Results 11  20
of
365
A Simple Option Formula for General JumpDiffusion and Other Exponential Levy Processes
 Other Exponential Lévy Processes,” Environ Financial Systems and OptionCity.net
, 2001
"... Option values are wellknown to be the integral of a discounted transition density times a payoff function; this is just martingale pricing. It's usually done in 'Sspace', where S is the terminal security price. But, for L6vy processes the Sspace transition densities are often very ..."
Abstract

Cited by 89 (2 self)
 Add to MetaCart
Option values are wellknown to be the integral of a discounted transition density times a payoff function; this is just martingale pricing. It's usually done in 'Sspace', where S is the terminal security price. But, for L6vy processes the Sspace transition densities are often very complicated, involving many special functions and infinite summations. Instead, we show that it's much easier to compute the option value as an integral in Fourier space  and interpret this as a Parseval identity. The formula is especially simple because (i) it's a single integration for any payoff and (ii) the integrand is typically a compact expressions with just elementary functions. Our approach clarifies and generalizes previous work using characteristic functions and Fourier inversions. For example, we show how the residue calculus leads to several variation formulas, such as a wellknown, but less numerically efficient, 'BlackScholes style' formula for call options. The result applies to any Europeanstyle, simple or exotic option (without pathdependence) under any L6vy process with a known characteristic function.
Russian and American put options under exponential phasetype Lévy models
, 2002
"... Consider the American put and Russian option [33, 34, 17] with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phasetype jumps in both directions. The solution rests on the reduction to the first passage ..."
Abstract

Cited by 89 (4 self)
 Add to MetaCart
Consider the American put and Russian option [33, 34, 17] with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phasetype jumps in both directions. The solution rests on the reduction to the first passage time problem for (reflected) Lévy processes and on an explicit solution of the latter in the phasetype case via martingale stopping and WienerHopf factorisation. Also the first passage time problem is studied for a regime switching Lávy process with phasetype jumps. This is achieved by an embedding into a a semiMarkovian regime switching Brownian motion.
Pricing and Hedging in Incomplete Markets
 Journal of Financial Economics
, 2001
"... We present a new approach for positioning, pricing, and hedging in incomplete markets that bridges standard arbitrage pricing and expected utility maximization. Our approach for determining whether an investor should undertake a particular position involves specifying a set of probability measures a ..."
Abstract

Cited by 81 (8 self)
 Add to MetaCart
(Show Context)
We present a new approach for positioning, pricing, and hedging in incomplete markets that bridges standard arbitrage pricing and expected utility maximization. Our approach for determining whether an investor should undertake a particular position involves specifying a set of probability measures and associated °oors which expected payo®s must exceed in order for the investor to consider the hedged and ¯nanced investment to be acceptable. By assuming that the liquid assets are priced so that each portfolio of assets has negative expected return under at least one measure, we derive a counterpart to the ¯rst fundamental theorem of asset pricing. We also derive a counterPricing and Hedging in Incomplete Markets 2 part to the second fundamental theorem, which leads to unique derivative security pricing and hedging even though markets are incomplete. For products that are not spanned by the liquid assets of the economy, we show how our methodology provides more realistic bidask spreads.
Lévy Processes in Finance: Theory, Numerics, and Empirical Facts
, 2000
"... Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible distribution can be chosen as the increment distribution over periods of time ∆t. On the other hand, they have ..."
Abstract

Cited by 81 (2 self)
 Add to MetaCart
Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible distribution can be chosen as the increment distribution over periods of time ∆t. On the other hand, they have a simple structure in comparison with general semimartingales. Thus stochastic models based on Lévy processes often allow for analytically or numerically tractable formulas. This is a key factor for practical applications. This thesis is divided into two parts. The first, consisting of Chapters 1, 2, and 3, is devoted to the study of stock price models involving exponential Lévy processes. In the second part, we study term structure models driven by Lévy processes. This part is a continuation of the research that started with the author's diploma thesis Raible (1996) and the article Eberlein and Raible (1999). The content of the chapters is as follows. In Chapter 1, we study a general stock price model where the price of a single stock follows an exponential Lévy process. Chapter 2 is devoted to the study of the Lévy measure of infinitely divisible distributions, in particular of generalized hyperbolic distributions. This yields information about what changes in the distribution of a generalized hyperbolic Lévy motion can be achieved by a locally equivalent change of the underlying probability measure. Implications for
Specification Analysis of Option Pricing Models Based on TimeChanged Lévy Processes
, 2003
"... We analyze the specifications of option pricing models based on timechanged Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. O ..."
Abstract

Cited by 67 (12 self)
 Add to MetaCart
We analyze the specifications of option pricing models based on timechanged Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. Our estimation of a variety of model specifications indicates that to better capture the behavior of the S&P 500 index options, we must incorporate a high frequency jump component in the return process and generate stochastic volatilities from two different sources, the jump component and the diffusion component.
Optimal stopping and perpetual options for Lévy processes
, 2000
"... Solution to the optimal stopping problem for a L'evy process and reward functions (e x \Gamma K) + and (K \Gamma e x ) + , discounted at a constant rate is given in terms of the distribution of the overall supremum and infimum of the process killed at this rate. Closed forms of this sol ..."
Abstract

Cited by 67 (8 self)
 Add to MetaCart
Solution to the optimal stopping problem for a L'evy process and reward functions (e x \Gamma K) + and (K \Gamma e x ) + , discounted at a constant rate is given in terms of the distribution of the overall supremum and infimum of the process killed at this rate. Closed forms of this solutions are obtained under the condition of positive jumps mixedexponentially distributed. Results are interpreted as admissible pricing of perpetual American call and put options on a stock driven by a L'evy process, and a BlackScholes type formula is obtained. Keywords and Phrases: Optimal stopping, L'evy process, mixtures of exponential distributions, American options, Derivative pricing. JEL Classification Number: G12 Mathematics Subject Classification (1991): 60G40, 60J30, 90A09. 1 Introduction and general results 1.1 L'evy processes Let X = fX t g t0 be a real valued stochastic process defined on a stochastic basis(\Omega ; F ; F = (F t ) t0 ; P ) that satisfy the usual conditions. A...
Telling From Discrete Data Whether the Underlying ContinuousTime Model is a Diffusion
, 2001
"... ..."
Stochastic Skew in Currency Options
 Journal of Financial Economics
, 2007
"... ours. We welcome comments, including references to related papers we have inadvertently overlooked. ..."
Abstract

Cited by 64 (5 self)
 Add to MetaCart
ours. We welcome comments, including references to related papers we have inadvertently overlooked.
Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation
 JOURNAL OF FINANCIAL ECONOMETRICS, 2009, 1–41
, 2009
"... We propose a dynamically consistent framework that allows joint valuation and estimation of stock options and credit default swaps written on the same reference company. We model default as controlled by a Cox process with a stochastic arrival rate. When default occurs, the stock price drops to zero ..."
Abstract

Cited by 58 (9 self)
 Add to MetaCart
We propose a dynamically consistent framework that allows joint valuation and estimation of stock options and credit default swaps written on the same reference company. We model default as controlled by a Cox process with a stochastic arrival rate. When default occurs, the stock price drops to zero. Prior to default, the stock price follows a jumpdiffusion process with stochastic volatility. The instantaneous default rate and variance rate follow a bivariate continuous process, with its joint dynamics specified to capture the observed behavior of stock option prices and credit default swap spreads. Under this joint specification, we propose a tractable valuation methodology for stock options and credit default swaps. We estimate the joint risk dynamics using data from both markets for eight companies that span five sectors and six major credit rating classes from B to AAA. The estimation highlights the interaction between market risk (return variance) and credit risk (default arrival) in pricing stock options and credit default swaps.