The Interval Market Model in Mathematical Finance

Inhaltsverzeichnis

1. Frontmatter

2. Revisiting two classical results in dynamic portfolio management

   1. Chapter 1. Merton’s Optimal Dynamic Portfolio Revisited

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      In this chapter, we first derive the classic Merton optimal portfolio theory as a reference for the classic use of the continuous Samuelson market model. Then we show a different approach to fundamentally the same problem, but in discrete trading – deemed more realistic than the classic continuous trading fiction – generalizing somewhat Samuelson’s solution and using an interval market model. This is still a stochastic approach, which seems necessary for the problem at hand.

   2. Chapter 2. Option Pricing: Classic Results

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      We recall here the basics of the most classic result of option pricing, perhaps the most famous result in mathematical finance: the Black–Scholes theory for the pricing of “European options” in a perfect market, infinitely divisible and liquid, with no “friction” such as transaction costs or information lag. However, in keeping with the spirit of this volume, we derive it via a game-theoretic approach, devoid of any probabilities.

3. Hedging in Interval Models

   1. Frontmatter

   2. Chapter 3. Introduction

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      In this chapter, we cover some basic facts about hedging: the motivation for so doing, the simplest hedging scheme, called stop-loss, and, more importantly, the binomial tree model of Cox, Ross, and Rubinstein and the “delta strategy” it leads to. This very classic model has a strong relationship with the interval model to be discussed in the remainder of this volume. Familiarity with it will be assumed in the sequel. Finally, the chapter concluded with a discussion of risk assessment models, both classic and in the interval market model.

   3. Chapter 4. Fair Price Intervals

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      In this chapter, we introduce the fair price interval of a European claim and characterize it in various ways for the interval model, giving explicit computational algorithms. We also provide a detailed comparison with the simpler binomial tree model. We show that under unfavorable conditions, this simpler model may severely underestimate the cost of a given hedging strategy.

   4. Chapter 5. Optimal Hedging Under Robust-Cost Constraints

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      In this chapter, we consider an original problem: that of maximizing the best-case return of a trading strategy with a hard bound on the worst-case loss. We provide an explicit numerical algorithm for solving that problem.

   5. Chapter 6. Appendix: Proofs

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      This is a technical appendix giving the complete proofs of all propositions, lemmas, theorems, and algorithms of the preceding three chapters.

4. Robust Control Approach to option Pricing

   1. Chapter 7. Continuous and Discrete-Time Option Pricing and Interval Market Model

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      In this chapter, we introduce the continuous-time interval market model and derive the game-theoretic problem whose solution gives the pricing function as a function of the option type (the terminal payment), maturity, and current underlying stock’s market price. We deal with both continuous and discrete trading and include transaction costs.

   2. Chapter 8. Vanilla Options

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      This chapter develops the full theory of the pricing of classic, or “vanilla,” European options. We detail the theory for a call with closure in kind. The theory for closure in cash is less symmetric, and we give the few differences. The theory for a put is very similar and will be briefly discussed when necessary. Our theory encompasses continuous and discrete trading, the convergence of the latter to the former when the step size vanishes, and via a representation formula gives fast algorithms to compute a premium and a hedging strategy.

   3. Chapter 9. Digital Options

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      We investigate here options of the type “cash or nothing,” which we call digital options. We did not attempt, in the previous chapter, to be so general as to cover digital options every time a general treatment was possible. Yet this chapter will make many references to the previous one in order to avoid repetitions whenever possible. The theory is seemingly as complete as for vanilla options, but actually two difficult technical results are here only conjectured.

   4. Chapter 10. Validation

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      In this chapter, we initiate a discussion on the strengths and weaknesses of the new theory and of its applicability.

5. Game-theoretic analysis of rainbow options in incomplete markets

   1. Chapter 11. Introduction

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      In this chapter, we give a general introduction to our game-theoretic pricing and relate it to the rest of the literature on option pricing

   2. Chapter 12. Emergence of Risk-Neutral Probabilities from a Game-Theoretic Origin

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      This chapter lays the geometric foundations for our approach to option pricing establishing a link between certain game-theoretic problems and risk-neutral probabilities. We show how these probabilities, in their abstract geometric representation, appear automatically in the evaluation formula for minimax expressions that depend linearly on one of the control parameters. The introductory section lays the groundwork by introducing geometric risk-neutral probabilities and describing their extreme points. The next two sections develop our basic risk-neutral evaluation formula for games. The following sections describe, respectively, nonlinear, infinite-dimensional, stochastic, and mixed strategies with linear constraint extensions of the basic formula. These extensions are described independently of each other and used later for different financial models.

   3. Chapter 13. Rainbow Options in Discrete Time, I

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      This chapter is central for our exposition. It contains the main results describing the applications of the basic risk-neutral evaluation formula for games, developed in the previous chapter, to pricing rainbow (or colored) options under various market conditions. The first section sets the stage by defining the game of an investor with Nature leading to the basic game-theoretic expression for the hedging price in the simplest case of a standard European (rainbow) option without transaction costs. We then evaluate this expression using our risk-neutral evaluation formula, yielding various multidimensional extensions of the classic Cox–Ross–Rubinstein formula for a binomial model. The next sections show the essential simplifications that become available for submodular payoffs. In particular, even a unique risk-neutral selector can be specified sometimes, say, in the case of two-color options (for a still incomplete market). This is of key importance, as the major examples of real-life rainbow payoffs turn out to be submodular. Finally, transaction costs are fitted to our model.

   4. Chapter 14. Rainbow Options in Discrete Time, II

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      This chapter extends the analysis of discrete-time models in various directions. In passing, it is also meant to demonstrate that the standard stochastic models of financial dynamics come very naturally into play in our game-theoretic setting.

   5. Chapter 15. Continuous-Time Limits

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      This chapter is devoted to continuous-time limits. These limits are described without any probability, but only assuming that the magnitude of jumps per time period τ is of order \({\tau }^{\alpha }\), \(\alpha\in[1/2,1]\). Details related to models with transaction costs are explained. The last two sections address the modifications that are available in continuous-time limits when additional probability laws are specified. First, stochastic volatilities are dealt with, showing, in particular, how the standard stochastic volatility models for option pricing are obtained in our context. Then a model with waiting times having power decay is discussed, leading naturally to a fractional in time version (degenerate or nonlinear) of the Black–Scholes equation.

   6. Chapter 16. Credit Derivatives

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      The main geometric condition of completeness (uniqueness of risk-neutral probability) requiring d+1 possible vector jumps in dimension d was not very natural for stock prices (resulting in incompleteness of many real-life models). For credit derivatives this condition is, in contrast, very natural. We will begin with an illustration of this statement by discussing the basic model for credit default swaps without simultaneous jumps from our game-theoretic point of view. Then we discuss models with simultaneous jumps showing (1) how our general geometric results produce hedging (dominated or superreplicating) prices under basic assumptions and (2) how naturally the market can be completed by introducing tranches (e.g., collateralized debt obligations). Finally, to indicate the possible direction of qualitative and quantitative analysis of this model for large numbers of underlying assets, we briefly discuss the mean-field and stochastic limits for this model as the number of underlying securities tends to infinity.

6. Viability Approach to Complex Options Pricing and Portfolio Insurance

   1. Chapter 17. Computational Methods Based on the Guaranteed Capture Basin Algorithm

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      This chapter is devoted to numerical approximation methods for managing replicating portfolios and more complex financial instruments. One aim is to regulate evolutions under uncertainty in order not only to reach a target in finite time but also to fulfill constraints (known as viability constraints) until that time. Considering the portfolio evaluation problem, the target is defined by the payoff function at maturity time, which usually expresses the financial goal to be achieved. Constraints appear when one wants to take into account any limitations on prices, quantities to share, or other restrictive conditions that affect asset prices as well as quantities of time or other characteristics of an agreement. The extension of viability theory to hybrid or impulse systems allows us to evaluate more complex financial instruments. Since reaching a target while remaining in a given set for impulse dynamics can be characterized by a nondeterministic controlled differential equation and a controlled instantaneous reset equation, the set of initial conditions from which a given objective can be reached is computed using the Hybrid Guaranteed Capture Basin Algorithm, which extends the Guaranteed Capture Basin and the Viability Kernel algorithms to more complex dynamic systems. This algorithm can be applied to evaluate options in the presence of barriers or transaction costs, given a lack of observation, or in the frame of Ngarch modeling or other processes of price prevision.

   2. Chapter 18. Asset and Liability Insurance Management (ALIM) for Risk Eradication

      Pierre Bernhard, Jacob C. Engwerda, Berend Roorda, J. M. Schumacher, Vassili Kolokoltsov, Patrick Saint-Pierre, Jean-Pierre Aubin

      Abstract

      Unlike traditional valuation methods, viability theory provides tools for eradicating risk by determining the minimum initial capital required to meet the commitments of the investor, regardless of forecasted market developments. This capital can play the role of risk assessment within a framework of asset and liability management (ALM) of a guaranteed fund with the tools of viability theory. Using viability theory, this study offers a practical example of asset and liability insurance management (ALIM) of a radically different philosophy, eradicating risk as a function of any forecasting mechanism providing a prevision of the future risk returns, from chartist to the VIMADES Extrapolator presented in this chapter.

7. Backmatter