Jin E. Zhang

A new well-posed algorithm to recover implied local volatility

Abstract This paper presents a new algorithm to calibrate the option pricing model, i.e. the algorithm that recovers the implied local volatility function from market option prices in the optimal control framework. A unique optimal control is shown to exist. Our algorithm is well-posed. Our numerical experiments show that, with the help of the techniques developed in the field of optimal control, the local volatility function is recovered very well.

The implied volatility smirk

This paper provides an industry standard on how to quantify the shape of the implied volatility smirk in the equity index options market. Our local expansion method uses a second-order polynomial to describe the implied volatility–moneyness function and relates the coefficients of the polynomial to the properties of the implied risk-neutral distribution of the equity index return. We present a formal, two-way representation of the link between the level, slope and curvature of the implied volatility smirk and the risk-neutral standard deviation, skewness and excess kurtosis. We then propose a new semi-analytical method to calibrate option-pricing models based on the quantified implied volatility smirk, and investigate the applicability of two option-pricing models.

Equilibrium Asset and Option Pricing Under Jump Diffusion

This paper develops an equilibrium asset and option pricing model in a production economy under jump diffusion. The model provides analytical formulas for an equity premium and a more general pricing kernel that links the physical and risk-neutral densities. The model explains the two empirical phenomena of the negative variance risk premium and implied volatility smirk if market crashes are expected. Model estimation with the S&P 500 index from 1985 to 2005 shows that jump size is indeed negative and the risk aversion coefficient has a reasonable value when taking the jump into account.

Bidirectional soliton solutions of the classical Boussinesq system and AKNS system

Abstract In this letter, an alternative method is used to generate the bidirectional soliton solutions of the classical Boussinesq system. We point out that the classical Boussinesq system is gauge equivalent to a member of the AKNS system, give a general formula of the multi-soliton solutions by using Darboux transformation, and present some examples for different cases.

Pricing and Hedging American Options Analytically: A Perturbation Method

This paper studies the critical stock price of American options with continuous dividend yield. We solve the integral equation and derive a new analytical formula in a series form for the critical stock price. American options can be priced and hedged analytically with the help of our critical-stock-price formula. Numerical tests show that our formula gives very accurate prices. With the error well controlled, our formula is now ready for traders to use in pricing and hedging the S&P 100 index options and for the Chicago Board Options Exchange to use in computing the VXO volatility index.

The Relation Between Physical and Risk-Neutral Cumulants

Variance swaps are natural instruments for investors taking directional bets on volatility and are often used for portfolio protection. But the crucial observation suggests that derivative professionals may desire to hedge beyond volatility risk and there exists the need to hedge higher-moment market risks, such as skewness and kurtosis risks. We propose new derivative contracts: skewness swap and kurtosis swap, which trade the forward realized third and fourth cumulants. Using S&P 500 index options data from 1996 to 2005, we document the returns of these swap contracts, i.e., skewness risk premium and kurtosis risk premium. We find that the skewness risk premium is significantly negative and kurtosis risk premium for 90 day maturity is significantly positive.

Coastal hydrodynamics of ocean waves on beach

Publisher Summary This chapter describes the coastal hydrodynamics of ocean waves on beach. A comprehensive study on modeling three-dimensional ocean waves coming from an open ocean of uniform depth and obliquely incident on beach with arbitrary offshore slope distribution, while evolving under balanced effects of nonlinearity and dispersion is presented. A family of beach configurations that is uniform in the long-shore direction as a first approximation for beaches with negligible long-shore curvature is considered. The beach slope variation is assumed to have such distributions that the ocean waves will evolve on beach without breaking. The overall approach adopted begins with development of a three-dimensional linear shallow-water wave theory, followed by taking, step by step, the nonlinear and dispersive effects into account. The linear theory is shown to provide a fundamental solution involving a central function, called the beach-wave function that delineates the evolution of the incoming train of simple waves during interaction with any beach belonging to this broad family of beach configurations. This linear theory can easily afford to cover such factors as oblique wave incidence, arbitrary distribution of offshore beach slope, and wavelength variations with respect to beach breadth.

Option pricing with Weyl-Titchmarsh theory

In the Black–Merton–Scholes framework, the price of an underlying asset is assumed to follow a pure diffusion process. No-arbitrage theory shows that the price of an option contract written on the asset can be determined by solving a linear diffusion equation with variable coefficients. Applying the separating variable method, the problem of option pricing under state-dependent deterministic volatility can be transformed into a Schrödinger spectral problem, which has been well studied in quantum mechanics. With Weyl–Titchmarsh theory, we are able to determine the boundary condition and the nature of the eigenvalues and eigenfunctions. The solution can be written analytically in a Stieltjes integral. A few case studies demonstrate that a new analytical option pricing formula can be produced with our method.

Expected Stock Returns and Forward Variance

Bakshi, Panayotov, and Skoulakis (2011) show that forward variances are predictive of real economic activity and asset returns. In this paper, we study this relation by using CBOE VIX term structure data between January 1992 and August 2009. We find that certain combinations of the 3-, 6-, and 9-month forward variances (single forward variance factor) are predictive of stock market returns at 1-, 3-, and 6-month horizons. Forward variances constructed from seven out of nine sectors are also predictive of market returns and growth in measures of real economic activity. Out-of-sample analysis confirms the prediction power of the single forward variance factor.

New analytical option pricing models with Weyl--Titchmarsh theory

where r is the risk-free rate, q the dividend yield, the volatility (St) is a general function of St, and Bt is a standard Brownian motion. Our purpose is to search for those volatility functions that give closed-form option pricing formulas. By closed form, we mean that the solution can be written in an analytical form, such as a summation or an integration of some special functions. The method proposed in this paper is based on eigenfunction expansion, which has been applied in continuous-time finance by Lewis (1998) and Davydov and Linetsky (2003). Other related literature include Carr et al. (1999, 2002) and Haven (2005). A more general problem of pricing options on one-dimensional diffusions with non-zero drift has been analysed by Linetsky (2004). Henry-Labordère (2005, 2008) provides some new solvable option pricing formulas under local and stochastic volatility models using supersymmetric methods. Option pricing using the Lie group analysis approach suggested by Lo and Hui (2001) and Yam and Yang (2006) can be regarded as a macroscopic theory. Its implementation needs more concrete computations and depends on the structure of the underlying undetermined functions. Weyl–Titchmarsh theory, presented by Li and Zhang (2004) and this paper, is an example of realizing the general theory in commonly encountered examples. In order to concentrate on studying the impact of the volatility functions, we assume r1⁄4 0 and q1⁄4 0. The price of a European call option satisfies the following partial differential equation with a final condition (Black and Scholes 1973, Merton 1973)