Local volatility

Overview

A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility as a function of both the current asset level

S

t

{\displaystyle S_{t}}

and of time

t

{\displaystyle t}

. As such, it is a generalisation of the Black–Scholes model, where the volatility is a constant (i.e. a trivial function of

S

t

{\displaystyle S_{t}}

and

t

{\displaystyle t}

). Local volatility models are often compared with stochastic volatility models, where the instantaneous volatility is not just a function of the asset level

S

t

{\displaystyle S_{t}}

but depends also on a new "global" randomness coming from an additional random component.

Formulation

In mathematical finance, the asset St that underlies a financial derivative is typically assumed to follow a stochastic differential equation of the form

d

S

t

= (

r

t

−

d

t

)

S

t

d t +

σ

t

S

t

d

W

Development

The concept of a local volatility fully consistent with option markets was developed when Bruno Dupire and Emanuel Derman and Iraj Kani noted that there is a unique diffusion process consistent with the risk neutral densities derived from the market prices of European options. Derman and Kani described and implemented a local volatility function to model instantaneous volatility. They used this function at each node in a binomial options pricing model. The tree successfully produced option valuations consistent with all market prices across strikes and expirations. The Derman-Kani model was thus formulated with discrete time and stock-price steps. (Derman and Kani produced what is called an "implied binomial tree"; with Neil Chriss they extended this to an implied trinomial tree. The implied binomial tree fitting process was numerically unstable.) The key continuous-time equations used in local volatility models were developed by Bruno Dupire in 1994. Dupire's equation states

∂ C

∂ T

=

1 2

σ

2

( K , T ;

S

0

)

K

2

∂

2

C

∂

K

2

− ( r − d ) K

Derivation

Given the price of the asset

S

t

{\displaystyle S_{t}}

governed by the risk neutral SDE

d

S

t

= ( r − d )

S

t

d t + σ ( t ,

S

t

)

S

t

d

W

t

Parametric local volatility models

Dupire's approach is non-parametric. It requires to pre-interpolate the data to obtain a continuum of traded prices and the choice of a type of interpolation. As an alternative, one can formulate parametric local volatility models. A few examples are presented below.

Bachelier model

The Bachelier model has been inspired by Louis Bachelier's work in 1900. This model, at least for assets with zero drift, e.g. forward prices or forward interest rates under their forward measure, can be seen as a local volatility model

d

F

t

= v

d

W

t

{\displaystyle dF_{t}=v\,dW_{t}}

. In the Bachelier model the diffusion coefficient is a constant

v

{\displaystyle v}

, so we have

σ (

F

t

, t )

F

t

= v

Displaced diffusion model

This model was introduced by Mark Rubinstein. For a stock price, it follows the dynamics

d

S

t

= r

S

t

d t + σ (

S

t

− β

e

r t

)

d

W

t

{\displaystyle dS_{t}=rS_{t}\,dt+\sigma (S_{t}-\beta e^{rt})\,dW_{t}}

where for simplicity we assume zero dividend yield. The model can be obtained with a change of variable from a standard Black-Scholes model as follows. By setting

Y

CEV model

The constant elasticity of variance model (CEV) is a local volatility model where the stock dynamics is, under the risk neutral measure and assuming no dividends,

d

S

t

= r

S

t

d

t + σ

S

t

γ

d

W

t

,

{\displaystyle \mathrm {d} S_{t}=rS_{t}\mathrm {d} t+\sigma S_{t}^{\gamma }\mathrm {d} W_{t},}

for a constant interest rate r, a positive constant

σ > 0

{\displaystyle \sigma >0}

The lognormal mixture dynamics model

This model has been developed from 1998 to 2021 in several versions by Damiano Brigo, Fabio Mercurio and co-authors. Carol Alexander studied the short and long term smile effects. The starting point is the basic Black Scholes formula, coming from the risk neutral dynamics

d

S

t

= r

S

t

d t + σ

S

t

d

W

t

,

{\displaystyle dS_{t}=rS_{t}dt+\sigma S_{t}dW_{t},}

with constant deterministic volatility

σ

{\displaystyle \sigma }

and with lognormal probability density function denoted by

p

Use

Local volatility models are useful in any options market in which the underlying's volatility is predominantly a function of the level of the underlying, interest-rate derivatives for example. Time-invariant local volatilities are supposedly inconsistent with the dynamics of the equity index implied volatility surface, but see Crepey (2004), who claims that such models provide the best average hedge for equity index options, and note that models like the mixture dynamics allow for time dependent local volatilities, calibrating also the term structure of the smile. Local volatility models are also useful in the formulation of stochastic volatility models. Local volatility models have a number of attractive features. Because the only source of randomness is the stock price, local volatility models are easy to calibrate. Numerous calibration methods are developed to deal with the McKean-Vlasov processes including the most used particle and bin approach. Also, they lead to complete markets where hedging can be based only on the underlying asset. As hinted above, the general non-parametric approach by Dupire is problematic, as one needs to arbitrarily pre-interpolate the input implied volatility surface before applying the method. Alternative parametric approaches with a rich and sound parametrization, as the above tractable mixture dynamical local volatility models, can be an alternative. Since in local volatility models the volatility is a deterministic function of the random stock price, local volatility models are not very well used to price cliquet options or forward start options, whose values depend specifically on the random nature of volatility itself. In such cases, stochastic volatility models are preferred.