The interval market model in mathematical finance. Game-theoretic methods
The interval market model, introduced independently by the authors of this book, postulates that for a market consisting of \(n\) assets, there is a compact convex set in \(\mathbb R^{n}\) which contains the one-step relative price changes (in a discrete-time setting) or the relative asset price velocities (in a continuous-time setting). E.g.\ for a single asset in discrete time, this means there are constants \(d < 1 < u\) such that \(S(t + \Delta t) \in [dS(t), uS(t)]\) for all possible price trajectories; compare this to the \(S(t + \Delta t) \in \{dS(t), uS(t)\}\) specification of the binary tree model. In contrast with classic mathematical finance, the interval market model is not of probabilistic-stochastic nature. Instead, as the authors state it, ``\dots most developments to be reported here belong to the realm of robust control, i.e.\ minimax approaches to decision making in the presence of uncertainty. These take several forms: the discrete Isaacs equation, Isaacs' and Breakwell's geometric analysis of extremal fields, Aubin's viability approach, Crandall and Lions' viscosity solutions as extended to differential games by Evans and Souganidis, Bardi, and others, Frankowska's non-smooth analysis approach to viscosity solutions of positively complete sets. As a consequence, we will not attempt to give here a general introduction to dynamic game theory, as different parts of the book use different approaches. We will, however, strive to make each part self-contained. Nor will we try to uniform the notation, although some of these works deal with closely related topics. As a matter of fact, the developments we report here evolved, relatively independently, over more than a decade. As a result, they have developed independent, consistent notation systems. Merging them at this late stage would have been close to impossible. We will provide a concise `dictionary' between the notations of Parts II--V.'' Part I of the book is written by P.\ Bernhard. Chapter 1 revisits Merton's optional dynamic portfolio problem, in which the investor aims to maximize a utility function over a finite or infinite time horizon by investing into a market consisting of one risk-free asset and a finite number of risky assets. Two market models are considered: first the classic one where the prices of risky assets follow Itō processes with constant drifts and volatilities; and then the uniform interval market model where the relative price increment of risky assets in one discrete time step is constrained to a symmetric interval around the mean. In Chapter 2, the classic Black-Scholes theory is recalled with emphasis on the self-financing replicating portfolio determining the no-arbitrage price of options. The authors of Part II are J.\ Engwerda, B.\ Roorda, and H.\ Schumacher. In Chapter 3, the classic binary tree model for pricing and hedging European options in discrete time is contrasted to its interval model counterpart. In Chapter 4, the concept of fair price interval for options is introduced as the set of hedging costs which may show up along an asset price path for \textit{every} hedging strategy. The main results of the chapter are that, under mild technical assumptions, the fair price interval is indeed an interval; and that in the model \(S(t + \Delta t) \in [dS(t), uS(t)]\) mentioned above the left and right endpoints of the fair price interval correspond to the so-called subgradient strategy and the classic delta hedging strategy. The authors describe how the fair price interval can be determined for path-independent strategies and examine the asset price paths leading to the worst-case hedging costs. In Chapter 5 the analysis of Chapter 4 is partially repeated for the same interval market model but with admissible hedging strategies constrained to those for which the worst-case cost doesn't exceed a given limit; thus the problem studied can be considered as the interval model analogue of the classic risk hedging under a value-at-risk constraint. Chapter 6 is an appendix containing the omitted proofs of some results. Part III, written by P.\ Bernhard, delivers the full theory of European option hedging in the interval model with transaction costs. Chapter 7 states the (standard) key assumptions about market and trading (constant risk-free interest rate, infinite divisibility of assets, no price impact of trade, etc.); transaction costs, which fit naturally into the model and are in a sense necessary to make it non-trivial, are proportional to the value of the transaction. The seller's price of an option is defined as the worth of the cheapest hedging portfolio (i.e.\ the one having the lowest worst-case cost) at inception, which can be formalized in a minimax formula. Chapter 8 presents the following key results on European call and put options. Let \(W(t,u,v)\) denote the value of the option at time \(t\) with the underlying stock's discounted price \(u\) and discounted worth in the hedging portfolio \(v\). It is shown that \(W\) is the unique Lipschitz continuous viscosity solution of the following differential quasivariational inequality: \(\forall (t,u,v) \in [0,T)\times \mathbb R^{+} \times \mathbb R\), \[ \max\left\{ -\frac{\partial W}{\partial t} - \tau^{\varepsilon}\left\langle \frac{\partial W}{\partial u}u +\frac{\partial W}{\partial v}v-v\right\rangle, -\frac{\partial W}{\partial v} - C^{+}, \frac{\partial W}{\partial v} + C^{-}\right\} = 0, \] \[ W(T,u,v) = N(u,v), \] where \(T\) is the expiry date of the option, \(N(u,v)\) is the terminal payoff, \(C^{+},C^{-}\) are the trading cost multipliers for buy and sell, and \(\tau^{\varepsilon}\langle x \rangle = \tau^{\operatorname{sgn} (x)} \cdot x\) with \(\tau^{-},\tau^{+}\) denoting the extreme relative rates of change of the underlying stock price in the interval model. The function \(W\) can be expressed explicitly using the solutions of a system of linear partial differential equations. The corresponding results for the discrete-time trading case are also obtained, and it is shown that the solutions to the discrete problem converge uniformly on compact intervals for \((u,v)\) as the step size of the (sub)divisions goes to zero. The proof of these results is based on a subtle geometric analysis of the asset path trajectories in the interval model. Chapter 9 repeats the same analysis and obtains the analogous results, in part without proofs, for digital options. In Chapter 10, option prices in the interval model are numerically computed and contrasted to the Black-Scholes prices. The author concludes that the analytics from the two modeling approaches resemble qualitatively and quantitatively, as well, and the higher premia observed in the interval model naturally follow from the presence of transaction costs. It is also shown that Black-Scholes implied volatilities computed from interval model option prices exhibit a good-looking smile, and that the hedging strategies based on the interval model are robust to misspecification of model parameters (e.g.\ of the \(\tau^{\pm}\)). So without claiming superiority over the Black-Scholes model, the interval model is proposed as its useful alternative or complement, e.g.\ as a support tool in investment decision making, especially in markets where the ``no transaction cost'' and ``continuous trading'' assumptions underpinning the Black-Scholes theory are strongly violated. Part IV is written by V.\ N.\ Kolokoltsov. As outlined in the short introductory Chapter 11, the results of Part III are extended to options on multiple underlying instruments, and it is shown how the game-theoretic techniques developed for interval models can be applied in the classic stochastic framework for incomplete markets to identify the martingale measures yielding the upper (or seller's) prices or lower (or buyer's) prices of options. In Chapter 12, the author presents an abstract approach to risk neutrality, where risk-neutral measures naturally arise in minimax results. As an illustration, it is obtained that if \(E \subset \mathbb R^{d}\) is a compact set which does not belong to any closed half-space of \(\mathbb R^{d}\) and \(f\) is a continuous function on \(E\) then \(\inf_{\gamma \in \mathbb R^{d}}\sup_{\xi \in E} [f(\xi) - (\xi,\gamma)] = \max_{\mu}\mathbb E _{\mu}(f)\) where \(\max\) is taken over all extreme points \(\mu\) of risk-neutral laws on \(E\). Chapter 13 is about European options on multiple underlying instruments in discrete time. It is shown that, under mild technical assumptions, the hedging (or seller's) price of an option, defined by a game-theoretic minimax problem, can be obtained by taking \(\max\) over expectations w.r.t.\ risk-neutral measures. This result facilitates the computation of option prices in interval models, which turn out to be given by simple analytical formulae for a special kind of option payoff (submodular and convex). Chapter 14 extends the main results of Chapter 13 to path dependent (e.g.\ American or real) options. The techniques developed in Chapter 13 are also applied to incomplete markets in the classic stochastic framework, e.g.\ where stock prices may jump or interest rates and volatilities may be stochastic, to determine the upper/lower prices of options. In Chapter 15 the continuous-time counterpart of the theory is outlined; Chapter 16 briefly discusses the modeling of (instantaneous digital) credit default swaps in this framework. The two chapters of Part V are written by P.\ Saint-Pierre and J.-P.\ Aubin. Chapter 17 describes how the Guaranteed Capture Basin Algorithm and some of its extensions can be used to numerically evaluate the price of a wide variety of options (European vanilla, Bermudan, barrier, etc.) with or without transaction costs, in classic stochastic or interval models. Chapter 18 discusses portfolio insurance: the Constant Proportion Portfolio Insurance (CPPI) method, leaving a positive probability for large losses (gap risk), is contrasted to another method, called Viabilist Portfolio Performance and Insurance (VPPI), which ``eradicates'' the gap risk (conditionally on a given assumption of asset price dynamics). VPPI in action, as implemented at VIMADES (see {\texttt{http://vimades.com}}), is illustrated using the CAC 40 as portfolio for the June 2 -- September 26, 2011 time period.
discrete-time and continuous-time hedging
Viabilist Portfolio Performance and Insurance
scientific article; zbMATH DE number 6106769