Dupire Formula In Terms Of Implied Volatility

[Alana Fekety]

QuestionIf I compare the graphs in the paper of the implied vol surface and the local vol surface why is it so different? The local vol should be consistent with the liquid option prices. i.e. Term 1.0, 550 level: implied surface 13.5% vol, local vol surface 18% vol.If there is a liquid strike in the market they should have the same vol, am i right?

You should not expect the local vol to be equal to the implied vol except in the trivial case where both are constant (Black-Scholes model). I haven't read the Derman articles but it is quite clear using Dupire's formula (see Gatheral's book for example).

Local volatility can be computed in terms of call prices using Dupire's formula$$ \sigma^2(T,K) = \frac\frac\partial C\partial T + (r - q)K \frac\partial C\partial K + qC \frac12 K^2 \frac\partial^2C\partial K^2$$To get the relationship with implied volatility, it is better to think in terms of the log-moneyness forward $y = \ln(K/F_0^T)$ rather than strike. Writing $w(T,y) = T\Sigma^2(T,y)$ for the total implied variance, the Black-Scholes formula reads$$ C(T,K) = C_BS(T,K,\Sigma(T,\ln(K/F)),r,q) = S_0 \left( N(-\fracy\sqrtw + \frac12\sqrtw) - e^y N(-\fracy\sqrtw - \frac12\sqrtw) \right)$$Plugging it into the Dupire formula, one gets

$$\sigma_\mathrmDup(T,K)^2 = \frac \frac\partial w\partial T 1 - \fracyw \frac\partial w\partial y+ \frac14\left( - \frac14 + \frac1w + \fracy^2w^2 \right) \left(\frac\partial w\partial y\right)^2 + \frac12\frac\partial^2 w\partial y^2 $$

No skew: in this case, $\sigma_\mathrmDup(T)^2 = \frac\partial w\partial T = \Sigma(T)^2 + 2T\Sigma\frac\partial \Sigma\partial T$. Local vol is already different from implied vol unless they are both constant (Black-Scholes model).

Adding as answer as I don't have enough reputation to comment -- there is a typo in AFK's local vol formula, it should be:$$\sigma_\mathrmDup(T,K)^2 = \frac \frac\partial w\partial T 1 - \fracyw \frac\partial w\partial y+ \frac14\left( - \frac14 - \frac1w + \fracy^2w^2 \right) \left(\frac\partial w\partial y\right)^2 + \frac12\frac\partial^2 w\partial y^2 $$see Gatheral's "The Volatility Surface", p.13, eq. (1.10) (i.e., the sign of $\frac1w$ in the denominator is wrong).

One interesting fact that explains a lot at least to me is that implied vol has to do with only one option price. (One data point) Whereas local volatility is not determined from one unique call price in particular, but rather by many data points. To make things simple consider 4 neighbor data points in the surface. From this four is derived one local vol. The idea here is that it has to do with how we get from one call price to another so the local vol doesn't belong to a particular datapoint.

Dominic Brecher, a Quantitative Analyst at FINCAD, has a PhD in high-energy theoretical physics from the University of Cambridge. In addition, he has conducted postdoctoral research in String Theory at the Universities of Durham and British Columbia. With several papers published in the Journal of High Energy Physics, Nuclear Physics and Physical Review, he has recently brought his extensive research experience in physics to the world of financial engineering. Combining theoretical requirements and industry practice, Dominic designs derivative pricing models and explores their consequences in the financial markets. In this article, he explains what he has learned about the theory of local volatility in option pricing, and discusses its practical implementation in FINCAD Analytics Suite and FINCAD Analytics Suite Developer.

This will include functions to price European and American options, option strategies, portfolios of options, and options with arbitrary piecewise linear payoffs, from the smile as described in this article. All numerical pricing is done through the no-arbitrage PDE. Other functions start instead with a given local volatility process defined either by standard parameterizations (such as the CEV, normal and shifted lognormal processes), or a user-defined data table. These functions also price European and American options, option strategies, portfolios of options, and options with arbitrary piecewise linear payoffs, and can be used to back out the corresponding implied volatility smile. Using a combination of the two sets of functions, further issues such as the corresponding implied volatility smile. Using a combination of the two sets of functions, further issues such as the smile dynamics can be analyzed. Starting with either a given local volatility function, or with a given implied volatility smile, the interplay between these functions is shown below.

Despite its many deficiencies, the Black-Scholes model of option pricing [1] remains widely used, some 30 years after its inception. It is still the first model that both quants and traders reach for when given a deal to price. Quants like it because it provides a theoretically consistent framework to price options in almost any asset class, given some (possibly dubious) assumptions. Traders like it because of the tractability of the Black-Scholes formula for option prices, and because it is easily tweaked to account for their intuition about the market.

The Black-Scholes formula allows the fair price of a European option to be determined, given various observables - current asset price, strike price, time till maturity, risk-free rate and dividend yield - and a single unobservable constant - the volatility, σ, of the underlying asset price. Historical volatility is easily determined, though this is not necessarily a good guide as to how the future asset price will change. One is ultimately left with making a (more or less) educated guess as to the magnitude of σ, the likely average future volatility.

Traders have turned this drawback of the Black-Scholes model into a useful feature, by quoting European option prices not in terms of their dollar value, but in terms of the equivalent implied volatility. This is the value of σ which must be used in the Black-Scholes formula to give back the market price of that specific option. If the market price of some call option is C, then the implied volatility (a function of asset price S, strike K and maturity T) is defined through the relationship

It is thus always possible to determine the market's view of the future volatility of some asset from quotes for options written on that asset: it is just the implied volatility at which the options trade.

The ability to quote option prices in terms of a constant volatility is partly why the Black-Scholes model is still so widely used in the market. However, it makes the most important deficiency of the Black-Scholes framework transparent: the market-observed volatility is simply not a constant.

At least since the crash of 1987, the market-observed implied volatilities of European equity options have exhibited a distinct "skew" structure. Deep out-of-the-money puts generally trade at higher implied volatilities than out-of-themoney calls. An example - for options on the S&P 500 index - is shown in Figure 1. One reason is that this reflects the market's appetite for insurance against stock prices falling. Options on interest rates - caps, floors, swaptions - also generally exhibit such a volatility skew. Other options exhibit a volatility "smile", in which both deep out-of-the-money calls and puts trade at higher implied volatilities than their at-the-money counterparts.

We refer to the variation of implied volatility with strike price as the "volatility smile" despite the fact that a skew, frown, smirk or other such structures are often seen in the market. The implied volatility of traded options also varies with the option maturity - the volatility term structure - and so one often talks of an implied volatility surface: σ = σ(K, T) is a function of two variables. Points on the implied volatility surface for options on the S&P 500 index are shown in Figure 2.

It is easy to modify the Black-Scholes framework to deal with a volatility term structure, but the model simply does not allow for volatilities to vary with strike price. Once one recognizes this it becomes clear that, in the words of Rebonato, implied volatility is really only the "... wrong number to put in the wrong formula to obtain the right price of plain vanilla options" [2].

Arguably the most important problem in the theory of option pricing is how to take account of the volatility smile. This is far from a purely academic exercise. Given the skew shown in Figure 1, for example, what would be the correct volatility to use to price a European option struck at 1295? Whilst fairly easy to answer (one would use some form of interpolation on the volatility smile), it is less clear what volatility to use to price the corresponding American option, and much, much less clear what value to use to price an exotic contract, say a knock-in barrier option.

The real motivation for considering the effects of the volatility smile in option pricing is precisely this: to calibrate one's pricing of exotic, or even just American, options to the market-observed prices of European plain vanillas. Various models have been developed over the years to go beyond Black-Scholes in this way. They include:

The transition probability function3 associated with the stochastic process (2) satisfies both the backward (Kolmogorov) and forward (Fokker-Planck) equations. The noarbitrage equation (3) is essentially the backward equation. As shown by Dupire [7] (see also [8] and [12]), the local volatility function can in principle be determined from the forward equation. From the latter, one can derive

Both equations (3) and (4) apply in complete generality, to any option written on the asset S. The local volatility function can now be derived by specializing to European plain vanillas, say call options with prices C. Applying equation (4) to a complete set of such options (for all strikes and maturities) and upon re-arranging, we get Dupire's equation

3a8082e126