Asymptotics and calibration of local volatility models

@article{Berestycki2002AsymptoticsAC,
  title={Asymptotics and calibration of local volatility models},
  author={Henri Berestycki and J{\'e}r{\^o}me Busca and Igor Florent},
  journal={Quantitative Finance},
  year={2002},
  volume={2},
  pages={61 - 69}
}
Abstract We derive a direct link between local and implied volatilities in the form of a quasilinear degenerate parabolic partial differential equation. Using this equation we establish closed-form asymptotic formulae for the implied volatility near expiry as well as for deep in- and out-of-the-money options. This in turn leads us to propose a new formulation near expiry of the calibration problem for the local volatility model, which we show to be well posed. 

ASYMPTOTICS OF IMPLIED VOLATILITY IN LOCAL VOLATILITY MODELS

Using an expansion of the transition density function of a one‐dimensional time inhomogeneous diffusion, we obtain the first‐ and second‐order terms in the short time asymptotics of European call

The regularized implied local volatility equations-A new model to recover the volatility of underlying asset fromobserved market option price

In this paper, we propose a new continuous time model to recover the volatility of underlying asset from observed market European option price. The model is a couple of fully nonlinear parabolic

Calibration of Local Volatility Using the Local and Implied Instantaneous Variance

We document the calibration of the local volatility in terms of local and implied instantaneous variances; we first explore the theoretical properties of the method for a particular class of

Local Volatility from American Options

In this paper, we focus on short-time asymptotics for American options in the case of local and stochastic volatility models. As a byproduct, we obtain:(1) an efficient algorithm for calibrating

Implied Volatility at Expiration

The main result of the paper is a formula for zero time-to-maturity limit of implied volatilities of European options under a broad class of stochastic volatility models. Based on this formula, we

Explicit Implied Volatilities for Multifactor Local-Stochastic Volatility Models

We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices

Implied Volatility from Local Volatility: A Path Integral Approach

Assuming local volatility, we derive an exact Brownian bridge representation for the transition density; an exact expression for the transition density in terms of a path integral then follows. By

Asymptotic Methods for Computing Implied Volatilities Under Stochastic Volatility

In this paper we propose an analytical formula for computing implied volatilities of European options based on their short term asymptotics. The analysis is performed in a general framework with

ON THE RELATIONSHIP BETWEEN THE CALL PRICE SURFACE AND THE IMPLIED VOLATILITY SURFACE CLOSE TO EXPIRY

We examine the asymptotic behaviour of the call price surface and the associated Black-Scholes implied volatility surface in the small time to expiry limit under the condition of no arbitrage. In the

MATURITY ASYMPTOTICS FOR A FAST MEAN-REVERTING HESTON STOCHASTIC VOLATILITY MODEL

In this paper, we study the Heston stochastic volatility model in the regime where the maturity is small but large compared to the mean-reversion time of the stochastic volatility factor. We derive a
...

References

SHOWING 1-10 OF 43 REFERENCES

TOPICAL REVIEW: Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets

Market prices of financial derivatives such as options are directly observable. This information can be used to recover an unobservable local volatility function for the underlying stochastic

Implied and local volatilities under stochastic volatility

For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices,

The inverse problem of option pricing

Valuation of options and other financial derivatives critically depends on the underlying stochastic process specified for a particular market. An inverse problem of option pricing is to determine

A technique for calibrating derivative security pricing models: numerical solution of an inverse problem

A technique is presented for calibrating derivative security pricing models with respect to observed market prices. This technique can be applied in a very general multifactor setting where model

Computation of Deterministic Volatility Surfaces

The 'volatility smile' is one of the well-known biases of Black-Scholes models for pricing options. In this paper, we introduce a robust method of reducing this bias by pricing subject to a

Calibrating Volatility Surfaces Via Relative-Entropy Minimization

TLDR
The algorithm yields an arbitrage-free diffusion process that minimizes the Kullback-Leibler relative entropy distance to a prior diffusion, which can be used to interpolate, both in strike and expiration date, between implied volatilities of traded options and to price exotics.

Volatility and Correlation: In the Pricing of Equity, FX, and Interest-Rate Options

FOUNDATIONS. Volatility: Fundamental Concepts and Definitions. Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds. Instantaneous and Terminal Correlations. DEALING WITH SMILES.

Martingale Methods in Financial Modelling

Spot and Futures Markets.- An Introduction to Financial Derivatives.- Discrete-time Security Markets.- Benchmark Models in Continuous Time.- Foreign Market Derivatives.- American Options.- Exotic

Equivalent Black volatilities

We consider European calls and puts on an asset whose forward price F(t) obeys dF(t)=α(t)A(F)dW(t,) under the forward measure. By using singular perturbation techniques, we obtain explicit algebraic