In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market, a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free. A risk-neutral measure is a probability measure The easiest way to remember what the risk-neutral measure is, or to explain it to a probability generalist who might not know much about finance, is to realize that it is:

The probability measure of a transformed random variable. Typically this transformation is the utility function of the payoff. The risk-neutral measure would be the measure co

Three Essays on Asset Pricing

Emmanuel Leclercq

In the first chapter,which is a joint work with Mathieu Cambou and Philippe H.A. Charmoy, we study the distribution of the hedging errors of a European call option for the delta and variance-minimizing strategies. Considering the setting proposed by Heston (1993), we assess the error distribution by computing its moments under the real-world probability measure. It turns out that one is better off implementing either a delta hedging or a variance-minimizing strategy, depending on the strike and maturity of the option under consideration. In the second paper, which is a joint work with Damir Filipovic and Loriano Mancini, we develop a practicable continuous-time dynamic arbitrage-free model for the pricing of European contingent claims. Using the framework introduced by Carmona and Nadtochiy (2011, 2012), the stock price is modeled as a semi-martingale process and, at each time t , the marginal distribution of the European option prices is coded by an auxiliary process that starts at t and follows an exponential additive process. The jump intensity that characterizes these auxiliary processes is then set in motion by means of stochastic dynamics of Itô's type. The model is a modification of the one proposed by Carmona and Nadtochiy, as only finitely many jump sizes are assumed. This crucial assumption implies that the jump intensities are taken values in only a finitedimensional space. In this setup, explicit necessary and sufficient consistency conditions that guarantee the absence of arbitrage are provided. A practicable dynamic model verifying them is proposed and estimated, using options on the S&P 500. Finally, the hedging of variance swap contracts is considered. It is shown that under certain conditions, a variance-minimizing hedging portfolio gives lower hedging errors on average, compared to a model-free hedging strategy. In the third and last chapter, which is a joint work with Rémy Praz, we concentrate on the commodity markets and try to understand the impact of financiers on the hedging decisions. We look at the changes in the spot price, variance, production and hedging choices of both producers and financiers, when the mass of financiers in the economy increases. We develop an equilibrium model of commodity spot and futures markets in which commodity production, consumption, and speculation are endogenously determined. Financiers facilitate hedging by the commodity suppliers. The entry of new financiers thus increases the supply of the commodity and decreases the expected spot prices, to the benefits of the end-users. However, this entry may be detrimental to the producers, as they do not internalize the price reduction due to greater aggregate supply. In the presence of asymmetric information, speculation on the futures market serves as a learning device. The futures price and open interest reveal different pieces of private information regarding the supply and demand side of the spot market, respectively. When the accuracy of private information is low, the entry of new financiers makes both production and spot prices more volatile. The entry of new financiers typically increases the correlation between financial and commodity markets.

EPFL2014

The non-linear stochastic wave equation in high dimensions

Daniel Conus

The main topic of this thesis is the study of the non-linear stochastic wave equation in spatial dimension greater than 3 driven by spatially homogeneous Gaussian noise that is white in time. We are interested in questions of existence and uniqueness of solutions, as well as in properties of solutions, such as existence of high order moments and Hölder-continuity properties. The stochastic wave equation is formulated as an integral equation in which appear stochastic integrals with respect to martingale measures (in the sense of J.B. Walsh). Since, in dimensions greater than 3, the fundamental solution of the wave equation is neither a function nor a non-negative measure, but a general Schwartz distribution, we first develop an extension of the Dalang-Walsh stochastic integral that makes it possible to integrate a wide class of Schwartz distributions. This class contains the fundamental solution of the wave equation, under a hypothesis on the spectral measure of the noise that has already been used in the literature. With this extended stochastic integral, we establish existence of a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension. Uniqueness of the solution is established within a specific class of processes. In the case of a fine multiplicative noise, we obtain a series representation of the solution and estimates on the p-th moments of the solution (p ≥ 1). From this, we deduce Hölder-continuity of the solution under standard assumptions. The Hölder exponent that we obtain is optimal. For the case of general multiplicative noise, we construct a framework for working with appropriate iterated stochastic integrals and then derive a truncated Itô-Taylor expansion for the solution of the stochastic wave equation. The convergence of this expansion remains an open problem, so we conclude with some remarks that suggest an Itô-Taylor series expansion for the solution.

EPFL2008

A General Stochastic Volatility Model for the Pricing of Interest Rate Derivatives

Anders Trolle

We develop a tractable and flexible stochastic volatility multifactor model of the term structure of interest rates. It features unspanned stochastic volatility factors, correlation between innovations to forward rates and their volatilities, quasi-analytical prices of zero coupon bond options, and dynamics of the forward rate curve, under both the actual and risk-neutral measures, in terms of a finite-dimensional affine state vector. The model has a very good fit to an extensive panel data set of interest rates, swaptions, and caps. In particular, the model matches the implied cap skews and the dynamics of implied volatilities.

Oxford University Press2009

Three Essays on Asset Pricing

Emmanuel Leclercq

EPFL2014

The non-linear stochastic wave equation in high dimensions

Daniel Conus

EPFL2008

Risk-neutral measure

The probability measure of a transformed random variable. Typically this transformation is the utility function of the payoff. The risk-neutral measure would be the measure co

Three Essays on Asset Pricing

The non-linear stochastic wave equation in high dimensions

A General Stochastic Volatility Model for the Pricing of Interest Rate Derivatives

Three Essays on Asset Pricing

A General Stochastic Volatility Model for the Pricing of Interest Rate Derivatives

The non-linear stochastic wave equation in high dimensions