Huyên Pham

Bio: Huyên Pham is an academic researcher from University of Paris. The author has contributed to research in topics: Stochastic control & Bellman equation. The author has an hindex of 38, co-authored 122 publications receiving 4248 citations. Previous affiliations of Huyên Pham include Paris Diderot University & ENSAE ParisTech.

Mean‐Variance Hedging and Numéraire

Abstract: We consider the mean-variance hedging problem when the risky assets price process is a continuous semimartingale. The usual approach deals with self-financed portfolios with respect to the primitive assets family. By adding a numeraire as an asset to trade in, we show how self-financed portfolios may be expressed with respect to this extended assets family, without changing the set of attainable contingent claims. We introduce the hedging numeraire and relate it to the variance-optimal martingale measure. Using this numeraire both as a deflator and to extend the primitive assets family, we are able to transform the original mean-variance hedging problem into an equivalent and simpler one; this transformed quadratic optimization problem is solved by the Galtchouk–Kunita–Watanabe projection theorem under a martingale measure for the hedging numeraire extended assets family. This gives in turn an explicit description of the optimal hedging strategy for the original mean-variance hedging problem.

Optimal stopping, free boundary, and American option in a jump-diffusion model

Abstract: This paper considers the American put option valuation in a jump-diffusion model and relates this optimal-stopping problem to a parabolic integro-differential free-boundary problem, with special attention to the behavior of the optimal-stopping boundary. We study the regularity of the American option value and obtain in particular a decomposition of the American put option price as the sum of its counterpart European price and the early exercise premium. Compared with the Black-Scholes (BS) [5] model, this premium has an additional term due to the presence of jumps. We prove the continuity of the free boundary and also give one estimate near maturity, generalizing a recent result of Barleset al. [3] for the BS model. Finally, we study the effect of the market price of jump risk and the intensity of jumps on the American put option price and its critical stock price.

Optimal quantization methods and applications to numerical problems in finance

Abstract: We review optimal quantization methods for numerically solving nonlinear problems in higher dimensions associated with Markov processes. Quantization of a Markov process consists in a spatial discretization on finite grids optimally fitted to the dynamics of the process. Two quantization methods are proposed: the first one, called marginal quantization, relies on an optimal approximation of the marginal distributions of the process, while the second one, called Markovian quantization, looks for an optimal approximation of transition probabilities of the Markov process at some points. Optimal grids and their associated weights can be computed by a stochastic gradient descent method based on Monte Carlo simulations. We illustrate this optimal quantization approach with four numerical applications arising in finance: European option pricing, optimal stopping problems and American option pricing, stochastic control problems and mean-variance hedging of options and filtering in stochastic volatility models.

Dynamic programming and mean-variance hedging

Abstract: We consider the mean-variance hedging problem when asset prices follow Ito processes in an incomplete market framework. The hedging numeraire and the variance-optimal martingale measure appear to be a key tool for characterizing the optimal hedging strategy (see Gourieroux et al. 1996; Rheinlander and Schweizer 1996). In this paper, we study the hedging numeraire $\tilde a$ and the variance-optimal martingale measure $\tilde P$ using dynamic programming methods. We obtain new explicit characterizations of $\tilde a$ and $\tilde P$ in terms of the value function of a suitable stochastic control problem. We provide several examples illustrating our results. In particular, for stochastic volatility models, we derive an explicit form of this value function and then of the hedging numeraire and the variance-optimal martingale measure. This provides then explicit computations of optimal hedging strategies for the mean-variance hedging problem in usual stochastic volatility models.

Optimal high frequency trading with limit and market orders

Abstract: We propose a framework for studying optimal market-making policies in a limit order book (LOB). The bid–ask spread of the LOB is modeled by a tick-valued continuous-time Markov chain. We consider a small agent who continuously submits limit buy/sell orders at best bid/ask quotes, and may also set limit orders at best bid (resp. ask) plus (resp. minus) a tick for obtaining execution order priority, which is a crucial issue in high-frequency trading. The agent faces an execution risk since her limit orders are executed only when they meet counterpart market orders. She is also subject to inventory risk due to price volatility when holding the risky asset. The agent can then also choose to trade with market orders, and therefore obtain immediate execution, but at a less favorable price. The objective of the market maker is to maximize her expected utility from revenue over a short-term horizon by a trade-off between limit and market orders, while controlling her inventory position. This is formulated as a mi...

Stochastic Equations in Infinite Dimensions

Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Valuing American Options by Simulation: A Simple Least-Squares Approach

Abstract: This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniques cannot be used. We illustrate this technique with several realistic examples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an American swaption in a 20-factor string model of the term structure.