arXiv subjects
Publications and source records attributed to Ivan Guo.
We extend the Benamou-Brenier formula from classical optimal transport to weak optimal transport and show that the barycentric optimal transport problem studied by Gozlan and Juillet has a dynamic analogue. We also investigate a martingale relaxation of this problem, and relate it to the martingale Benamou-Brenier formula of Backhoff-Veraguas, Beiglb\"ock, Huesmann and K\"allblad.
We consider the robust pricing and hedging of American options in a continuous time setting. We assume asset prices are continuous semimartingales, but we allow for general model uncertainty specification via adapted closed convex constraints on the volatility. We prove the robust pricing-hedging duality. When European options with given prices are available for static trading, we show that duality holds against richer models where these options are traded dynamically. Our proofs rely on probabilistic treatment of randomised stopping times and suitable measure decoupling, and on optimal transport duality. In addition, similarly to the approach of Aksamit et al. (2019) in discrete time, we identify American options with European options on an enlarged space.
We propose a stochastic game modelling the strategic interaction between market makers and traders. From the trader's perspective, the conventional exogenous permanent price impact is replaced by the endogenous quoting strategies of the market makers. Conversely, from the market maker's perspective, order flows are no longer assumed to be exogenous, but are driven endogenously by the strategic traders. Characterizing the Nash equilibria via forward-backward stochastic differential equations (FBSDEs), we establish a local well-posedness result for the general game. For the specific `Almgren-Chriss-Avellaneda-Stoikov' model, the decoupling approach guarantees the global well-posedness of the FBSDEs by reducing it to a backward stochastic Riccati equation with $M_+$-matrix coefficients. Finally, by introducing small diffusion terms into the inventory processes as an approximation to the general game, we establish its global well-posedness. Simulations reveal a negative correlation between quotes and strategic orders, in contrast to the positive correlation observed between quotes and noise orders.
We consider a continuum of carbon-emitting firms who seek to maximise their stock price, and a regulator (e.g., Government) who wishes for the economy to flourish, whilst simultaneously punishing firms who behave non-green. Interpreting the regulator as a major player and the firms as the minor players, we model this setting through a mean field game with major and minor players. We extend the stochastic maximum principle derived by Carmona & Zhu [A probabilistic approach to mean field games with major and minor players. Annals of Applied Probability, 2016, 94, 745--788] by relaxing the assumptions on the forms of the minimisers for the Hamiltonians, allowing them to depend on more arguments. This allows the major and representative minor player to interact in a more natural fashion, thereby permitting us to consider more realistic models for our green and sustainable finance problem. Through our stochastic maximum principle, we derive explicit Nash equilibria for a number of examples.
Building on the macroscopic market making framework as a control problem, this paper investigates its extension to stochastic games. In the context of price competition, each agent is benchmarked against the best quote offered by the others. We begin with the linear case. While constructing the solution directly, the \textit{ordering property} and the dimension reduction in the equilibrium are revealed. For the non-linear case, we extend the decoupling approach by introducing a multidimensional \textit{characteristic equation} to analyse the well-posedness of the forward-backward stochastic differential equations. Properties of the coefficients in this characteristic equation are derived using tools from non-smooth analysis. Several new well-posedness results are presented.
We propose a macroscopic market making model \`a la Avellaneda-Stoikov, using continuous processes for orders instead of discrete point processes. The model intends to bridge the gap between market making and optimal execution problems, while shedding light on the influence of order flows on the optimal strategies. We demonstrate our model through three problems. The study provides a comprehensive analysis from Markovian to non-Markovian noises and from linear to non-linear intensity functions, encompassing both bounded and unbounded coefficients. Mathematically, the contribution lies in the existence and uniqueness of the optimal control, guaranteed by the well-posedness of the strong solution to the Hamilton-Jacobi-Bellman equation and the (non-)Lipschitz forward-backward stochastic differential equation. Finally, the model's applications to price impact and optimal execution are discussed.
In this paper, we introduce two novel methods to solve the American-style option pricing problem and its dual form at the same time using neural networks. Without applying nested Monte Carlo, the first method uses a series of neural networks to simultaneously compute both the lower and upper bounds of the option price, and the second one accomplishes the same goal with one global network. The avoidance of extra simulations and the use of neural networks significantly reduce the computational complexity and allow us to price Bermudan options with frequent exercise opportunities in high dimensions, as illustrated by the provided numerical experiments. As a by-product, these methods also derive a hedging strategy for the option, which can also be used as a control variate for variance reduction.
We consider the superhedging price of an exotic option under nondominated model uncertainty in discrete time in which the option buyer chooses some action from an (uncountable) action space at each time step. By introducing an enlarged space we reformulate the superhedging problem for such an exotic option as a problem for a European option, which enables us to prove the pricing-hedging duality. Next, we present a duality result that, when the option buyers action is observed by the seller up to $l$ periods later, the superhedging price equals the model-based price where the option buyer has the power to look into the future for $l$ periods.
We provide a survey of recent results on model calibration by Optimal Transport. We present the general framework and then discuss the calibration of local, and local-stochastic, volatility models to European options, the joint VIX/SPX calibration problem as well as calibration to some path-dependent options. We explain the numerical algorithms and present examples both on synthetic and market data.
The price of a financial derivative can be expressed as an iterated conditional expectation, where the inner term conditions on the future of an auxiliary process. We show that this inner conditional expectation solves an SPDE (a 'conditional Feynman-Kac formula'). The problem requires conditioning on a backward filtration generated by the noise of the auxiliary process and enlarged by its terminal value, leading us to search for a backward Brownian motion here. This adds a source of irregularity to the SPDE which we tackle with new techniques. Lastly, we establish a new class of mixed Monte-Carlo PDE numerical methods.
We propose two deep neural network-based methods for solving semi-martingale optimal transport problems. The first method is based on a relaxation/penalization of the terminal constraint, and is solved using deep neural networks. The second method is based on the dual formulation of the problem, which we express as a saddle point problem, and is solved using adversarial networks. Both methods are mesh-free and therefore mitigate the curse of dimensionality. We test the performance and accuracy of our methods on several examples up to dimension 10. We also apply the first algorithm to a portfolio optimization problem where the goal is, given an initial wealth distribution, to find an investment strategy leading to a prescribed terminal wealth distribution.
This paper studies a portfolio allocation problem, where the goal is to prescribe the wealth distribution at the final time. We study this problem with the tools of optimal mass transport. We provide a dual formulation which we solve by a gradient descent algorithm. This involves solving an associated HJB and Fokker--Planck equation by a finite difference method. Numerical examples for various prescribed terminal distributions are given, showing that we can successfully reach attainable targets. We next consider adding consumption during the investment process, to take into account distribution that either not attainable, or sub-optimal.
A Dynkin game is a zero-sum, stochastic stopping game between two players where either player can stop the game at any time for an observable payoff. Typically the payoff process of the max-player is assumed to be smaller than the payoff process of the min-player, while the payoff process for simultaneous stopping is in between the two. In this paper, we study general Dynkin games whose payoff processes are in arbitrary positions. In both discrete and continuous time settings, we provide necessary and sufficient conditions for the existence of pure strategy Nash equilibria and epsilon-optimal stopping times in all possible subgames.
This paper addresses the joint calibration problem of SPX options and VIX options or futures. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints, in the spirit of [arXiv:1906.06478]. We introduce a PDE formulation along with its dual counterpart. The solution, a calibrated diffusion process, can be represented via the solutions of Hamilton-Jacobi-Bellman equations arising from the dual formulation. The method is tested on both simulated data and market data. Numerical examples show that the model can be accurately calibrated to SPX options, VIX options and VIX futures simultaneously.
This paper studies the valuation of European contingent claims with short selling bans under the equal risk pricing (ERP) framework proposed in Guo and Zhu (2017) where analytical pricing formulae were derived in the case of monotonic payoffs under risk-neutral measures. We establish a unified framework for this new pricing approach so that its range of application can be significantly expanded. The results of Guo and Zhu (2017) are extended to the case of non-monotonic payoffs (such as a butterfly spread option) under risk-neutral measures. We also provide numerical schemes for computing equal-risk prices under other measures such as the original physical measure. Furthermore, we demonstrate how short selling bans can affect the valuation of contingent claims by comparing equal-risk prices with Black-Scholes prices.
In this paper we consider the question of when a strongly regular graph with parameters $((s+1)(st+1),s(t+1),s-1,t+1)$ can exist. These parameters arise when the graph is derived from a generalized quadrangle, but there are other examples which do not arise in this manner, and we term these {\it pseudo-generalized quadrangles}. If the graph is a generalized quadrangle then $t \leq s^2$ and $s \leq t^2$, while for pseudo-generalized quadrangles we still have the former bound but not the latter. Previously, Neumaier has proved a bound for $s$ which is cubic in $t$, but we improve this to one which is quadratic. The proof involves a careful analysis of cliques and cocliques in the graph. This improved bound eliminates many potential parameter sets which were otherwise feasible.
This paper addresses the problem of utility maximization under uncertain parameters. In contrast with the classical approach, where the parameters of the model evolve freely within a given range, we constrain them via a penalty function. We show that this robust optimization process can be interpreted as a two-player zero-sum stochastic differential game. We prove that the value function satisfies the Dynamic Programming Principle and that it is the unique viscosity solution of an associated Hamilton-Jacobi-Bellman-Isaacs equation. We test this robust algorithm on real market data. The results show that robust portfolios generally have higher expected utilities and are more stable under strong market downturns. To solve for the value function, we derive an analytical solution in the logarithmic utility case and obtain accurate numerical approximations in the general case by three methods: finite difference method, Monte Carlo simulation, and Generative Adversarial Networks.
In this paper, we study a semi-martingale optimal transport problem and its application to the calibration of Local-Stochastic Volatility (LSV) models. Rather than considering the classical constraints on marginal distributions at initial and final time, we optimise our cost function given the prices of a finite number of European options. We formulate the problem as a convex optimisation problem, for which we provide a PDE formulation along with its dual counterpart. Then we solve numerically the dual problem, which involves a fully non-linear Hamilton-Jacobi-Bellman equation. The method is tested by calibrating a Heston-like LSV model with simulated data and foreign exchange market data.