A curious feature of quantitative finance is the way pricing models are calibrated to the market. For bonds, one adds a Z-spread to the rates which matches market quotes. The Z-spread allows the model to incorporate the market’s views about credit default risk and future interest rates. For options priced using the Black-Scholes model, one prices using the implied volatilities that reproduce market prices. In this way, pricing models are really just interpolation tools to interpolate between the prices of products trading in the market in order to price different contracts with different parameters which are not currently represented in the market.

For options priced using the constant volatility Black-Scholes model, this creates the strange situation where options with different strikes and expiries are imagined to be based on underlying assets with different volatilities – even though they all have the same underlying which can only realise a single volatility.

Another issue concerns the pricing of more exotic derivatives like Asian options and barrier options. When adding a barrier to a European option, it’s no longer clear whether it’s appropriate to use implied volatilities backed out from European options. Unfortunately, barrier options may not themselves be sufficiently liquid in the market to generate their own implied volatilities (such an implied volatility surface would now have three dimensions – strike, expiry and barrier level). Furthermore, unlike vanilla options, barrier option valuation depends on volatility term structure, so that replacing a variable volatility with it’s constant average value isn’t appropriate. Similarly, for Asian options, valuation depends on the path taken by the underlying and by its volatility, not just on their average values.

We know that volatilities are not, in reality, constant. And the assumption that they are leads to difficulties in reproducing the market prices of European options, and meaningfully pricing less liquid exotic derivatives. Furthermore, empirical analysis of stock data may indicate that volatility has, statistically speaking, a dependence on the current underlying level. This naturally leads us to consider various volatility option pricing models where the volatility depends on both time and the underlying level.

## Local Volatility Models

In the local volatility model (LV), the constant volatility of Black-Scholes is replaced by an instantaneous volatility \(\sigma(S_t,t)\) which depends on both the current underlying value and time. The underlying evolves under the stochastic differential equation

\[dS_t = (r_t – d_t)S_t dt + \sigma(S_t,t) S_t dW_t,\]

where \(r\) is the risk free rate, \(d\) is the dividend yield, and \(W_t\) is a Wiener process (i.e. a brownian motion or random walk).

As long as the volatility surface is arbitrage free, local volatility models can be calibrated to exactly reproduce the market option prices and volatility smile at each pillar maturity trading in the market. Calibration can be done computationally efficiently using Dupire’s formula. Dupire’s formula gives the volatility \(\sigma(K,T)\) at some strike \(K\) and expiry \(T\), in terms of the price \(C(K,T)\) of a call option. The formula is

\[\sigma(K,T)^2 = \frac{\frac{\partial C}{\partial K} – (r-q)(C – K\frac{\partial C}{\partial K})}{\frac{1}{2}K^2\frac{\partial^2 C}{\partial K^2}},\]

where \(r\) is the risk free rate and \(q\) is the dividend yield. Slightly confusingly, this function seems to define the local volatility function \(\sigma(S,t)\) at a strike \(K\) instead of at some underlying level \(S\). However, the assumption is simply that, in order to calculate the local volatility at \(S\), you set \(K=S\) in this formula.

Thus, the problem of calibrating the model to market prices (i.e. obtaining the function \(\sigma(S,t)\)), is reduced to the problem of generating the function of call prices \(C(K,T)\), for every strike and expiry. This of course must be done starting with market prices for only a finite number of strikes and expiries. There are a variety of methods proposed for constructing an arbitrage free surface of call prices, and we refer to Rasmussen for a survey.

Once the local volatility function has been obtained, one proceeds to price derivatives numerically. For example, one can use Monte Carlo method to generate a large number of possible paths using the stochastic differential equation above. Alternatively, one might use PDE finite differencing methods to price the derivative.

A significant downside of local vol models is that \(dV/dS\) has the opposite from that observed in the market. When spot increases, a local vol model predicts that volatility should decrease, and vice versa. Since the observed market behaviour is the opposite, this makes the model unsuitable for hedging.. Also, smile flattens with maturity.

Another issue which arises when pricing certain types of derivatives, is the flattening of the forward smile. Local volatility models are fitted to the prices of European options. It’s known that the vol smile of European options becomes flatter for longer maturities (in other words, the implied vol is more constant, having less dependence on strike). This means that when you consider the forward volatility between \(T_1\) and \(T_2\), you find that it is also quite flat. This is not what’s observed in practice, where volatility smiles in the future are not in general less flat than those observed today.

To address these issues, we have stochastic volatility models.

## Stochastic Volatility Models

Local volatility models are determinist in that, given future values for \(t\) and \(S_t\), we know the volatility will be precisely that prescribed by the local volatility function \(\sigma(S,T)\). Stochastic volatility (SV) models allow the volatility to evolve according to its own stochastic differential equation with a random term, just like the underlying \(S\). Stochastic volatility models are of the form

\[dS_t = (r_t – d_t)S_t dt + \sigma_t S_t dW_t,\]

\[d \sigma_t^2 = \alpha(\sigma_t^2,t) dt + \beta(\sigma_t^2,t) dB_t,\]

where \(B\) is a second Wiener process, correlated to \(W\) with some correlation \(\rho\).

Stochastic volatility models are typically more challenging to calibrate to market data than a local volatility model. With few exceptions (such as the Hagen formula for the SABR model), stochastic volatility models are solved numerically, and calibrated to the market numerically. Standard optimization routines can be used to solve the model, check the level of agreement with market prices, adjust the model parameters, and repeat until a sufficiently close fit is obtained. For example, a least squares objective function may be used to minimise the difference between the prices generated by the model and those seen in the market.

A prominent example of a stochastic volatility model which is the market standard for vanilla interest rate derivatives, see our main article on the SABR model. Another stochastic volatility model, which is commonly used for equities, is the Heston model, which we discuss in the next section.

## Heston Model

The Heston model is a particular example of a stochastic volatility model which is popular for equity derivatives. The second equation above takes the form

\[d \sigma_t^2 = \theta (\omega – \sigma_t^2) dt + \varepsilon \sigma_t dB_t,\]

where \(\theta\), \(\omega\) and \(\varepsilon\) are constants. Note that the first term causes the variance to tend to mean revert to the value \(\omega\).

## Local Stochastic Volatility Models

Finally, we mention models which are a kind of hybrid between local volatility models and stochastic volatility models. These are sometimes known as “local stochastic volatililty” or LSV models. They aim to incorporate the best characteristics of both models. See for example the following papers: