Philip Protter

Solving forward-backward stochastic differential equations explicitly - a four step scheme

SummaryIn this paper we investigate the nature of the adapted solutions to a class of forward-backward stochastic differential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an “ordinary” sense over an arbitrarily prescribed time duration, via a direct “Four Step Scheme”. Using this scheme, we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic PDE system. Moreover the uniqueness of the adapted solutions (over an arbitrary time duration), as well as the continuous dependence of the solutions on the parameters, can all be proved within this unified framework. Some special cases are studied separately. In particular, we derive a new form of the integral representation of the Clark-Haussmann-Ocone type for functionals (or functions) of diffusions, in which the conditional expectation is no longer needed.

An analysis of a least squares regression method for American option pricing

Abstract. Recently, various authors proposed Monte-Carlo methods for the computation of American option prices, based on least squares regression. The purpose of this paper is to analyze an algorithm due to Longstaff and Schwartz. This algorithm involves two types of approximation. Approximation one: replace the conditional expectations in the dynamic programming principle by projections on a finite set of functions. Approximation two: use Monte-Carlo simulations and least squares regression to compute the value function of approximation one. Under fairly general conditions, we prove the almost sure convergence of the complete algorithm. We also determine the rate of convergence of approximation two and prove that its normalized error is asymptotically Gaussian.

Asset Price Bubbles in Incomplete Markets

This paper reviews and extends the mathematical finance literature on bubbles in complete markets. We provide a new characterization theorem for bubbles under the standard no-arbitrage framework, showing that bubbles can be of three types. Type 1 bubbles are uniformly integrable martingales, and these can exist with an infinite lifetime. Type 2 bubbles are nonuniformly integrable martingales, and these can exist for a finite, but unbounded, lifetime. Last, Type 3 bubbles are strict local martingales, and these can exist for a finite lifetime only. When one adds a no-dominance assumption (from Merton [24]), only Type 1 bubbles remain. In addition, under Merton’s no-dominance hypothesis, put–call parity holds and there are no bubbles in standard call and put options. Our analysis implies that if one believes asset price bubbles exist and are an important economic phenomena, then asset markets must be incomplete.

Modeling Credit Risk with Partial Information

This paper provides an alternative approach to Duffie and Lando [Econometrica 69 (2001) 633-664] for obtaining a reduced form credit risk model from a structural model. Duffie and Lando obtain a reduced form model by constructing an economy where the market sees the managers information set plus noise. The noise makes default a surprise to the market. In contrast, we obtain a reduced form model by constructing an economy where the market sees a reduction of the managers information set. The reduced information makes default a surprise to the market. We provide an explicit formula for the default intensity based on an Azema martingale, and we use excursion theory of Brownian motions to price risky debt.

The Monte-Carlo method for filtering with discrete-time observations

The problem which we address in this work is the following we have an IRd valued state process X Xt t which evolves according to a stochastic di erential equation of the form dXt a Xt dt b Xt dWt L X where is a known distribution on IRd and a b are known functions and W is a d dimensional Wiener process We have noisy observations Y YN at N regularly spaced times and without loss of generality we will assume that these times are N We wish to compute the conditional expectations Y Nf E f XN jY YN for all reasonable functions f on IRd

A Dysfunctional Role of High Frequency Trading in Electronic Markets

This paper shows that high frequency trading may play a dysfunctional role in financial markets. Contrary to arbitrageurs who make financial markets more efficient by taking advantage of and thereby eliminating mispricings, high frequency traders can create a mispricing that they unknowingly exploit to the disadvantage of ordinary investors. This mispricing is generated by the collective and independent actions of high frequency traders, coordinated via the observation of a common signal.

Wong-Zakai Corrections, Random Evolutions, and Simulation Schemes for SDE's

Abstract A general weak limit theorem for solutions of stochastic differential equations driven by arbitrary semimartingales is applied to give a unified treatment of limit theorems for random evolutions and consistency results for numerical schemes for stochastic differential equations. The asymptotic distribution of the error in an Euler scheme is studied. The Wong-Zakai correction in the random evolution limit arises through an integration by parts.

No Arbitrage and General Semimartingales

No free lunch with vanishing risk (NFLVR) is known to be equivalent to the existence of an equivalent martingale measure for the price process semimartingale. We give necessary conditions for such a semimartingale to have the property NFLVR. We also extend Novikov’s criterion for the stochastic exponential of a local martingale to be a martingale to the general case; that is, the case where the paths need not be continuous.

Risk-neutral compatibility with option prices

A common problem is to choose a “risk-neutral” measure in an incomplete market in asset pricing models. We show in this paper that in some circumstances it is possible to choose a unique “equivalent local martingale measure” by completing the market with option prices. We do this by modeling the behavior of the stock price X, together with the behavior of the option prices for a relevant family of options which are (or can theoretically be) effectively traded. In doing so, we need to ensure a kind of “compatibility” between X and the prices of our options, and this poses some significant mathematical difficulties.

How to Detect an Asset Bubble

After the 2007 credit crisis, financial bubbles have once again emerged as a topic of current concern. An open problem is to determine in real time whether or not a given assets price process exhibits a bubble. Due to recent progress in the characterization of asset price bubbles using the arbitrage-free martingale pricing technology, we are able to propose a new methodology for answering this question based on the assets price volatility. We limit ourselves to the special case of a risky assets price being modeled by a Brownian driven stochastic differential equation. Such models are ubiquitous both in theory and in practice. Our methods use sophisticated volatility estimation techniques combined with the method of reproducing kernel Hilbert spaces. We illustrate these techniques using several stocks from the alleged Internet dot-com episode of 1998-2001, where price bubbles were widely thought to have existed. Our results confirm the suspicions of the presence of bubbles in many of the dot-com stocks of 1998-2001.