10} with a given volatility = 0.232, varying the Maturity and Strikes, and if we calculate the local and implied volatilities with these option prices, we expect to obtain a flat volatility surface with the original value (0.232). However, this is not what happens using {11} (see figure 1.3).

Figure 1.3.- Comparison of using the local volatility formulas {11} & {12}.

If we increase the time steps in the finite difference approximations 10 times, the option price is in the money, and the expiry is greater than 3 months, the local volatility as a function of call prices can give some acceptable results (see figure 1.4). Otherwise we still obtain the results and behavior shown above.

Figure 1.4.- Error of using the local volatility formulas {11} & {12}.

In summary, calculating the local volatility with the implied volatility gives us a more accurate and stable result. Furthermore we can save huge amounts of computation time (30 times less). The last results tell us that a flat implied volatility surface automatically yields a flat local volatility surface.

Comments around the world

"Implied volatility is the wrong number to put into wrong formulae to obtain the correct price. Local volatility on the other hand has the distinct advantage of being logically consistent. It is a volatility function which produces, via the Black-Scholes equation, prices which agree with those of the exchange traded options".

Rebonato 1999

Prof. Klaus Schmitz