A local volatility model, in
mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
and
financial engineering, is an option pricing model that treats
volatility as a function of both the current asset level
and of time
. As such, it is a generalisation of the
Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black� ...
, where the volatility is a constant (i.e. a trivial function of
and
).
Formulation
In
mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
, the asset ''S''
''t'' that
underlies a
financial derivative
In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be u ...
is typically assumed to follow a
stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
of the form
:
,
under the risk neutral measure, where
is the instantaneous
risk free rate, giving an average local direction to the dynamics, and
is a
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
, representing the inflow of randomness into the dynamics. The amplitude of this randomness is measured by the instant volatility
. In the simplest model i.e. the
Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black� ...
,
is assumed to be constant; in reality, the realised volatility of an underlying actually varies with time.
When such volatility has a randomness of its own—often described by a different equation driven by a different ''W''—the model above is called a
stochastic volatility model. And when such volatility is merely a function of the current asset level ''S''
''t'' and of time ''t'', we have a local volatility model. The local volatility model is a useful simplification of the
stochastic volatility model.
"Local volatility" is thus a term used in
quantitative finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
to denote the set of diffusion coefficients,
, that are consistent with market prices for all options on a given underlying. This model is used to calculate
exotic option
In finance, an exotic option is an option which has features making it more complex than commonly traded vanilla options. Like the more general exotic derivatives they may have several triggers relating to determination of payoff. An exotic op ...
valuations which are consistent with observed prices of
vanilla options.
Development
The concept of a local volatility was developed when
Bruno Dupire
Bruno Dupire (born 1958) is a researcher and lecturer in quantitative finance. He is currently Head of Quantitative Research at Bloomberg LP. He is best known for his contributions to local volatility modeling and Functional Itô Calculus. He is ...
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
In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" ( lattice based) model of the varying price over time of the underlying f ...
. 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
Bruno Dupire (born 1958) is a researcher and lecturer in quantitative finance. He is currently Head of Quantitative Research at Bloomberg LP. He is best known for his contributions to local volatility modeling and Functional Itô Calculus. He is ...
in 1994. Dupire's equation states
:
In order to compute the partial derivatives, there exist few known parameterizations of the implied volatility surface based on the Heston model: Schönbucher, SVI and gSVI. Other techniques include mixture of lognormal distribution and stochastic collocation.
Derivation
Given the price of the asset
governed by the risk neutral SDE
:
The transition probability
conditional to
satisfies the forward Kolmogorov equation (also known as
Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, ...
)
:
Because of the
Martingale pricing theorem, the price of a call option with maturity
and strike
is
:
Differentiating the price of a call option with respect to
:
and replacing in the formula for the price of a call option and rearranging terms
:
Differentiating the price of a call option with respect to
twice
:
Differentiating the price of a call option with respect to
yields
:
using the Forward Kolmogorov equation
:
integrating by parts the first integral once and the second integral twice
:
using the formulas derived differentiating the price of a call option with respect to
:
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 , who claims that such models provide the best average hedge for equity index options. Local volatility models are nonetheless 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. The general non-parametric approach by Dupire is however problematic, as one needs to arbitrarily pre-interpolate the input implied
volatility surface before applying the method. Alternative parametric approaches have been proposed, notably the highly tractable mixture dynamical local volatility models by
Damiano Brigo and
Fabio Mercurio.
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 option A cliquet option or ratchet option is an exotic option consisting of a series of consecutive forward start options. The first is active immediately. The second becomes active when the first expires, etc. Each option is struck at-the-money when it ...
s or
forward start option
In finance, a forward start option is an option that starts at a specified future date with an expiration date set further in the future.
A forward start option starts at a specified date in the future; however, the premium is paid in advance, a ...
s, whose values depend specifically on the random nature of volatility itself.
References
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{{derivatives market, state=collapsed
Derivatives (finance)