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arXiv2017

Super-hedging American Options with Semi-static Trading Strategies under Model Uncertainty

We consider the super-hedging price of an American option in a discrete-time market in which stocks are available for dynamic trading and European options are available for static trading. We show that the super-hedging price $π$ is given by the supremum over the prices of the American option under randomized models. That is, $π=\sup_{(c_i,Q_i)_i}\sum_ic_iϕ^{Q_i}$, where $c_i \in \mathbb{R}_+$ and the martingale measure $Q^i$ are chosen such that $\sum_i c_i=1$ and $\sum_i c_iQ_i$ prices the European options correctly, and $ϕ^{Q_i}$ is the price of the American option under the model $Q_i$. Our result generalizes the example given in ArXiv:1604.02274 that the highest model based price can be considered as a randomization over models.

q-fin.MF↗

arXiv2017

Optimal Execution of Limit and Market Orders with Trade Director, Speed Limiter, and Fill Uncertainty

We study the optimal execution of market and limit orders with permanent and temporary price impacts as well as uncertainty in the filling of limit orders. Our continuous-time model incorporates a trade speed limiter and a trader director to provide better control on the trading rates. We formulate a stochastic control problem to determine the optimal dynamic strategy for trade execution, with a quadratic terminal penalty to ensure complete liquidation. In addition, we identify conditions on the model parameters to ensure optimality of the controls and finiteness of the associated value functions. For comparison, we also solve the schedule-following optimal execution problem that penalizes deviations from an order schedule. Numerical results are provided to illustrate the optimal market and limit orders over time.

q-fin.MF↗

arXiv2016

Gap Risk KVA and Repo Pricing: An Economic Capital Approach in the Black-Scholes-Merton Framework

Although not a formal pricing consideration, gap risk or hedging errors are the norm of derivatives businesses. Starting with the gap risk during a margin period of risk of a repurchase agreement (repo), this article extends the Black-Scholes-Merton option pricing framework by introducing a reserve capital approach to the hedging error's irreducible variability. An extended partial differential equation is derived with two new terms for expected gap loss and economic capital charge, leading to the gap risk economic value adjustment and capital valuation adjustment (KVA) respectively. Practical repo pricing formulae is obtained showing that the break-even repo rate decomposes into cost of fund and economic capital charge in KVA. At zero haircut, a one-year term repo on main equities could command a capital charge as large as 50 basis points for a 'BBB' rated borrower.

q-fin.PR↗

arXiv2017

Robust pricing--hedging duality for American options in discrete time financial markets

We investigate pricing-hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, e.g. a family of European options, only statically. In the first part of the paper we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a non-dominated family of probability measures. Our first insight is that by considering a (universal) enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This may be seen as a weak formulation of the original problem. Our second insight is that lack of duality is caused by the lack of dynamic consistency and hence a different enlargement with dynamic consistency is sufficient to recover duality: it is enough to consider (fictitious) extensions of the market in which all the assets are traded dynamically. In the second part of the paper we study two important examples of robust framework: the setup of Bouchard and Nutz (2015) and the martingale optimal transport setup of Beiglböck et al. (2013), and show that our general results apply in both cases and allow us to obtain pricing-hedging duality for American options.

math.OC↗

arXiv2016

Optimal investment and consumption with downside risk constraint in jump-diffusion models

This paper extends the results of the article [C. Klüppelberg and S. M. Pergamenchtchikov. Optimal consumption and investment with bounded downside risk for power utility functions. In Optimality and Risk: {\it Modern Trends in Mathematical Finance. The Kabanov Festschrift}, pages 133-169, 2009] to a jump-diffusion setting. We show that under the assumption that only positive jumps in the asset prices are allowed, the explicit optimal strategy can be found in the subset of admissible strategies satisfying the same risk constraint as in the pure diffusion setting. When negative jumps probably happen, the regulator should be more conservative. In that case, we suggest to impose on the investor's portfolio a stricter constraint which depends on the probability of having negative jumps in the assets during the whole considered horizon.

q-fin.MF↗

arXiv2017

Concurrent Credit Portfolio Losses

We consider the problem of concurrent portfolio losses in two non-overlapping credit portfolios. In order to explore the full statistical dependence structure of such portfolio losses, we estimate their empirical pairwise copulas. Instead of a Gaussian dependence, we typically find a strong asymmetry in the copulas. Concurrent large portfolio losses are much more likely than small ones. Studying the dependences of these losses as a function of portfolio size, we moreover reveal that not only large portfolios of thousands of contracts, but also medium-sized and small ones with only a few dozens of contracts exhibit notable portfolio loss correlations. Anticipated idiosyncratic effects turn out to be negligible. These are troublesome insights not only for investors in structured fixed-income products, but particularly for the stability of the financial sector.

q-fin.MF↗

arXiv2016

Utility maximization problem with random endowment and transaction costs: when wealth may become negative

In this paper we study the problem of maximizing expected utility from the terminal wealth with proportional transaction costs and random endowment. In the context of the existence of consistent price systems, we consider the duality between the primal utility maximization problem and the dual one, which is set up on the domain of finitely additive measures. In particular, we prove duality results for utility functions supporting possibly negative values. Moreover, we construct the shadow market by the dual optimal process and consider the utility based pricing for random endowment.

q-fin.MF↗

arXiv2016

Robustness of mathematical models and technical analysis strategies

The aim of this paper is to compare the performances of the optimal strategy under parameters mis-specification and of a technical analysis trading strategy. The setting we consider is that of a stochastic asset price model where the trend follows an unobservable Ornstein-Uhlenbeck process. For both strategies, we provide the asymptotic expectation of the logarithmic return as a function of the model parameters. Finally, numerical examples find that an investment strategy using the cross moving averages rule is more robust than the optimal strategy under parameters mis-specification.

q-fin.PM↗

arXiv2016

Semi-analytic path integral solution of SABR and Heston equations: pricing Vanilla and Asian options

We discuss a semi-analytical method for solving SABR-type equations based on path integrals. In this approach, one set of variables is integrated analytically while the second set is integrated numerically via Monte-Carlo. This method, known in the literature as Conditional Monte-Carlo, leads to compact expressions functional on three correlated stochastic variables. The methodology is practical and efficient when solving Vanilla pricing in the SABR, Heston and Bates models with time depending parameters. Further, it can also be practically applied to pricing Asian options in the $β=0$ SABR model and to other $β=0$ type models.

q-fin.CP↗

arXiv2016

Revisiting a Theorem of L.A. Shepp on Optimal Stopping

Using a bondholder who seeks to determine when to sell his bond as our motivating example, we revisit one of Larry Shepp's classical theorems on optimal stopping. We offer a novel proof of Theorem 1 from from \cite{Shepp}. Our approach is that of guessing the optimal control function and proving its optimality with martingales. Without martingale theory one could hardly prove our guess to be correct.

q-fin.MF↗

arXiv2016

On Optimal Retirement (How to Retire Early)

We pose an optimal control problem arising in a perhaps new model for retirement investing. Given a control function $f$ and our current net worth as $X(t)$ for any $t$, we invest an amount $f(X(t))$ in the market. We need a fortune of $M$ "superdollars" to retire and want to retire as early as possible. We model our change in net worth over each infinitesimal time interval by the Ito process $dX(t)= (1+f(X(t))dt+ f(X(t))dW(t)$. We show how to choose the optimal $f=f_0$ and show that the choice of $f_0$ is optimal among all nonanticipative investment strategies, not just among Markovian ones.

q-fin.ST↗

arXiv2018

No-arbitrage and hedging with liquid American options

Since most of the traded options on individual stocks is of American type it is of interest to generalize the results obtained in semi-static trading to the case when one is allowed to statically trade American options. However, this problem has proved to be elusive so far because of the asymmetric nature of the positions of holding versus shorting such options. Here we provide a unified framework and generalize the fundamental theorem of asset pricing (FTAP) and hedging dualities in arXiv:1502.06681 (to appear in Annals of Applied Probability) to the case where the investor can also short American options. Following arXiv:1502.06681, we assume that the longed American options are divisible. As for the shorted American options, we show that the divisibility plays no role regarding arbitrage property and hedging prices. Then using the method of enlarging probability spaces proposed in arXiv:1604.05517, we convert the shorted American options to European options, and establish the FTAP and sub- and super-hedging dualities in the enlarged space both with and without model uncertainty.

q-fin.MF↗

arXiv2016

Generalized semi-Markovian dividend discount model: risk and return

The article presents a general discrete time dividend valuation model when the dividend growth rate is a general continuous variable. The main assumption is that the dividend growth rate follows a discrete time semi-Markov chain with measurable space. The paper furnishes sufficient conditions that assure finiteness of fundamental prices and risks and new equations that describe the first and second order price-dividend ratios. Approximation methods to solve equations are provided and some new results for semi-Markov reward processes with Borel state space are established. The paper generalizes previous contributions dealing with pricing firms on the basis of fundamentals.

q-fin.MF↗

arXiv2018

Robust framework for quantifying the value of information in pricing and hedging

We investigate asymmetry of information in the context of robust approach to pricing and hedging of financial derivatives. We consider two agents, one who only observes the stock prices and another with some additional information, and investigate when the pricing--hedging duality for the former extends to the latter. We introduce a general framework to express the superhedging and market model prices for an informed agent. Our key insight is that an informed agent can be seen as a regular agent who can restrict her attention to a certain subset of possible paths. We use results of Hou & Obłój on robust approach with beliefs to establish the pricing--hedging duality for an informed agent. Our results cover number of scenarios, including information arriving before trading starts, arriving after static position in European options is formed but before dynamic trading starts or arriving at some point before the maturity. For the latter we show that the superhedging value satisfies a suitable dynamic programming principle, which is of independent interest.

q-fin.MF↗

arXiv2016

Stochastic Portfolio Theory: A Machine Learning Perspective

In this paper we propose a novel application of Gaussian processes (GPs) to financial asset allocation. Our approach is deeply rooted in Stochastic Portfolio Theory (SPT), a stochastic analysis framework introduced by Robert Fernholz that aims at flexibly analysing the performance of certain investment strategies in stock markets relative to benchmark indices. In particular, SPT has exhibited some investment strategies based on company sizes that, under realistic assumptions, outperform benchmark indices with probability 1 over certain time horizons. Galvanised by this result, we consider the inverse problem that consists of learning (from historical data) an optimal investment strategy based on any given set of trading characteristics, and using a user-specified optimality criterion that may go beyond outperforming a benchmark index. Although this inverse problem is of the utmost interest to investment management practitioners, it can hardly be tackled using the SPT framework. We show that our machine learning approach learns investment strategies that considerably outperform existing SPT strategies in the US stock market.

q-fin.PM↗

arXiv2016

Heat Kernels, Solvable Lie Groups, and the Mean Reverting SABR Stochastic Volatility Model

We use commutator techniques and calculations in solvable Lie groups to investigate certain evolution Partial Differential Equations (PDEs for short) that arise in the study of stochastic volatility models for pricing contingent claims on risky assets. In particular, by restricting to domains of bounded volatility, we establish the existence of the semi-groups generated by the spatial part of the operators in these models, concentrating on those arising in the so-called "SABR stochastic volatility model with mean reversion." The main goal of this work is to approximate the solutions of the Cauchy problem for the SABR PDE with mean reversion, a parabolic problem the generator of which is denoted by $L$. The fundamental solution for this problem is not known in closed form. We obtain an approximate solution by performing an expansion in the so-called volvol or volatility of the volatility, which leads us to study a degenerate elliptic operator $L_0$, corresponding the the zero-volvol case of the SABR model with mean reversion, to which the classical results do not apply. However, using Lie algebra techniques we are able to derive an exact formula for the solution operator of the PDE $\partial_t u - L_0 u = 0$. We then compare the semi-group generated by $L$--the existence of which does follows from standard arguments--to that generated by $L_0$, thus establishing a perturbation result that is useful for numerical methods for the SABR PDE with mean reversion. In the process, we are led to study semigroups arising from both a strongly parabolic and a hyperbolic problem.

math.AP↗

arXiv2016

Optimality of two-parameter strategies in stochastic control

In this note, we study a class of stochastic control problems where the optimal strategies are described by two parameters. These include a subset of singular control, impulse control, and two-player stochastic games. The parameters are first chosen by the two continuous/smooth fit conditions, and then the optimality of the corresponding strategy is shown by verification arguments. Under the setting driven by a spectrally one-sided Levy process, these procedures can be efficiently done thanks to the recent developments of scale functions. In this note, we illustrate these techniques using several examples where the optimal strategy as well as the value function can be concisely expressed via scale functions.

math.OC↗

arXiv2016

Wrong-Way Risk Models: A Comparison of Analytical Exposures

In this paper, we compare static and dynamic (reduced form) approaches for modeling wrong-way risk in the context of CVA. Although all these approaches potentially suffer from arbitrage problems, they are popular (respectively) in industry and academia, mainly due to analytical tractability reasons. We complete the stochastic intensity models with another dynamic approach, consisting in the straight modeling of the survival (Azéma supermartingale) process using the $Φ$-martingale. Just like the other approaches, this method allows for automatic calibration to a given default probability curve. We derive analytically the positive exposures $V^+_t$ "conditional upon default" associated to prototypical market price processes of FRA and IRS in all cases. We further discuss the link between the "default" condition and change-of-measure techniques. The expectation of $V^+_t$ conditional upon $τ=t$ is equal to the unconditional expectation of $V^+_tζ_t$. The process $ζ$ is explicitly derived in the dynamic approaches: it is proven to be positive and to have unit expectation. Unfortunately however, it fails to be a martingale, so that Girsanov machinery cannot be used. Nevertheless, the expectation of $V^+_tζ_t$ can be computed explicitly, leading to analytical expected positive exposure profiles in the considered examples.

q-fin.MF↗