The minimal entropy measure and an Esscher transform in an incomplete market model

We consider an incomplete market model with one traded stock and two correlated Brownian motions $W$,$\widetilde{W}$. The Brownian motion $W$ drives the stock price, whose volatility and Sharpe ratio are adapted to the filtration $\mathbb{F} := (\widetilde{\mathcal{F}}_{t})_{0 \le t \le T}$ generated by $\widetilde{W}$. We show that the projections of the minimal entropy and minimal martingale measures onto $\widetilde{\mathcal{F}}_{T}$ are related by an Esscher transform involving the correlation between $W$,$\widetilde{W}$, and the mean-variance trade-off process. The result leads to a new formula for the marginal exponential utility-based price of an $\widetilde{\mathcal{F}}_{T}$-measurable European claim

The minimal entropy measure and an Esscher transform in an incomplete market model

We consider an incomplete market model with one traded stock and two correlated Brownian motions $W$,$\widetilde{W}$. The Brownian motion $W$ drives the stock price, whose volatility and Sharpe ratio are adapted to the filtration $\mathbb{F} := (\widetilde{\mathcal{F}}_{t})_{0 \le t \le T}$ generated by $\widetilde{W}$. We show that the projections of the minimal entropy and minimal martingale measures onto $\widetilde{\mathcal{F}}_{T}$ are related by an Esscher transform involving the correlation between $W$,$\widetilde{W}$, and the mean-variance trade-off process. The result leads to a new formula for the marginal exponential utility-based price of an $\widetilde{\mathcal{F}}_{T}$-measurable European claim

Which Method for Pricing Weather Derivatives ?

Since the introduction of the first weather derivative in the United-States in 1997, a significant number of work was directed towards the pricing of this product and the modelling of the daily average temperature which characterizes most of the traded weather instruments. The weather derivatives were created to enable companies to hedge against climate risks. They respond more to a need to cover seasonal variations which may cause loss of profits for companies than to a coverage need in property damage. Despite the abundance of work on the topic, no consensus has emerged so far about the methodology for evaluating weather derivatives. The major problems of these instruments are on one hand, they are based on an meteorological index that is not traded on financial market which does not allow the use of traditional pricing methods and on the other hand, it is difficult to get round this obstacle by susbtituting the underlying for a linked exchanged security since the weather index is weakly correlated with prices of other financial assets. To further the question of evaluation, we propose in this paper to, firstly, shed light on the difficulties of implementing the three major pricing approaches suggested in the literature for the weather derivatives (actuarial, arbitrage-free and consumption-based methods) and, secondly, to compute the prices of a weather contract by the three methodologies for comparison.weather derivatives; arbitrage-free pricing method; actuarial pricing approach; consumption-based pricing model; risk-neutral distribution; market price of risk; finite difference method; Monte-Carlo simulations.

Determination of Risk Pricing Measures from Market Prices of Risk

A new insurance provider or a regulatory agency may be interested in determining a risk measure consistent with observed market prices of a collection of risks. Using a relationship between distorted coherent risk measures and spectral risk measures, we provide a method for reconstruction distortion functions from the observed prices of risk. The technique is based on an appropriate application of the method on maximum entropy in the mean.

Which Method for Pricing Weather Derivatives ?

Since the introduction of the first weather derivative in the United-States in 1997, a significant number of work was directed towards the pricing of this product and the modelling of the daily average temperature which characterizes most of the traded weather instruments. The weather derivatives were created to enable companies to hedge against climate risks. They respond more to a need to cover seasonal variations which may cause loss of profits for companies than to a coverage need in property damage. Despite the abundance of work on the topic, no consensus has emerged so far about the methodology for evaluating weather derivatives. The major problems of these instruments are on one hand, they are based on an meteorological index that is not traded on financial market which does not allow the use of traditional pricing methods and on the other hand, it is difficult to get round this obstacle by susbtituting the underlying for a linked exchanged security since the weather index is weakly correlated with prices of other financial assets. To further the question of evaluation, we propose in this paper to, firstly, shed light on the difficulties of implementing the three major pricing approaches suggested in the literature for the weather derivatives (actuarial, arbitrage-free and consumption-based methods) and, secondly, to compute the prices of a weather contract by the three methodologies for comparison

A Hedged Monte Carlo Approach to Real Option Pricing

In this work we are concerned with valuing optionalities associated to invest or to delay investment in a project when the available information provided to the manager comes from simulated data of cash flows under historical (or subjective) measure in a possibly incomplete market. Our approach is suitable also to incorporating subjective views from management or market experts and to stochastic investment costs. It is based on the Hedged Monte Carlo strategy proposed by Potters et al (2001) where options are priced simultaneously with the determination of the corresponding hedging. The approach is particularly well-suited to the evaluation of commodity related projects whereby the availability of pricing formulae is very rare, the scenario simulations are usually available only in the historical measure, and the cash flows can be highly nonlinear functions of the prices.Comment: 25 pages, 14 figure

Extracting Information from the Market to Price the Weather Derivatives

Weather derivatives were first launched in 1996 in the United-States to allow companies to protect themselves against weather fluctuations. Even now their valuation still remains tricky. Because their underlying is not a traded asset, the weather options cannot be priced by using the Black and Scholes formula. Other pricing methods were proposed but they cannot be calibrated to the market since there are no available weather option price. However, quoted prices exist for the weather futures. The purpose of this paper is to extract two types of information from these prices, the risk-neutral distribution and the market price of risk, to value the weather derivatives. The prices are calculated by assuming that the daily average temperature obeys a mean-reverting jump-EGARCH process since it is shown that the temperature is not normally distributed and exhibits a time-varying volatility.weather derivatives; incomplete market; mean-reverting jump diffusion process; EGARCH process; PIDE; inversion problem

Three essays on pricing and hedging in incomplete markets

The thesis focuses on valuation and hedging problems when the market is incomplete. The Örst essay considers the quadratic hedging strategy. We propose a generalized quadratic hedging strategy which can balance a short-term risk (additional cost) with a long-term risk (hedging errors). The traditional quadratic hedging strategies, i.e. self-Önancing strategy and risk-minimization strategy, can be seen as special cases of the generalized quadratic hedging strategy. This is applied to the insurance derivatives market. The second essay compares parametric and nonparametric measure-changing techniques. The essay discusses three pricing approaches: pricing via Esscher measure, via calibration and via nonparametric risk-neutral density; and empirically compares the performance of the three approaches in the metal futures markets. The last essay establishes the concept of stochastic volatility of volatility and proposes several estimation methods