Short Maturity Forward Start Asian Options in Local Volatility Models

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

We propose analytical approximations for the sensitivities (Greeks) of the Asian options in the Black–Scholes model, following from a small maturity/volatility approximation for the option prices which has the exact short maturity limit, obtained using large deviations theory. Numerical tests demonstrate good agreement of the proposed approximation with alternative numerical simulation results for cases of practical interest. We also study the qualitative properties of Asian Greeks, including new results for Rho, the sensitivity with respect to changes in the risk-free rate, and Psi, the sensitivity with respect to the dividend yield. In particular, we show that the Rho of a fixed-strike Asian option and the Psi of a floating-strike Asian option can change sign.

Because of its very general formulation, the local volatility model does not have an analytical solution for European options. In this article, we present a new methodology to derive closed form solutions for the price of any European options. The formula results from an asymptotic expansion, terms of which are Black-Scholes price and related Greeks. The accuracy of the formula depends on the payoff smoothness and it converges with very few terms.

Stochastic Local Volatility

In This Issue

Stochastic differential equation (SDE) models are the foundation for pricing and hedging financial derivatives. The drift and volatility functions in SDE models are typically chosen to be algebraic functions with a small number ( < 5 ) of parameters which can be calibrated to market data. A more flexible approach is to use neural networks to model the drift and volatility functions, which provides more degrees-of-freedom to match observed market data. Training of models requires optimizing over an SDE, which is computationally challenging. For European options, we develop a fast stochastic gradient descent (SGD) algorithm for training the neural network-SDE model. Our SGD algorithm uses two independent SDE paths to obtain an unbiased estimate of the direction of steepest descent. For American options, we optimize over the corresponding Kolmogorov partial differential equation (PDE). The neural network appears as coefficient functions in the PDE. Models are trained on large datasets (many contracts), requiring either large simulations (many Monte Carlo samples for the stock price paths) or large numbers of PDEs (a PDE must be solved for each contract). Numerical results are presented for real market data including S&P 500 index options, S&P 100 index options, and single-stock American options. The neural-network-based SDE models are compared against the Black-Scholes model, the Dupire's local volatility model, and the Heston model. Models are evaluated in terms of how accurate they are at pricing out-of-sample financial derivatives, which is a core task in derivative pricing at financial institutions. Specifically, we calibrate a neural network-SDE model to market data for a financial derivative on an asset with price S t with a payoff function g ( s ) , and we then evaluate its generalization accuracy for a financial derivative on the same asset S t but with a different payoff function f ( s ) . In addition to comparing out-of-sample pricing accuracy, we evaluate the hedging performance of the neural network-SDE model.

Static super-replicating strategies for a class of exotic options

The goal of this chapter is to give a concise account of the connection between Bessel processes and the integral of geometric Brownian motion. The latter appears in the pricing of Asian options. Bessel processes are defined and some of their properties are given. The known expressions for the probability density function of the integral of geometric Brownian motion are stated, and other related results are given, in particular the Geman and Yor (1993) Laplace transform for Asian option prices.

Calibrating and Pricing with Stochastic-Local Volatility Model

The averaging principle for non-autonomous slow-fast stochastic differential equations and an application to a local stochastic volatility model

Local Volatility Enhanced by a Jump to Default

A general framework is proposed for pricing both continuously and discretely monitored Asian options under one-dimensional Markov processes. For each type (continuously monitored or discretely monitored), we derive the double transform of the Asian option price in terms of the unique bounded solution to a related functional equation. In the special case of continuous-time Markov chain (CTMC), the functional equation reduces to a linear system that can be solved analytically via matrix inversion. Thus the Asian option prices under a one-dimensional Markov process can be obtained by first constructing a CTMC to approximate the targeted Markov process model, and then computing the Asian option prices under the approximate CTMC by numerically inverting the double transforms. Numerical experiments indicate that our pricing method is accurate and fast under popular Markov process models, including the CIR model, the CEV model, Merton’s jump diffusion model, the double-exponential jump diffusion model, the variance gamma model, and the CGMY model.

In this paper, we study the impact of introducing uncertainty volatility into Asian options pricing, with emphasis on the use of Hamilton-Jacobi-Bellman (HJB) equation. The traditional Asian option pricing model usually assumes that volatility is known and constant, but in the actual market, volatility is often uncertain and volatile. This paper first reviews the pricing theory of Asian options, and then introduces the hypothesis of uncertain volatility. By constructing the HJB equation based on uncertainty volatility, a new pricing method is proposed and verified by numerical simulation. The results show that after the introduction of uncertain volatility, the price range of Asian options expands significantly, reflecting higher market uncertainty and risk.

Calibrating stochastic local volatility (SLV) models to market implied volatility surfaces remains an inherently complex and ill-posed inverse problem, as it involves solving nonlinear PDEs under sparse and noisy data, while simultaneously ensuring consistency with the underlying risk-neutral probability density. We present a probability density-consistent PINNs (PDC-PINNs) that embeds the forward Kolmogorov Fokker Planck equation and the risk-neutral density constraints whose joint validity ensures the absence of static and dynamic arbitrage. A self-adaptive bilevel optimiser is developed to balance the PDE, data-fit, and density losses by rescaling them according to gradient magnitudes for stabilising learning. In synthetic and real foreign exchange datasets, PDC-PINNs match or exceed the accuracy of conventional calibrators while halving computational time through TPU vectorisation and producing reasonable surfaces that remain all constraints. The framework further enables consistent pricing of path-dependent options and generalises to other constrained financial PDE problems.