Smart Trading Under Fire: Portfolio Defense Against Market Crashes

Table of Links

Abstract and 1. Introduction

2. Financial Market Model and Worst-Case Optimization Problem

3. Solution to the Post-Crash Problem

4. Solution to the Pre-Crash Problem

5. A BSDE Characterization of Indifferences Strategies

6. The Markovian Case

7. Numerical Experiments

Acknowledgments and References

Appendix A. Proofs from Section 3

Appendix B. Proofs of BASDE Results from Section 5

Appendix C. Proofs of (CIR) Results from Section 6

Abstract. We study and solve the worst-case optimal portfolio problem as pioneered by Korn and Wilmott in [39] of an investor with logarithmic preferences facing the possibility of a market crash with stochastic market coefficients by enhancing the martingale approach developed by Seifried in [49]. With the help of backward stochastic differential equations (BSDEs), we are able to characterize the resulting indifference optimal strategies in a fairly general setting. We also deal with the question of existence of those indifference strategies for market models with an unbounded market price of risk. We therefore solve the corresponding BSDEs via solving their associated PDEs using a utility crash-exposure transformation. Our approach is subsequently demonstrated for Heston’s stochastic volatility model, Bates’ stochastic volatility model including jumps, and Kim-Omberg’s model for a stochastic excess return.

1. Introduction

An important aspect that is neglected in the pure Merton type portfolio optimization setting is the presence of so-called crash scenarios as first introduced by Hua and Wilmott [25] in discrete time. In this setting, parameters are subject to Knightian uncertainty in the sense of Knight [32], which consequently does not impose any distributional assumptions. In particular, in these worst-case optimization models, a financial crash is identified with an instantaneous jump in asset prices.

The literature strand on worst-case portfolio optimization possess by now a long history. In their seminal work [39], Korn and Wilmott have solved the worst-case scenario portfolio problem under the threat of a crash for logarithmic utility in continuous time. Their results have been extended in [34] by using the so-called indifference principle. Korn and Steffensen [38] then derive (classical) HJB systems for the worst-case portfolio problem. What all these works have in common from a conceptual point of view, is that the resulting worst-case optimal strategies are characterized by the requirement that the investor is indifferent between the worst crash happening immediately and no crash happening at all.

Based on a controller-vs-stopper game, Seifried introduced fundamental concepts for the worst-case portfolio optimization in [49], namely the indifference frontier, the indifference optimality principle and the change-of-measure device (see also [37]), in order to generalize the results to multi-asset frameworks and in particular discontinuous price dynamics. During the course of this paper, we will heavily rely on these methods and enhance them by allowing the market coefficients - in particular the volatility and the excess return - to be stochastic.

Further generalizations of the worst-case approach comprise, among others, proportional transaction costs, cf. [7], a random number of crashes, cf. [6], lifetime-consumption, cf. [16], a second layer of robustness, cf. [15], explicit solutions for the multi-asset framework, cf. [33], and more recently stress scenarios, cf. [36] and dynamic reinsurance, cf. [35].

From an abstract point of view, the worst-case approach shares common features with classical robust portfolio optimization, which typically focus on the financial markets’ parameters. For instance in [50], the market acts against the trader and chooses the worst possible market coefficients. Among others, another example is Schied, cf. [48], who considers a set of probability measures to maximize the robust utility of the terminal wealth. We refer to [21] for an overview on this literature. We however wish to stress that in the worst-case portfolio optimization problem, the jump times and the jump intensity are unknown, which renders the problem more delicate than standard portfolio optimization problems.

Another strand of literature, to which our work is related, is portfolio optimization with unhedegable risks, which typically comes along with incomplete markets. The seminal paper of Zariphopoulou [51] introduces the so-called martingale distortion, which is able to deal with stochastic volatility models in a very general factor model setting. Concerning Heston’s model, Kraft [40] finds explicit solutions for power utility using stochastic control methods. Martingale methods are then employed in [29] in an affine setting. Related works in that context include as well [45, 12, 44]. In a setting with stochastic excess return, Kim and Omberg [31] find optimal trading strategies for HARA utility functions. For a concise overview of asset allocation in the presence of jumps, both in the asset price and the volatility, we refer to [8] and the references therein. In the context of worst-case portfolio optimization, so far market coefficients are assumed to be constant - with the exception of Engler and Korn [20], who solve the worst-case optimization problem for a Vasicek short rate process.

In this work, we combine the strands of worst-case portfolio optimization and optimal investment with unhedgeable risks as follows:

• We solve the worst-case optimal investment problem of an investor with logarithmic utility, facing both structural crashes and jump risk, in a setting that allows as well for stochastic market coefficients.

• We enhance the concepts indifference frontier, the indifference optimality principle and the change-of-measure device to the case of stochastic market coefficients for logarithmic utility.

• We characterize the resulting indifference strategies via the unique solutions of BSDEs, using the so-called utility crash exposure.

• We exemplify and analyze the resulting indifference strategies for the Heston model, the Bates model and the Kim-Omberg model.

We also contribute to the general theory of backward stochastic differential equations, since the equation that emerges when describing indifference strategies, leads to a BSDE coefficient that does not satisfy a Lipschitz condition with deterministic constants. Rather, we are confronted with a stochastic Lipschitz constant that satisfies an exponential integrability condition. Additionally, the generator is exponentially integrable itself. In the setting without jumps, similar conditions, sufficient for existence and uniqueness, have e.g. been treated in [9], [18] and [43]. In the case including jumps, BSDEs with stochastic Lipschitz condition have been treated in [47, Chapter II]. However, the spaces for solutions used there are different than those in the standard theory with deterministic Lipschitz constants. Our results in this article still allow the use of the standard spaces. The approach we follow is based on an approximation argument building on the L´evy settings used e.g. in [42] and [41]. We obtain existence, uniqueness of a solution and a comparison theorem, necessary to guarantee the one-to-one relation between BSDEs and PDEs (see [3]). This relation between the deterministic and stochastic world needs several requirements, e.g. a local Lipschitz continuity in the initial value of the stochastic process that models the volatility of the asset price. To be applicable to the popular model choice of the CIR process, we extend the existing result from [13] to a wider parameter range.

The remainder of the paper is organized as follows: In Section 2 we define the financial market and and the worst-case optimization problem in incomplete markets. In Section 3 and and Section 4, we solve the worst-case portfolio optimization problem by disentangling the problem in the post-crash and pre-crash problem. Section 5 then develops the BSDE machinery which is needed for the characterization of indifference strategies when market coefficients are stochastic. Section 6 and Section 7 deal with the concrete examples, i.e. the Heston model, the Bates model and the Kim-Omberg model. Appendices A, B and C contain several proofs and auxiliary results.