Michał Barski
In the paper a problem of risk measures on a discrete-time market model with transaction costs is studied. Strategy effectiveness and shortfall risk is introduced. This paper is a generalization of quantile hedging presented in [4].
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With model uncertainty characterized by a convex, possibly non-dominated set of probability measures, the agent minimizes the cost of hedging a path dependent contingent claim with given expected success ratio, in a discrete-time, semi-static market of stocks and options. Based on duality results which link quantile hedging to a randomized composite hypothesis test, an arbitrage-free discretization of the market is proposed as an approximation. The discretized market has a dominating measure, which guarantees the existence of the optimal hedging strategy and helps numerical calculation of the …
The pricing, hedging, optimal exercise and optimal cancellation of game or Israeli options are considered in a multi-currency model with proportional transaction costs. Efficient constructions for optimal hedging, cancellation and exercise strategies are presented, together with numerical examples, as well as probabilistic dual representations for the bid and ask price of a game option.
In this paper, we consider the problem of hedging Asian options in financial markets with transaction costs. For this, we use the asymptotic hedging approach. The main task of asymptotic hedging in financial markets with transaction costs is to prove the probability convergence of the terminal value of the investment portfolio to the payment function when the number of portfolio revisions tends to be $n$ to infinity. In practice, this means that the investor, using such a strategy, is able to compensation payments for all financial transactions, even if their number increases unlimitedly.
This paper studies the problem of option replication in general stochastic volatility markets with transaction costs, using a new specification for the volatility adjustment in Leland's algorithm \cite{Leland}. We prove several limit theorems for the normalized replication error of Leland's strategy, as well as that of the strategy suggested by Lépinette. The asymptotic results obtained not only generalize the existing results, but also enable us to fix the under-hedging property pointed out by Kabanov and Safarian. We also discuss possible methods to improve the convergence rate and to reduce…
We consider so-called regular invertible Gaussian Volterra processes and derive a formula for their prediction laws. Examples of such processes include the fractional Brownian motions and the mixed fractional Brownian motions. As an application, we consider conditional-mean hedging under transaction costs in Black-Scholes type pricing models where the Brownian motion is replaced with a more general regular invertible Gaussian Volterra process.
Dynamic hedging of an European option under a general local volatility model with small linear transaction costs is studied. A continuous control version of Leland's strategy that asymptotically replicates the payoff is constructed. An associated central limit theorem of hedging error is proved. The asymptotic error variance is minimized by an explicit trading strategy.
Hedging in Jump Diffusion Model with Transaction Costs
本研究针对跳跃扩散模型，提出条件最小二乘对冲策略并推导闭式解，特别处理交易成本下的欧式看涨期权，通过递归策略和决策树等工具验证，为投资者提供考虑交易成本的对冲方法。
We study the problem of option replication under constant proportional transaction costs in models where stochastic volatility and jumps are combined to capture the market's important features. Assuming some mild condition on the jump size distribution we show that transaction costs can be approximately compensated by applying the Leland adjusting volatility principle and the asymptotic property of the hedging error due to discrete readjustments is characterized. In particular, the jump risk can be approximately eliminated and the results established in continuous diffusion models are recovere…
Convex duality for two two different super--replication problems in a continuous time financial market with proportional transaction cost is proved. In this market, static hedging in a finite number of options, in addition to usual dynamic hedging with the underlying stock, are allowed. The first one the problems considered is the model--independent hedging that requires the super--replication to hold for every continuous path. In the second one the market model is given through a probability measure P and the inequalities are understood P almost surely. The main result, using the convex duali…
The QLBS Model within the presence of feedback loops through the impacts of a large trader
本研究扩展QLBS模型，考虑大交易者永久影响汇率的情形，通过批量强化学习提出低交易成本且收敛至公平价格的最优对冲策略。
The classical discrete time model of proportional transaction costs relies on the assumption that a feasible portfolio process has solvent increments at each step. We extend this setting in two directions, allowing for convex transaction costs and assuming that increments of the portfolio process belong to the sum of a solvency set and a family of multivariate acceptable positions, e.g. with respect to a dynamic risk measure. We describe the sets of superhedging prices, formulate several no (risk) arbitrage conditions and explore connections between them. In the special case when multivariate …
Explicit Recursive Construction of Super-Replication Prices under Proportional Transaction Costs
针对离散时间比例交易成本下的欧式衍生品超对冲问题，提出基于凸对偶与扭曲Legendre--Fenchel变换的显式递归方案，无需鞅论证即可计算超对冲价格与最优策略，并在真实数据上验证了其实用性。
The probability minimizing problem of large losses of portfolio in discrete and continuous time models is studied. This gives a generalization of quantile hedging presented in [3].
Hedging Options on Asset Portfolios against Just One Underlying Asset in the Presence of Transaction Costs
本文研究在交易成本下，用与标的资产相关的低成本资产对冲资产组合期权，通过模拟数据计算风险调整值以决定对冲资产选择。
Model-Free Deep Hedging with Transaction Costs and Light Data Requirements
本文提出一种仅需256条轨迹即可训练的无模型深度对冲方法，在 Geometric Brownian Motion 框架下显著优于 Black & Scholes 公式与 Leland 模型，降低了数据需求并提升了实时金融序列上的实用性。
A risk-averse agent hedges her exposure to a non-tradable 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 non-tradable risk factor $ψ(U_T)$ is non-linear, we provide an approximation to the optimal strategy in closed-form, and prove that the value function is correctly approximated…
We provide a Fundamental Theorem of Asset Pricing and a Superhedging Theorem for a model independent discrete time financial market with proportional transaction costs. We consider a probability-free version of the Robust No Arbitrage condition introduced in Schachermayer ['04] and show that this is equivalent to the existence of Consistent Price Systems. Moreover, we prove that the superhedging price for a claim g coincides with the frictionless superhedging price of g for a suitable process in the bid-ask spread.
Duality Theory for Exponential Utility--Based Hedging in the Almgren--Chriss Model
研究Almgren–Chriss模型下含二次交易成本且到期强制清算的指数效用最大化问题，建立对偶理论，并据此给出Bachelier模型中二次收益欧式或有权益的显式最优对冲策略。
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 …
We consider Geometric Mean Market Makers -- a special type of Decentralized Exchange -- with two types of users: liquidity takers and arbitrageurs. Liquidity takers trade at prices that can create arbitrage opportunities, while arbitrageurs align the exchange's price with the external market price. We show that in Geometric Mean Market Makers charging proportional transaction fees, Impermanent Loss can be super-hedged by a model-free rebalancing strategy. Moreover, we demonstrate that in such a DEX, the exchange rate is of finite variation, so that loss-versus-rebalancing (the shortfall of pro…
In this paper we derive robust super- and subhedging dualities for contingent claims that can depend on several underlying assets. In addition to strict super- and subhedging, we also consider relaxed versions which, instead of eliminating the shortfall risk completely, aim to reduce it to an acceptable level. This yields robust price bounds with tighter spreads. As examples we study strict super- and subhedging with general convex transaction costs and trading constraints as well as risk-based hedging with respect to robust versions of the average value at risk and entropic risk measure. Our …
We study partial hedging for game options in markets with transaction costs bounded from below. More precisely, we assume that the investor's transaction costs for each trade are the maximum between proportional transaction costs and a fixed transaction costs. We prove that in the continuous time Black--Scholes (BS) model, there exists a trading strategy which minimizes the shortfall risk. Furthermore, we use binomial models in order to provide numerical schemes for the calculation of the shortfall risk and the corresponding optimal portfolio in the BS model.
This work studies the deep learning-based numerical algorithms for optimal hedging problems in markets with general convex transaction costs on the trading rates, focusing on their scalability of trading time horizon. Based on the comparison results of the FBSDE solver by Han, Jentzen, and E (2018) and the Deep Hedging algorithm by Buehler, Gonon, Teichmann, and Wood (2019), we propose a Stable Transfer Hedging (ST-Hedging) algorithm, to aggregate the convenience of the leading-order approximation formulas and the accuracy of the deep learning-based algorithms. Our ST-Hedging algorithm achieve…
The problem of hedging and pricing sequences of contingent claims in large financial markets is studied. Connection between asymptotic arbitrage and behavior of the $α$~-~quantile price is shown. The large Black-Scholes model is carefully examined.
This paper studies the utility maximization on the terminal wealth with random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system (CPS) such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility ma…
On convergence of the Mayer problems arising in the theory of financial markets with transaction cost
本文在带有交易成本的金融市场理论框架下，探讨了随机Mayer控制问题，研究了投资组合终端财富期望效用最大化问题的连续性及最优控制。
Paper is based on "The cost of illiquidity and its effects on hedging", L. C. G. Rogers and Surbjeet Singh, 2010. We generalize its thesis to constant elasticity model, which own previously used Black-Schoels model as a special case. The Goal of this article is to find optimal hedging strategy of European call/put option in illiquid environment. We understand illiquidity as a non linear transaction cost function depending only on rate of change of our portfolio. In case this function is quadratic, optimal policy is given by system of 3 PDE. In addition we show, that for small $ε$ costs of sell…
We prove the superhedging duality for a discrete-time financial market with proportional transaction costs under model uncertainty. Frictions are modeled through solvency cones as in the original model of [Kabanov, Y., Hedging and liquidation under transaction costs in currency markets. Fin. Stoch., 3(2):237-248, 1999] adapted to the quasi-sure setup of [Bouchard, B. and Nutz, M., Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab., 25(2):823-859, 2015]. Our approach allows to remove the restrictive assumption of No Arbitrage of the Second Kind considered in [Bouchar…
We propose a hedging approach for general contingent claims when liquidity is a concern and trading is subject to transaction cost. Multiple assets with different liquidity levels are available for hedging. Our risk criterion targets a tradeoff between minimizing the risk against fluctuations in the stock price and incurring low liquidity costs. Following Çetin U., Jarrow R.A., and Protter P. (2004) we work in an arbitrage-free setting assuming a supply curve for each asset. In discrete time, following the ideas in Schweizer M. (1998) and Lamberton D., Pham H., Schweizer M. (1998) we prove the…
Continuous-time Equilibrium Returns in Markets with Price Impact and Transaction Costs
本文研究具有价格冲击和交易成本的伊藤金融市场，通过市场出清条件内生推导均衡收益，比较无摩擦与有摩擦情形，并刻画纳什均衡为FBSDEs的解，发现噪声交易者存在时交易成本影响被放大。
Applying the Cherny-Shiryaev-Yor invariance principle, we introduce a generalized Jarrow-Rudd (GJR) option pricing model with uncertainty driven by a skew random walk. The GJR pricing tree exhibits skewness and kurtosis in both the natural and risk-neutral world. We construct implied surfaces for the parameters determining the GJR tree. Motivated by Merton's pricing tree incorporating transaction costs, we extend the GJR pricing model to include a hedging cost. We demonstrate ways to fit the GJR pricing model to a market driver that influences the price dynamics of the underlying asset. We sup…
Proactive Market Making and Liquidity Analysis for Everlasting Options in DeFi Ecosystems
本文研究永续期权市场的流动性问题，通过动态主动做市模型分析资金费率和交易成本，在低流动性高交易成本环境下展示流动性提供者如何通过有效对冲策略实现正收益，揭示流动性提供者的激励机制及永续期权对交易者的价值。
Market Making and Transient Impact in Spot FX
本文研究外汇市场中做市商在考虑风险管理和市场影响时的最优策略，探讨瞬时影响与永久影响之间的平衡。
We consider the framework proposed by Burgard and Kjaer (2011) that derives the PDE which governs the price of an option including bilateral counterparty risk and funding. We extend this work by relaxing the assumption of absence of transaction costs in the hedging portfolio by proposing a cost proportional to the amount of assets traded and the traded price. After deriving the nonlinear PDE, we prove the existence of a solution for the corresponding initial-boundary value problem. Moreover, we develop a numerical scheme that allows to find the solution of the PDE by setting different values f…
We study utility indifference prices and optimal purchasing quantities for a contingent claim, in an incomplete semi-martingale market, in the presence of vanishing hedging errors and/or risk aversion. Assuming that the average indifference price converges to a well defined limit, we prove that optimally taken positions become large in absolute value at a specific rate. We draw motivation from and make connections to Large Deviations theory, and in particular, the celebrated Gärtner-Ellis theorem. We analyze a series of well studied examples where this limiting behavior occurs, such as fixed m…
One Currency, Two Forward Prices: The Onshore-Offshore Renminbi Puzzle
本文通过建立包含交易成本和分割供给的即期-远期联合均衡模型，研究了在岸人民币（CNY）与离岸人民币（CNH）远期价格持续且经济上显著差异的原因，并发现随机离岸流动性压力（建模为交易成本跳跃）可以解释这一差异。
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