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Register a new userModel-free Portfolio Theory: A Rough Path Approach
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Based on a rough path foundation, we develop a model-free approach to stochastic portfolio theory (SPT). Our approach allows to handle significantly more general portfolios compared to previous model-free approaches based on Follmer integration. Without the assumption of any underlying probabilistic model, we prove pathwise Master formulae analogous to those of classical SPT, describing the growth of wealth processes associated to functionally generated portfolios relative to the market portfolio. We show that the appropriately scaled asymptotic growth rate of a far reaching generalization of Covers universal portfolio based on controlled paths coincides with that of the best retrospectively chosen portfolio within this class. We provide several novel results concerning rough integration, and highlight the advantages of the rough path approach by considering (non-functionally generated) log-optimal portfolios in an ergodic It^o diffusion setting.
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Using rough path theory, we provide a pathwise foundation for stochastic It^o integration, which covers most commonly applied trading strategies and mathematical models of financial markets, including those under Knightian uncertainty. To this end, we introduce the so-called Property (RIE) for c`adl`ag paths, which is shown to imply the existence of a c`adl`ag rough path and of quadratic variation in the sense of Follmer. We prove that the corresponding rough integrals exist as limits of left-point Riemann sums along a suitable sequence of partitions. This allows one to treat integrands of non-gradient type, and gives access to the powerful stability estimates of rough path theory. Additionally, we verify that (path-dependent) functionally generated trading strategies and Covers universal portfolio are admissible integrands, and that Property (RIE) is satisfied by both (Young) semimartingales and typical price paths.
Questions regarding the continuity in $kappa$ of the $SLE_{kappa}$ traces and maps appear very naturally in the study of SLE. In order to study the first question, we consider a natural coupling of SLE traces: for different values of $kappa$ we use the same Brownian motion. It is very natural to assume that with probability one, $SLE_kappa$ depends continuously on $kappa$. It is rather easy to show that $SLE$ is continuous in the Caratheodory sense, but showing that $SLE$ traces are continuous in the uniform sense is much harder. In this note we show that for a given sequence $kappa_jtokappa in (0, 8/3)$, for almost every Brownian motion $SLE_kappa$ traces converge locally uniformly. This result was also recently obtained by Friz, Tran and Yuan using different methods. In our analysis, we provide a constructive way to study the $SLE_{kappa}$ traces for varying parameter $kappa in (0, 8/3)$. The argument is based on a new dynamical view on the approximation of SLE curves by curves driven by a piecewise square root approximation of the Brownian motion. The second question can be answered naturally in the framework of Rough Path Theory. Using this theory, we prove that the solutions of the backward Loewner Differential Equation driven by $sqrt{kappa}B_t$ when started away from the origin are continuous in the $p$-variation topology in the parameter $kappa$, for all $kappa in mathbb{R}_+$
We show that every $mathbb{R}^d$-valued Sobolev path with regularity $alpha$ and integrability $p$ can be lifted to a Sobolev rough path provided $alpha < 1/p<1/3$. The novelty of our approach is its use of ideas underlying Hairers reconstruction theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.
We propose a Markov regime switching GARCH model with multivariate normal tempered stable innovation to accommodate fat tails and other stylized facts in returns of financial assets. The model is used to simulate sample paths as input for portfolio optimization with risk measures, namely, conditional value at risk and conditional drawdown. The motivation is to have a portfolio that avoids left tail events by combining models that incorporates fat tail with optimization that focuses on tail risk. In-sample test is conducted to demonstrate goodness of fit. Out-of-sample test shows that our approach yields higher performance measured by Sharpe-like ratios than the market and equally weighted portfolio in recent years which includes some of the most volatile periods in history. We also find that suboptimal portfolios with higher return constraints tend to outperform optimal portfolios.
We derive a backward and forward nonlinear PDEs that govern the implied volatility of a contingent claim whenever the latter is well-defined. This would include at least any contingent claim written on a positive stock price whose payoff at a possibly random time is convex. We also discuss suitable initial and boundary conditions for those PDEs. Finally, we demonstrate how to solve them numerically by using an iterative finite-difference approach.
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