Indranil SenGupta

Option Pricing with Transaction Costs and Stochastic Interest Rate

Abstract In the case when transaction costs are associated with trading assets the option pricing problem is known to lead to solving nonlinear partial differential equations even when the underlying asset is modelled using a simple geometric Brownian motion. The nonlinear term in the resulting PDE corresponds to the presence of transaction costs. We generalize this model to a stochastic one-factor interest rate model. We show that the model follows a nonlinear parabolic type partial differential equation. Under certain assumption we prove the existence of classical solution for this model.

Detecting market crashes by analysing long-memory effects using high-frequency data

It is well known that returns for financial data sampled with high frequency exhibit memory effects, in contrast to the behavior of the much celebrated log-normal model. Herein, we analyse minute data for several stocks over a seven-day period which we know is relevant for market crash behavior in the US market, March 10–18, 2008. We look at the relationship between the Lévy parameter α characterizing the data and the resulting H parameter characterizing the self-similar property. We give an estimate of how close this model is to a self-similar model.

GENERALIZED BN–S STOCHASTIC VOLATILITY MODEL FOR OPTION PRICING

In this paper, a class of generalized Barndorff-Nielsen and Shephard (BN–S) models is investigated from the viewpoint of derivative asset analysis. Incompleteness of this type of markets is studied in terms of equivalent martingale measures (EMM). Variance process is studied in details for the case of Inverse-Gaussian distribution. Various structure preserving subclasses of EMMs are derived. The model is then effectively used for pricing European style options and fitting implied volatility smiles.

Barrier Option Under Lévy Model: A PIDE and Mellin Transform Approach

We propose a stochastic model to develop a partial integro-differential equation (PIDE) for pricing and pricing expression for fixed type single Barrier options based on the Ito-Levy calculus with the help of Mellin transform. The stock price is driven by a class of infinite activity Levy processes leading to the market inherently incomplete, and dynamic hedging is no longer risk free. We first develop a PIDE for fixed type Barrier options, and apply the Mellin transform to derive a pricing expression. Our main contribution is to develop a PIDE with its closed form pricing expression for the contract. The procedure is easy to implement for all class of Levy processes numerically. Finally, the algorithm for computing numerically is presented with results for a set of Levy processes.

Pricing Variance and Volatility Swaps for Barndorff-Nielsen and Shephard Process Driven Financial Markets

The objective of this paper is to study the arbitrage free pricing of variance and volatility swaps for Barndorff-Nielsen and Shephard type Levy process driven financial markets. One of the major challenges in arbitrage free pricing of swap is to obtain an accurate pricing expression which can be used with good computational accuracy. In this paper, we obtain various approximate expressions for the pricing of volatility and variance swaps. We show that with the approximate formulas obtained from the Barndorff-Nielsen and Shephard model the error estimation in fitting the delivery price is much less than the existing models with comparable parameters. Pricing formulas proposed in this paper are simple to compute in real time and hence can be efficiently used in practical applications. Numerical results are provided in support of the accuracy of approximate formulas presented in this paper.

Pricing Covariance Swaps for Barndorff-Nielsen and Shephard Process Driven Financial Markets

The objective of this paper is to study the arbitrage free pricing of the covariance swap for Barndorff–Nielsen and Shephard (BN–S) type Levy process driven financial markets. One of the major challenges in arbitrage free pricing of swap is to obtain an accurate pricing expression which can be used with good computational accuracy. In this paper, we obtain analytic expressions for the pricing of the covariance swap. We show that with the analytic expressions obtained from the BN–S model, the error estimation in fitting the delivery price is much less than the existing models with comparable parameters. The models and pricing formulas proposed in this paper are computable in real time and hence can be efficiently used in practical applications.

Numerical methods applied to option pricing models with transaction costs and stochastic volatility

In this paper, we solve a complex partial differential equation motivated by applications in finance where the solution of the system gives the price of European options, including transaction costs and stochastic volatility. The model is based on theoretical analysis, and the resulting differential equation is solved using PDE2D software. The stability analysis agrees well with experimental results.

Nonlinear problems modeling stochastic volatility and transaction costs

The option pricing problem when the asset is driven by a stochastic volatility process and in the presence of transaction costs leads to solving a nonlinear partial differential equation (PDE). The nonlinear term in the PDE reflects the presence of transaction costs. Under a particular market completion assumption we derive the nonlinear PDE whose solution may be used to find the price of options. Under suitable conditions, we give an algorithmic scheme to obtain the solution of the problem by an iterative method. We prove theoretically the existence of strong solutions to the problem.

A PIDE and Closed-Form Fourier Pricing Expression for Look-Back Option Under Lévy Process

We propose a PIDE and closed-form Fourier Pricing formula for floating type Look-back option when stock price follows exponential Lévy Process. We first developed a PIDE based on Martingale method and derived a closed-form Fourier formula for pricing contracts. The formula is simple, easy to compute and can work for any class of Lévy Process.

Feynman path integrals and asymptotic expansions for transition probability densities of some Lévy driven financial markets

In this paper we implement the method of Feynman path integral for the analysis of option pricing for certain Levy process driven financial markets. For such markets, we find closed form solutions of transition probability density functions of option pricing in terms of various special functions. Asymptotic analysis of transition probability density functions is provided. We also find expressions for transition probability density functions in terms of various special functions for certain Levy process driven market where the interest rate is stochastic.