Hedging nontradable risks with transaction costs and price impact

A risk‐averse agent hedges her exposure to a nontradable risk factor U using a correlated traded asset S and accounts for the impact of her trades on both factors. The effect of the agent's trades on U is referred to as cross‐impact. By solving the agent's stochastic control problem, we obtain a closed‐form expression for the optimal strategy when the agent holds a linear position in U. When the exposure to the nontradable risk factor is nonlinear, we provide an approximation to the optimal strategy in closed‐form, and prove that the value function is correctly approximated by this strategy when cross‐impact and risk‐aversion are small. We further prove that when is nonlinear, the approximate optimal strategy can be written in terms of the optimal strategy for a linear exposure with the size of the position changing dynamically according to the exposure's “Delta” under a particular probability measure.     

Small‐time,large‐time,and asymptotics for the Rough Heston model

We characterize the behavior of the Rough Heston model introduced by Jaisson and Rosenbaum (2016, Ann. Appl. Probab., 26, 2860–2882) in the small‐time, large‐time, and (i.e., ) limits. We show that the short‐maturity smile scales in qualitatively the same way as a general rough stochastic volatility model , and the rate function is equal to the Fenchel–Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in El Euch and Rosenbaum (2019, Math. Financ., 29, 3–38), but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space–time scaling property which means we only need to solve this equation for the moment values of and so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log‐moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity which is useful for calibration purposes. We later derive a formal saddle point approximation for call options in the Forde and Zhang (2017) large deviations regime which goes to higher order than previous works for rough models. Our higher‐order expansion captures the effect of both drift terms, and at leading order is of qualitatively the same form as the higher‐order expansion for a general model which appears in Friz et al. (2018, Math. Financ., 28, 962–988). The limiting asymptotic smile in the large‐maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in Forde and Jacquier (2011, Finance Stoch., 15, 755–780). Finally, using Lévy's convergence theorem, we show that the log stock price tends weakly to a nonsymmetric random variable as (i.e., ) whose moment generating function (MGF) is also the solution to the Rough Heston VIE with , and we show that tends weakly to a nonsymmetric random variable as , which leads to a nonflat nonsymmetric asymptotic smile in the Edgeworth regime, where the log‐moneyness as , and we compute this asymptotic smile numerically. We also show that the third moment of the log stock price tends to a finite constant as (in contrast to the Rough Bergomi model discussed in Forde et al. (2020, Preprint) where the skew flattens or blows up) and the process converges on pathspace to a random tempered distribution which has the same law as the hyper‐rough Heston model, discussed in Jusselin and Rosenbaum (2020, Math. Finance, 30, 1309–1336) and Abi Jaber (2019, Ann. Appl. Probab., 29, 3155–3200).     

Optimal portfolio under fractional stochastic environment

Rough stochastic volatility models have attracted a lot of attention recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non‐Markovian) fractional stochastic environment (for all values of the Hurst index ). We rigorously establish a first‐order approximation of the optimal value, when the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein–Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth‐ order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.     

Semitractability of optimal stopping problems via a weighted stochastic mesh algorithm

In this paper, we propose a Weighted Stochastic Mesh (WSM) algorithm for approximating the value of discrete‐ and continuous‐time optimal stopping problems. In this context, we consider tractability of such problems via a useful notion of semitractability and the introduction of a tractability index for a particular numerical solution algorithm. It is shown that in the discrete‐time case the WSM algorithm leads to semitractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by with being the dimension of the underlying Markov chain. Furthermore, we study the WSM approach in the context of continuous‐time optimal stopping problems and derive the corresponding complexity bounds. Although we cannot prove semitractability in this case, our bounds turn out to be the tightest ones among the complexity bounds known in the literature. We illustrate our theoretical findings by a numerical example.     

Azéma martingales for Bessel and CIR processes and the pricing of Parisian zero‐coupon bonds

In this paper, we study the excursions of Bessel and Cox–Ingersoll–Ross (CIR) processes with dimensions . We obtain densities for the last passage times and meanders of the processes. Using these results, we prove a variation of the Azéma martingale for the Bessel and CIR processes based on excursion theory. Furthermore, we study their Parisian excursions, and generalize previous results on the Parisian stopping time of Brownian motion to that of the Bessel and CIR processes. We obtain explicit formulas and asymptotic results for the densities of the Parisian stopping times, and develop exact simulation algorithms to sample the Parisian stopping times of Bessel and CIR processes. We introduce a new type of bond, the zero‐coupon Parisian bond. The buyer of such a bond is betting against zero interest rates, while the seller is effectively hedging against a period where interest rates fluctuate around 0. Using our results, we propose two methods for pricing these bonds and provide numerical examples.     

Optimal investment and pricing in the presence of defaults

We consider the optimal investment problem with random endowment in the presence of defaults. For an investor with constant absolute risk aversion, we identify the certainty equivalent, and compute prices for defaultable bonds and dynamic protection against default. This latter price is interpreted as the premium for a contingent credit default swap, and connects our work with earlier articles, where the investor is protected upon default. We consider a multiple risky asset model with a single default time, at which point each of the assets may jump in price. Investment opportunities are driven by a diffusion X taking values in an arbitrary region . We allow for stochastic volatility, correlation, and recovery; unbounded random endowments; and postdefault trading. We identify the certainty equivalent with a semilinear parabolic partial differential equation with quadratic growth in both function and gradient. Under minimal integrability assumptions, we show that the certainty equivalent is a classical solution. Numerical examples highlight the relationship between the factor process, market dynamics, utility‐based prices, and default insurance premium. In particular, we show that the holder of a defaultable bond has a strong incentive to short the underlying stock, even for very low default intensities.     

STABILITY OF THE EXPONENTIAL UTILITY MAXIMIZATION PROBLEM WITH RESPECT TO PREFERENCES

This paper studies stability of the exponential utility maximization when there are small variations on agent's utility function. Two settings are considered. First, in a general semimartingale model where random endowments are present, a sequence of utilities defined on converges to the exponential utility. Under a uniform condition on their marginal utilities, convergence of value functions, optimal payoffs, and optimal investment strategies are obtained, their rate of convergence is also determined. Stability of utility‐based pricing is studied as an application. Second, a sequence of utilities defined on converges to the exponential utility after shifting and scaling. Their associated optimal strategies, after appropriate scaling, converge to the optimal strategy for the exponential hedging problem. This complements Theorem 3.2 in [Nutz, M. (2012): Risk aversion asymptotics for power utility maximization. Probab. Theory & Relat. Fields 152, 703–749], which establishes the convergence for a sequence of power utilities.     

OPTIMAL SELLING RULES FOR MONETARY INVARIANT CRITERIA: TRACKING THE MAXIMUM OF A PORTFOLIO WITH NEGATIVE DRIFT

Considering a positive portfolio diffusion X with negative drift, we investigate optimal stopping problems of the form where f is a nonincreasing function, τ is the next random time where the portfolio X crosses zero and θ is any stopping time smaller than τ. Hereby, our motivation is the obtention of an optimal selling strategy minimizing the relative distance between the liquidation value of the portfolio and its highest possible value before it reaches zero. This paper unifies optimal selling rules observed for the quadratic absolute distance criteria in this stationary framework with bang–bang type ones observed for monetary invariant criteria but in finite horizon. More precisely, we provide a verification result for the general stopping problem of interest and derive the exact solution for two classical criteria f of the literature. For the power utility criterion with , instantaneous selling is always optimal, which is consistent with previous observations for the Black‐Scholes model in finite observation. On the contrary, for a relative quadratic error criterion, , selling is optimal as soon as the process X crosses a specified function φ of its running maximum . These results reinforce the idea that optimal stopping problems of similar type lead easily to selling rules of very different nature. Nevertheless, our numerical experiments suggest that the practical optimal selling rule for the relative quadratic error criterion is in fact very close to immediate selling.     

MULTIVARIATE SUBORDINATION OF MARKOV PROCESSES WITH FINANCIAL APPLICATIONS

This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from d independent Markov processes with killing and an independent d‐dimensional time change , we construct a new process by time, changing each of the Markov processes with a coordinate . When is a d‐dimensional Lévy subordinator, the time changed process is a time‐homogeneous Markov process with state‐dependent jumps and killing in the product of the state spaces of . The dependence among jumps of its components is governed by the d‐dimensional Lévy measure of the subordinator. When is a d‐dimensional additive subordinator, Y is a time‐inhomogeneous Markov process. When with forming a multivariate Markov process, is a Markov process, where each plays a role of stochastic volatility of . This construction provides a rich modeling architecture for building multivariate models in finance with time‐ and state‐dependent jumps, stochastic volatility, and killing (default). The semigroup theory provides powerful analytical and computational tools for securities pricing in this framework. To illustrate, the paper considers applications to multiname unified credit‐equity models and correlated commodity models.     

HIGH‐ORDER SHORT‐TIME EXPANSIONS FOR ATM OPTION PRICES OF EXPONENTIAL LÉVY MODELS

The short‐time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In this work, a novel second‐order approximation for at‐the‐money (ATM) option prices is derived for a large class of exponential Lévy models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second‐order term, in time‐t, is of the form , with d₂ only depending on Y, the degree of jump activity, on σ, the volatility of the continuous component, and on an additional parameter controlling the intensity of the “small” jumps (regardless of their signs). This extends the well‐known result that the leading first‐order term is . In contrast, under a pure‐jump model, the dependence on Y and on the separate intensities of negative and positive small jumps are already reflected in the leading term, which is of the form . The second‐order term is shown to be of the form and, therefore, its order of decay turns out to be independent of Y. The asymptotic behavior of the corresponding Black–Scholes implied volatilities is also addressed. Our method of proof is based on an integral representation of the option price involving the tail probability of the log‐return process under the share measure and a suitable change of probability measure under which the pure‐jump component of the log‐return process becomes a Y‐stable process. Our approach is sufficiently general to cover a wide class of Lévy processes, which satisfy the latter property and whose Lévy density can be closely approximated by a stable density near the origin. Our numerical results show that the first‐order term typically exhibits rather poor performance and that the second‐order term can significantly improve the approximation's accuracy, particularly in the absence of a Brownian component.     

Small‐cost asymptotics for long‐term growth rates in incomplete markets

This paper provides a rigorous asymptotic analysis of long‐term growth rates under both proportional and Morton–Pliska transaction costs. We consider a general incomplete financial market with an unspanned Markov factor process that includes the Heston stochastic volatility model and the Kim–Omberg stochastic excess return model as special cases. Using a dynamic programming approach, we determine the leading‐order expansions of long‐term growth rates and explicitly construct strategies that are optimal at the leading order. We further analyze the asymptotic performance of Morton–Pliska strategies in settings with proportional transaction costs. We find that the performance of the optimal Morton–Pliska strategy is the same as that of the optimal one with costs increased by a factor of . Finally, we demonstrate that our strategies are in fact pathwise optimal, in the sense that they maximize the long‐run growth rate path by path.     

Local volatility under rough volatility

Several asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small-maturity regime), providing a better understanding of the shapes of the volatility surface induced by rough volatility models, supporting their calibration power to SP500 option data. Rough volatility models also generate a local volatility surface, via the so-called Markovian projection of the stochastic volatility. We complement the existing results on implied volatility by studying the asymptotic behavior of the local volatility surface generated by a class of rough stochastic volatility models, encompassing the rough Bergomi model. Notably, we observe that the celebrated “1/2 skew rule” linking the short-term at-the-money skew of the implied volatility to the short-term at-the-money skew of the local volatility, a consequence of the celebrated “harmonic mean formula” of [Berestycki et al. (2002). Quantitative Finance, 2, 61–69], is replaced by a new rule: the ratio of the at-the-money implied and local volatility skews tends to the constant $1/(H+3/2)$ (as opposed to the constant 1/2), where H is the regularity index of the underlying instantaneous volatility process.     

Neural network approximation for superhedging prices

This article examines neural network-based approximations for the superhedging price process of a contingent claim in a discrete time market model. First we prove that the α-quantile hedging price converges to the superhedging price at time 0 for α tending to 1, and show that the α-quantile hedging price can be approximated by a neural network-based price. This provides a neural network-based approximation for the superhedging price at time 0 and also the superhedging strategy up to maturity. To obtain the superhedging price process for $t>0$, by using the Doob decomposition, it is sufficient to determine the process of consumption. We show that it can be approximated by the essential supremum over a set of neural networks. Finally, we present numerical results.     

Trading with the crowd

We formulate and solve a multi-player stochastic differential game between financial agents who seek to cost-efficiently liquidate their position in a risky asset in the presence of jointly aggregated transient price impact, along with taking into account a common general price predicting signal. The unique Nash-equilibrium strategies reveal how each agent's liquidation policy adjusts the predictive trading signal to the aggregated transient price impact induced by all other agents. This unfolds a quantitative relation between trading signals and the order flow in crowded markets. We also formulate and solve the corresponding mean field game in the limit of infinitely many agents. We prove that the equilibrium trading speed and the value function of an agent in the finite N-player game converges to the corresponding trading speed and value function in the mean field game at rate $O(N^{-2})$. In addition, we prove that the mean field optimal strategy provides an approximate Nash-equilibrium for the finite-player game.     

The log-moment formula for implied volatility

We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that in the absence of arbitrage, if the underlying stock price at time T admits finite log-moments $\mathbb{E}[|\log S_T{|}^q]$ for some positive q, the arbitrage-free growth in the left wing of the implied volatility smile for T is less constrained than Lee's bound. The result is rationalized by a market trading discretely monitored variance swaps wherein the payoff is a function of squared log-returns, and requires no assumption for the underlying price to admit any negative moment. In this respect, the result can be derived from a model-independent setup. As a byproduct, we relax the moment assumptions on the stock price to provide a new proof of the notorious Gatheral–Fukasawa formula expressing variance swaps in terms of the implied volatility.     

MODEL‐INDEPENDENT LOWER BOUND ON VARIANCE SWAPS

It is well known that, under a continuity assumption on the price of a stock S, the realized variance of S for maturity T can be replicated by a portfolio of calls and puts maturing at T. This paper assumes that call prices on S maturing at T are known for all strikes but makes no continuity assumptions on S. We derive semiexplicit expressions for the supremum lower bound on the hedged payoff, at maturity T, of a long position in the realized variance of S. Equivalently, is the supremum strike K such that an investor with a long position in a variance swap with strike K can ensure a nonnegative payoff at T. We study examples with constant implied volatilities and with a volatility skew. In our examples, is close to the fair variance strike obtained under the continuity assumption.     

OPTIMAL INVESTMENT UNDER RELATIVE PERFORMANCE CONCERNS

We consider the problem of optimal investment when agents take into account their relative performance by comparison to their peers. Given N interacting agents, we consider the following optimization problem for agent i, : where is the utility function of agent i, his portfolio, his wealth, the average wealth of his peers, and is the parameter of relative interest for agent i. Together with some mild technical conditions, we assume that the portfolio of each agent i is restricted in some subset . We show existence and uniqueness of a Nash equilibrium in the following situations:

• ‐ unconstrained agents,
• ‐ constrained agents with exponential utilities and Black–Scholes financial market.

We also investigate the limit when the number of agents N goes to infinity. Finally, when the constraints sets are vector spaces, we study the impact of the s on the risk of the market.     

INDIFFERENCE PRICING FOR CONTINGENT CLAIMS: LARGE DEVIATIONS EFFECTS

We study utility indifference prices and optimal purchasing quantities for a nontraded contingent claim in an incomplete semimartingale market with vanishing hedging errors. We make connections with the theory of large deviations. We concentrate on sequences of semicomplete markets where in the nth market, the claim admits the decomposition . Here, is replicable by trading in the underlying assets , but is independent of . Under broad conditions, we may assume that vanishes in accordance with a large deviations principle (LDP) as n grows. In this setting, for an exponential investor, we identify the limit of the average indifference price , for units of , as . We show that if , the limiting price typically differs from the price obtained by assuming bounded positions , and the difference is explicitly identifiable using large deviations theory. Furthermore, we show that optimal purchase quantities occur at the large deviations scaling, and hence large positions arise endogenously in this setting.     

Weak equilibria for time-inconsistent control: With applications to investment-withdrawal decisions

This paper considers time-inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the time-inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair $(\widehat{u},C)$ of control-stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two-dimensional example with a hyperbolic discount.     

Cover Image,Volume 30, Issue 2

The cover image is based on the Original Article Semistatic and Sparse Variance‐Optimal Hedging by Di Tella et al., https://doi.org/10.1111/mafi.12235 .