Narrow Results

We briefly review the statistical properties of the escape times, or hitting times, for stock price returns by using different models which describe the stock market evolution. We compare the probability function (PF) of these escape times with that obtained from real market data. Afterwards we analyze in detail the effect both of noise and different initial conditions on the escape time in a market model with stochastic volatility and a cubic nonlinearity. For this model, we compare the PF of the stock price returns, the PF of the volatility and the return correlation with the same statistical characteristics obtained from real market data.

In this study, we present a new stochastic volatility model incorporating a flocking mechanism between individual volatilities of assets. Collective phenomena of asset pricing and volatilities in financial markets are often observed; these phenomena are more apparent when the market is in critical situations (market crashes). In the classical Heston model, the constant theoretical mean of the square of the volatility was employed, which can be assumed a priori. Our proposed model does not assume this mean value a priori, we instead use the flocking effect to continuously update the theoretical mean value using the local weighted average of individual volatility values. To perform this function, we use the Cucker–Smale flocking mechanism to calculate the local mean. For some classes of interaction weights such as all-to-all and symmetric coupling with a positive lower bound, we show that the fluctuations of the square process of volatility are uniformly bounded, such that the overall dynamics are mainly dictated by the averaged process. We also provide several numerical examples showing the dynamics of volatility.

We give a simple proof that in applications there is no need to track the branch-cut of the complex logarithm in the Heston model when using the Lewis-Lipton formula in the right way.

In this paper, we will explain how to perfectly hedge under Heston's stochastic volatility model with jump-to-default, which is in itself a generalization of the Merton jump-to-default model and a special case of the Heston model with jumps. The hedging instruments we use to build the hedge will be as usual the stock and the bond, but also the Variance Swap (VS) and a Credit Default Swap (CDS). These instruments are very natural choices in this setting as the VS hedges against changes in the instantaneous variance rate, while the CDS protects against the occurrence of the default event.

First, we explain how to perfectly hedge a power payoff under the Heston model with jump-to-default. These theoretical payoffs play an important role later on in the hedging of payoffs which are more liquid in practice such as vanilla options. After showing how to hedge the power payoffs, we show how to hedge newly introduced Gamma payoffs and Dirac payoffs, before turning to the hedge for the vanillas. The approach is inspired by the Post–Widder formula for real inversion of Laplace transforms. Finally, we will also show how power payoffs can readily be used to approximate any payoff only depending on the value of the underlier at maturity. Here, the theory of orthogonal polynomials comes into play and the technique is illustrated by replicating the payoff of a vanilla call option.

We propose a parsimonious multi-asset Heston model and provide an easy-to-implement calibration algorithm. The model is customized to pricing multi-asset options in markets with liquidly traded single-asset options but no liquidly traded cross-asset options. In this situation, single-asset model parameters can be calibrated from option price data, however, cross-asset parameters cannot. We formulate a parsimonious model specification such that all single-asset models are Heston models, which are affine allowing for efficient calibration of the respective parameters. The single-asset models are correlated using cross-asset correlations only. Cross-asset correlations are observable, in contrast to correlations of latent variables such as volatilities, and serve as basis for calibration. A hybrid calibration approach for identifying the model parameters consistent with option price data and asset price data is outlined and illustrated by a case study. In banking practice the approach is referred to as correlation adjustment.

In this paper, we present a correction to Merton (1973)'s well-known classical case of pricing perpetual American put options by considering the same pricing problem under a stochastic volatility model with the assumption that the volatility is slowly varying. Two analytic formulae for the option price and the optimal exercise price of a perpetual American put option are derived, respectively. Upon comparing the results obtained from our analytic approximations with those calculated by a spectral collocation method, it is shown that our current approximation formulae provide fast and reasonably accurate numerical values of both option price and the optimal exercise price of a perpetual American put option, within the validity of the assumption we have made for the asymptotic expansion. We shall also show that the range of applicability of our formulae is remarkably wider than it was initially aimed for, after the original assumption on the order of the "volatility of volatility" being somewhat relaxed. Based on the newly-derived formulae, the quantitative effect of the stochastic volatility on the optimal exercise strategy of a perpetual American put option has also been discussed. A most noticeable and interesting result is that there is a special cut-off value for the spot variance, below which a perpetual American put option priced under the Heston model should be held longer than the case of the same option priced under the traditional Black-Scholes model, when the price of the underlying is falling.

We suggest a general scheme for improvement of FT-pricing formulas for European options and give efficient recommendations for the choice of the parameters of the numerical scheme, which allow for very accurate and fast calculations. The efficiency of the method stems from the properties of functions analytical in a strip, which were introduced to finance by Feng and Linetsky (2008). We demonstrate that an indiscriminate choice of parameters of a numerical scheme leads to an inaccurate pricing and calibration. As applications, we consider the Heston model and its generalization. For many parameter sets documented in empirical studies of financial markets, relative accuracy better than 0.01% can be achieved by summation of less than 10-20 terms even in the situations in which the standard approach requires more than 200. In some cases, the one-term formula produces an error of several percent, and the summation of two terms — less than 0.5%. Typically, 10 terms and fewer suffice to achieve the error tolerance of several percent and smaller.

In this article, we study the price monotonicity in the parameters of the Heston model for a contract with a convex pay-off function; in particular we consider European put options. We show that the price is increasing in the constant term in the drift of the variance process and decreasing in the coefficient of the linear term in the drift of variance process. We also show that the price is increasing in the correlation for small values of the stock and decreasing for the large values.

Large-time asymptotics are established for the SABR model with β = 1, ρ ≤ 0 and β < 1, ρ = 0. We also compute large-time asymptotics for the constant elasticity of variance (CEV) model in the large-time, fixed-strike regime and a new large-time, large-strike regime, and for the uncorrelated CEV-Heston model. Finally, we translate these results into a large-time estimates for implied volatility using the recent work of Gao and Lee (2011) and Tehranchi (2009).

In this paper, we introduce a new form of asymptotic arbitrage, which we call a partial asymptotic arbitrage, half-way between those of Föllmer & Schachermayer (2007) [Mathematics and Financial Economics1 (34), 213–249] and Kabanov & Kramkov (1998) [Finance and Stochastics2, 143–172]. In the context of the Heston model, we establish a precise link between the set of equivalent martingale measures, the ergodicity of the underlying variance process and this partial asymptotic arbitrage. In contrast to Föllmer & Schachermayer (2007) [Mathematics and Financial Economics1 (34), 213–249], our result does not assume a suitable condition on the stock price process to allow for (partial) asymptotic arbitrage.

In this paper, we present an innovative hybrid model for the valuation of equity options. Our approach includes stochastic volatility according to Heston (1993) [Review of Financial Studies6 (2), 327–343] and features a stochastic interest rate that follows a three-factor short rate model based on Hull and White (1994) [Journal of Derivatives2 (2), 37–48]. Our model is of affine structure, allows for correlations between the stock, the short rate and the volatility processes and can be fitted perfectly to the initial term structure. We determine the zero bond price formula and derive the analytic solution for European type options in terms of characteristic functions needed for fast calibration. We highlight the flexibility of our approach, by comparing the price and implied volatility surfaces of our model with the Heston model, where we in particular focus on the correlation structure. Our approach forms a comprehensive market model with an intuitive correlation structure that connects both the equity and interest market to the market volatility.

The degree of relationship between financial products and financial institutions, e.g. must be considered for pricing and hedging. Usually, for financial products modeled with the specification of a system of stochastic differential equations, the relationship is represented by correlated Brownian motions (BMs). For example, the BM of the asset price and the BM of the stochastic volatility in the Heston model correlates with a deterministic constant. However, market observations clearly indicate that financial quantities are correlated in a strongly nonlinear way, correlation behaves even stochastically and unpredictably. In this work, we extend the Heston model by imposing a stochastic correlation given by the Ornstein–Uhlenbeck and the Jacobi processes. By approximating nonaffine terms, we find the characteristic function in a closed-form which can be used for pricing purposes. Our numerical results and experiment on calibration to market data validate that incorporating stochastic correlations improves the performance of the Heston model.

New simulation approaches to evaluating path-dependent options without matrix inversion issues nor Euler bias are evaluated. They employ three main contributions: (1) stochastic approximation replaces regression in the LSM algorithm; (2) explicit weak solutions to stochastic differential equations are developed and applied to Heston model simulation; and (3) importance sampling expands these explicit solutions. The approach complements Heston [(1993) A closed-form solutions for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies6, 327–343] and Broadie & Kaya [(2006) Exact simulation of stochastic volatility and other affine jump diffusion processes, Operations Research54 (2), 217–231] by handling the case of path-dependence in the option’s execution strategy. Numeric comparison against standard Monte Carlo methods demonstrates up to two orders of magnitude speed improvement. The general ideas will extend beyond the important Heston setting.

In this paper, the pricing problem of variance and volatility swaps is discussed under a two-factor stochastic volatility model. This model can be treated as a two-factor Heston model with one factor following the CIR process and another characterized by a Markov chain, with the motivation originating from the popularity of the Heston model and the strong evidence of the existence of regime switching in real markets. Based on the derived forward characteristic function of the underlying price, analytical pricing formulae for variance and volatility swaps are presented, and numerical experiments are also conducted to compare swap prices calculated through our formulae and those obtained under the Heston model to show whether the introduction of the regime switching factor would lead to any significant difference.

Characteristic functions of several popular classes of distributions and processes admit analytic continuation into unions of strips and open coni around $ℝ\subset ℂ$. The Fourier transform techniques reduce calculation of probability distributions and option prices in the evaluation of integrals whose integrands are analytic in domains enjoying these properties. In the paper, we suggest to use changes of variables of the form $\xi =\sqrt{-1}{\omega }_1+bsinh(\sqrt{-1}\omega +y)$ and the simplified trapezoid rule to evaluate the integrals accurately and fast. We formulate the general scheme, and apply the scheme for calculation probability distributions and pricing European options in Lévy models, the Heston model, the CIR model, and a Lévy model with the CIR-subordinator. We outline applications to fast and accurate calibration procedures and Monte Carlo simulations in Lévy models, regime switching Lévy models that can account for stochastic drift, volatility and skewness, the Heston model, other affine models and quadratic term structure models. For calculation of quantiles in the tails using the Newton or bisection method, it suffices to precalculate several hundreds of values of the characteristic exponent at points on an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf.

The use of sequential Monte Carlo within simulation for path-dependent option pricing is proposed and evaluated. Recently, it was shown that explicit solutions and importance sampling are valuable for efficient simulation of spot price and volatility, especially for purposes of path-dependent option pricing. The resulting simulation algorithm is an analog to the weighted particle filtering algorithm that might be improved by resampling or branching. Indeed, some branching algorithms are shown herein to improve pricing performance substantially while some resampling algorithms are shown to be less suitable in certain cases. A historical property is given and explained as the distinguishing feature between the sequential Monte Carlo algorithms that work on path-dependent option pricing and those that do not. In particular, it is recommended to use the so-called effective particle branching algorithm within importance-sampling Monte Carlo methods for path-dependent option pricing. All recommendations are based upon numeric comparison of option pricing problems in the Heston model.

Convexity correction is a well-known approximation technique used in pricing volatility swaps and VIX futures. However, the accuracy of the technique itself and the validity condition of this approximation have hardly been addressed and discussed in the literature. This paper shows that, through both theoretical analysis and numerical examples, this type of approximations is not necessarily accurate and one should be very careful in using it. We also show that a better accuracy cannot be achieved by extending the convexity correction approximation from a second-order Taylor expansion to third-order or fourth-order Taylor expansions. We then analyze why and when it deteriorates, and provide a validity condition of applying the convexity correction approximation. Finally, we propose a new approximation, which is an extension of the convexity correction approximation, to achieve better accuracies.

We propose a new methodology to evaluate VIX derivatives. The approach is based on a closed-form Hermite series expansion, and can be applied to general stochastic volatility models. We exemplify the proposed method using the Heston model, the mean-reverting CEV model and the 3/2 model. Numerical results show that the proposed method is accurate and efficient.

The two most popular equity and FX derivatives pricing models in banking practice are the local volatility model and the Heston model. While the former has the appealing property that it can be calibrated exactly to any given set of arbitrage free European vanilla option prices, the latter delivers more realistic smile dynamics. In this paper, we combine both modeling approaches to the Heston stochastic local volatility model. We build upon a theoretical framework that has been already developed and focus on the numerical model calibration which requires special care in the treatment of mixed derivatives and in cases where the Feller condition is not met in the Heston model leading to a singular transition density at zero volatility. We propose a finite volume scheme to calibrate the model after a suitable transformation of the model equation and demonstrate its accuracy in numerical test cases using real market data.

In this paper, a novel algorithm for determining the free exercise boundary for high-dimensional Bermudan option problems is presented. First, a rough estimate of the boundary is constructed on a fine (daily) time grid. This rough estimate is used to generate a more accurate estimate on a coarse time grid (exercise opportunities). Antithetic branching is used to reduce the computational workload. The method is validated by comparing it with other methods of solving the standard Black–Scholes problem. Finally, the method is applied to two cases of Bermudan options with a second stochastic variable: a stochastic interest rate and a stochastic volatility.