François Quittard-Pinon

A New Procedure for Pricing Parisian Options

In this article, we propose a new method to price numerically Parisian options by inversion of Laplace transform. We compare this method to other more traditional approaches (Monte-Carlo simulations and partial differential equation solving). We show that this method converges more rapidly and yields quasi-instantaneous answers to the valuation and hedging problem at stake.

The Valuation of Interest Rate Digital Options and Range Notes Revisited

The aim of this paper is to value interest rate structured products in a simpler and more intuitive way than Turnbull(1995). Considering some assumptions with respect to the evolution of the term structure of interest rates, the price of a European interest rate digital call option is given. Recall it is a contract designed to pay one dollar at maturity if a reference interest rate is above a prespecified level (the strike), and zero in all the others cases. Combining two options of this type enables us to value a European range digital option. Then using a one factor linear Gaussian model and the new well-known change of numeraire approach, a closed-form formula is found to value range notes which pay at the end of each defined period, a sum equal to prespecified interest rate times the number of days the reference interest rate lies inside a corridor.

On the Pricing of Power and Other Polynomial Options

There are limitless possibilities for new kinds of options, with payoff patterns of all shapes and contingencies related to any number of events. The simplest of these payoffs, like those for a call spread or a strangle, for example, can be constructed out of a straightforward combination of plain vanilla puts and calls. This leads to very easy formulas for pricing and hedging them, but the possible payoff patterns invariably consist of a collection of linear segments defined by the intervals between the strikes of the options in the set. This article provides a way to significantly extend the achievable payoff patterns to any polynomial function of the final asset price. The building block is no longer a plain vanilla option, but a power option, the payoff of which is a function of (Sa - X) for some exponent a. The authors first show how to price such options under several important price processes, including Heston-style stochastic volatility and a general Lévy process. They then derive the formula for an option with a general polynomial payoff, as a function of power options. Finally, they offer some examples to illustrate the technique, including parabolic payoff call options.

Development and Pricing of a New Participating Contract

Abstract This article designs and prices a new type of participating life insurance contract. Participating contracts are popular in the United States and European countries. They present many different covenants and depend on national regulations. In the present article we design a new type of participating contract very similar to the one considered in other studies, but with the guaranteed rate matching the return of a government bond. We prove that this new type of contract can be valued in closed form when interest rates are stochastic and when the company can default.

How to Price Efficiently European Options in Some Geometric Lévy Processes Models

This paper presents the implementation to the class of jump diffusion models of the approach used by Boyarchenko and Levendorskii (2002) in the case of exponential Levy models. We show that this approach is more computationally efficient than the semi closed form solutions derived by Kou (2002), especially compared to the latter case. A brand new model is then presented. It extends and generalizes Kou model.

Pricing, Hedging and Assessing Risk in a General Lévy Context

The goal of this paper is to suggest a general approach for risk management by allowing jumps to occur in the underlying of a European contingent claim. It gives a unified methodology for pricing, hedging and computing the standard risk measures, namely the Value-at-Risk (VaR) and the Conditional Tail Expectation (CTE). The core of the paper shows that such quantities as prices, hedging ratios and standard risk measures can be expressed as an integral form articularly suited for the calculation by FFT, simply by changing the integrand. The method can be applied as soon as the characteristic exponent of the Levy rocess is known. The suggested unified method is easier to implement than the numerical solution to PIDE and faster than Monte Carlo simulations. It gives a powerful tool to practitioners who want to price and control risk of European derivatives in a non-Gaussian setting.

Pricing Equity Index Annuities with Surrender Options in Four Models

Equity-indexed annuities (EIAs) have generated a great deal of interest since Keyport Life first launched “Key Index” in February 1995. They are considered to be the most innovative products to appear on the market in years. EIAs are, essentially equity-linked deferred annuities which provide the policyholder with a guaranteed accumulation rate on their premium, and also at maturity, benefits from an additional return based on the performance of an equity mutual fund or a family of mutual funds or a stock index, typically of the Standard and Poor’s (S&P500) index, so the customer can profit from the growth of the stock market. Product designs of EIAs can vary, depending on the companies that sell them. In this article, we focus on the pricing of one of the product designs in the market which gives the possibility to surrender their policy before maturity. Such options can be valued using Monte Carlo simulation method proposed for pricing American options but, here, we use the least squares Monte Carlo suggested by Longstaff and Schwartz added to the control variate tool in order to construct efficient estimators. We analyze these equity-linked life insurance contracts under four different models. The frameworks differ from the way we model the price of the fund associated with the contract. The first setting is the usual Black and Scholes model, the second is the environment of jump diffusions, especially a Kou process, the third is the regime switching log normal model (RSLN) developed by Hamilton and the fourth is a mixed of a regime switching and a jump diffusion. The surrender option is priced using the Longstaff and Schwartz methodology.

Pricing and Hedging Variable Annuities in a Lévy Market: A Risk Management Perspective: Pricing and Hedging Variable Annuities in a Lévy Market

Pricing and hedging life insurance contracts with minimum guarantees are major areas of concern for insurers and researchers. In this article, we propose a unified framework for pricing, hedging, and assessing the risk embedded in the guarantees offered by Variable Annuities in a Lévy market. We address these questions from a risk management perspective. This method proves to be fast, accurate, and efficient. For hedging, we use a local risk minimization to provide a concise formula for the optimal hedging ratio. We also consider hedging strategies that use a portfolio of standard options. For assessing risk, we introduce an accumulated discounted loss function that takes mortality, transaction costs, and fees into account. We apply our resulting unified framework to the Minimum Guarantees for Maturity Benefit, Death Benefit, and Accumulation Benefit contracts. We illustrate the whole method with CGMY and Kou processes, which prove to offer a realistic modeling for financial prices. From this application, we draw important practical implications. In particular, we show that the assumption of geometric Brownian motion leads to undervalue the actual economic capital necessary to hedge and gives an illusion of safety.

Protection of a Company Issuing a Certain Class of Participating Policies in a Complete Market Framework

Abstract In this article we examine to what extent policyholders buying a certain class of participating contracts (in which they are entitled to receive dividends from the insurer) can be described as standard bondholders. Our analysis extends the ideas of Biihlmann and sequences the fundamental advances of Merton, Longstaff and Schwartz, and Briys and de Varenne. In particular, we develop a setup where these participating policies are comparable to hybrid bonds but not to standard risky bonds (as done in most papers dealing with the pricing of participating contracts). In this mixed framework, policyholders are only partly protected against default consequences. Continuous and discrete protections are also studied in an early default Black and Cox-type setting. A comparative analysis of the impact of various protection schemes on ruin probabilities and severities of a life insurance company that sells only this class of contracts concludes this work.

On the Hedging of Variable Annuities with Ratchet in Jump Models

This paper suggests a unified methodology for assessing the risk embedded in ratchet guarantees offered in life insurance Variable Annuity contracts. Using a non-Gaussian setting in line with most stylized features observed in the market, we address these questions from an operational risk management perspective. Since the well-known and widely used delta-hedging ratio is not optimal, one of the most important problem raised is the hedging issue. The research suggests many theoretical solutions whose efficiency from a computational point of view is controversial and rarely studied. This paper tries to fill this gap by suggesting a unified method for fast pricing and hedging the market risk embedded in ratchet guarantees offered in VA. The authors also present a model-free recursive formula that facilitates the pricing and the hedging of Guaranteed Minimum Accumulation Benefits known as GMAB.