We consider a game-theoretic model of a market where investors compete for payoffs yielded by several assets. The main result consists in a proof of the existence and uniqueness of a strategy, called relative growth optimal, such that the logarithm of the share of its wealth in the total wealth of the market is a submartingale for any strategies of the other investors. It is also shown that this strategy is asymptotically optimal in the sense that it achieves the maximal capital growth rate when compared to competing strategies. Based on the results obtained, we study the asymptotic structure of the market when all the investors use the relative growth optimal strategy.
We present a family of multi-asset automated market makers whose liquidity curves are derived from the financial principles of self financing transactions and rebalancing. The constant product market maker emerges as a special case.
We propose a multi-agent model of an asset market and study conditions that guarantee that the strategy of an individual agent cannot outperform the market. The model assumes a mean-field approximation of the market by considering an infinite number of infinitesimal agents who use the same strategy and another infinitesimal agent with a different strategy who tries to outperform the market. We show that the optimal strategy for the market agents is to split their investment budgets among the assets proportionally to their discounted expected relative dividend intensities.
We consider a stochastic game-theoretic model of a discrete-time asset market with short-lived assets and endogenous asset prices. We prove that the strategy which invests in the assets proportionally to their expected relative payoffs asymptotically minimizes the expected time needed to reach a large wealth level. The result is obtained under the assumption that the relative asset payoffs and the growth rate of the total payoff during each time period are independent and identically distributed.
We consider a continuous-time game-theoretic model of an investment market with short-lived assets and endogenous asset prices. The first goal of the paper is to formulate a stochastic equation which determines wealth processes of investors and to provide conditions for the existence of its solution. The second goal is to show that there exists a strategy such that the logarithm of the relative wealth of an investor who uses it is a submartingale regardless of the strategies of the other investors, and the relative wealth of any other essentially different strategy vanishes asymptotically. Thi…
We study asset price bubbles in market models with proportional transaction costs $λ\in (0,1)$ and finite time horizon $T$ in the setting of [49]. By following [28], we define the fundamental value $F$ of a risky asset $S$ as the price of a super-replicating portfolio for a position terminating in one unit of the asset and zero cash. We then obtain a dual representation for the fundamental value by using the super-replication theorem of [50]. We say that an asset price has a bubble if its fundamental value differs from the ask-price $(1+λ)S$. We investigate the impact of transaction costs on a…
A market portfolio is a portfolio in which each asset is held at a weight proportional to its market value. Functionally generated portfolios are portfolios for which the logarithmic return relative to the market portfolio can be decomposed into a function of the market weights and a process of locally finite variation, and this decomposition is convenient for characterizing the long-term behavior of the portfolio. A permutation-weighted portfolio is a portfolio in which the assets are held at weights proportional to a permutation of their market values, and such a portfolio is functionally ge…
We study the range of prices at which a rational agent should contemplate transacting a financial contract outside a given securities market. Trading is subject to nonproportional transaction costs and portfolio constraints and full replication by way of market instruments is not always possible. Rationality is defined in terms of consistency with market prices and acceptable risk thresholds. We obtain a direct and a dual description of market-consistent prices with acceptable risk. The dual characterization requires an appropriate extension of the classical Fundamental Theorem of Asset Pricin…
I demonstrate that with the market return determined by the equilibrium returns of the CAPM, expected returns of an asset are affected by the risks of all assets jointly. Another implication is that the range of feasible market returns will be limited and dependent on the distribution of weights in the market portfolio. A large and well diversified market with no dominating asset will only return zero while a market dominated by a small number of assets will only return the risk-free rate. In the limiting case of atomistic assets, we recover the properties of the standard CAPM.
We define data-driven macroeconomic regimes by clustering the relative performance in time of indices belonging to different asset classes. We then investigate lead-lag relationships within the regimes identified. Our study unravels market features characteristic of different windows in time and leverages on this knowledge to highlight market trends or risks that can be informative with respect to recurrent market developments. The framework developed also lays the foundations for multiple possible extensions.
We obtain error estimates for strong approximations of a diffusion with a diffusion matrix $σ$ and a drift b by the discrete time process defined recursively X_N((n+1)/N) = X_N(n/N)+N^{1/2}σ(X_N(n/N))ξ(n+1)+N^{-1}b(XN(n/N)); where ξ(n); n\geq 1 are i.i.d. random vectors, and apply this in order to approximate the fair price of a game option with a diffusion asset price evolution by values of Dynkin's games with payoffs based on the above discrete time processes. This provides an effective tool for computations of fair prices of game options with path dependent payoffs in a multi asset market w…
We study the ex-ante minimization of market inefficiency, defined in terms of minimum deviation of market prices from fundamental values, from a centralized planner's perspective. Prices are pressured from exogenous trading actions of leverage targeting banks, which rebalance their portfolios in response to asset shocks. We characterize market inefficiency in terms of two key drivers, the banks' systemic significance and the statistical moments of asset shocks, and develop an explicit expression for the matrix of asset holdings which minimizes such inefficiency. Our analysis shows that to redu…
Given the promising results on joint modeling of SPX/VIX smiles of the recently introduced quadratic rough Heston model, we consider a multi-asset market making problem on SPX and its derivatives, e.g. VIX futures, SPX and VIX options. The market maker tries to maximize its profit from spread capturing while controlling the portfolio's inventory risk, which can be fully explained by the value change of SPX under the particular setting of the quadratic rough Heston model. The high dimensionality of the resulting optimization problem is relaxed by several approximations. An asymptotic closed-for…
We consider an incomplete multi-asset binomial market model. We prove that for a wide class of contingent claims the extremal multi-step martingale measure is a power of the corresponding single-step extremal martingale measure. This allows for closed form formulas for the bounds of a no-arbitrage contingent claim price interval. We construct a feasible algorithm for computing those boundaries as well as for the corresponding hedging strategies. Our results apply, for example, to European basket call and put options and Asian arithmetic average options.
J.P. Morgan（摩根大通）；University of Kaiserlautern（凯泽斯劳滕大学）
We construct realistic spot and equity option market simulators for a single underlying on the basis of normalizing flows. We address the high-dimensionality of market observed call prices through an arbitrage-free autoencoder that approximates efficient low-dimensional representations of the prices while maintaining no static arbitrage in the reconstructed surface. Given a multi-asset universe, we leverage the conditional invertibility property of normalizing flows and introduce a scalable method to calibrate the joint distribution of a set of independent simulators while preserving the dynam…
We consider a mean-variance portfolio selection problem in a financial market with contagion risk. The risky assets follow a jump-diffusion model, in which jumps are driven by a multivariate Hawkes process with mutual-excitation effect. The mutual-excitation feature of the Hawkes process captures the contagion risk in the sense that each price jump of an asset increases the likelihood of future jumps not only in the same asset but also in other assets. We apply the stochastic maximum principle, backward stochastic differential equation theory, and linear-quadratic control technique to solve th…
In this paper, we propose and study a novel continuous-time model, based on the well-known constant elasticity of variance (CEV) model, to describe the asset price process. The basic idea is that the volatility elasticity of the CEV model can not be treated as a constant from the perspective of stochastic analysis. To address this issue, we deduce the price process of assets from the perspective of volatility elasticity, propose the constant volatility elasticity (CVE) model, and further derive a more general variable volatility elasticity (VVE) model. Moreover, our model can describe the posi…
