Charles Bertucci, Louis Bertucci, Jean-Michel Lasry, Pierre-Louis Lions
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 960-987, September 2024. Abstract.We present an analysis of the Proof-of-Work consensus algorithm, used on the Bitcoin blockchain, using a mean field game framework. Using a master equation, we provide an equilibrium characterization of the total computational power devoted to mining the blockchain (hashrate). This class of models allows us to adapt to many different situations. The essential structure of the game is preserved across all the enrichments. In deterministic settings, the hashrate ultimately reaches a steady state in which it increases at the rate of technological progress only. In stochastic settings, there exists a target for the hashrate for every possible random state. As a consequence, we show that in equilibrium the security of the underlying blockchain and the energy consumption either are constant or increase with the price of the underlying cryptocurrency.
{"title":"A Mean Field Game Approach to Bitcoin Mining","authors":"Charles Bertucci, Louis Bertucci, Jean-Michel Lasry, Pierre-Louis Lions","doi":"10.1137/23m1617813","DOIUrl":"https://doi.org/10.1137/23m1617813","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 960-987, September 2024. <br/> Abstract.We present an analysis of the Proof-of-Work consensus algorithm, used on the Bitcoin blockchain, using a mean field game framework. Using a master equation, we provide an equilibrium characterization of the total computational power devoted to mining the blockchain (hashrate). This class of models allows us to adapt to many different situations. The essential structure of the game is preserved across all the enrichments. In deterministic settings, the hashrate ultimately reaches a steady state in which it increases at the rate of technological progress only. In stochastic settings, there exists a target for the hashrate for every possible random state. As a consequence, we show that in equilibrium the security of the underlying blockchain and the energy consumption either are constant or increase with the price of the underlying cryptocurrency.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 931-959, September 2024. Abstract. Constant product markets with concentrated liquidity (CL) are the most popular type of automated market makers. In this paper, we characterize the continuous-time wealth dynamics of strategic liquidity providers (LPs) who dynamically adjust their range of liquidity provision in CL pools. Their wealth results from fee income, the value of their holdings in the pool, and rebalancing costs. Next, we derive a self-financing and closed-form optimal liquidity provision strategy where the width of the LP’s liquidity range is determined by the profitability of the pool (provision fees minus gas fees), the predictable loss (PL) of the LP’s position, and concentration risk. Concentration risk refers to the decrease in fee revenue if the marginal exchange rate (akin to the midprice in a limit order book) in the pool exits the LP’s range of liquidity. When the drift in the marginal rate is stochastic, we show how to optimally skew the range of liquidity to increase fee revenue and profit from the expected changes in the marginal rate. Finally, we use Uniswap v3 data to show that, on average, LPs have traded at a significant loss, and to show that the out-of-sample performance of our strategy is superior to the historical performance of LPs in the pool we consider.
{"title":"Decentralized Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision","authors":"Álvaro Cartea, Fayçal Drissi, Marcello Monga","doi":"10.1137/23m1602103","DOIUrl":"https://doi.org/10.1137/23m1602103","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 931-959, September 2024. <br/> Abstract. Constant product markets with concentrated liquidity (CL) are the most popular type of automated market makers. In this paper, we characterize the continuous-time wealth dynamics of strategic liquidity providers (LPs) who dynamically adjust their range of liquidity provision in CL pools. Their wealth results from fee income, the value of their holdings in the pool, and rebalancing costs. Next, we derive a self-financing and closed-form optimal liquidity provision strategy where the width of the LP’s liquidity range is determined by the profitability of the pool (provision fees minus gas fees), the predictable loss (PL) of the LP’s position, and concentration risk. Concentration risk refers to the decrease in fee revenue if the marginal exchange rate (akin to the midprice in a limit order book) in the pool exits the LP’s range of liquidity. When the drift in the marginal rate is stochastic, we show how to optimally skew the range of liquidity to increase fee revenue and profit from the expected changes in the marginal rate. Finally, we use Uniswap v3 data to show that, on average, LPs have traded at a significant loss, and to show that the out-of-sample performance of our strategy is superior to the historical performance of LPs in the pool we consider.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142257346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 883-930, September 2024. Abstract. We study an optimal consumption, investment, and retirement decision of an economic agent with borrowing constraints under a general class of utility functions. We transform the problem into a dual two-person zero-sum game, which involves two players: a stopper who is a maximizer and chooses a stopping time and a controller who is a minimizer and chooses a nonincreasing process. We derive the Hamilton–Jacobi–Bellman quasi-variational inequality (HJBQVI) of a max-min type from the dual two-person zero-sum game. We provide a solution to the HJBQVI and verify that the solution to the HJBQVI is the value of the dual two-person zero-sum game. We establish the duality result which allows us to derive the optimal strategies and value function of the primal problem from those of the dual problem. We provide examples for a class of utility functions.
{"title":"A Two-Person Zero-Sum Game Approach for a Retirement Decision with Borrowing Constraints","authors":"Junkee Jeon, Hyeng Keun Koo, Minsuk Kwak","doi":"10.1137/22m1528124","DOIUrl":"https://doi.org/10.1137/22m1528124","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 883-930, September 2024. <br/> Abstract. We study an optimal consumption, investment, and retirement decision of an economic agent with borrowing constraints under a general class of utility functions. We transform the problem into a dual two-person zero-sum game, which involves two players: a stopper who is a maximizer and chooses a stopping time and a controller who is a minimizer and chooses a nonincreasing process. We derive the Hamilton–Jacobi–Bellman quasi-variational inequality (HJBQVI) of a max-min type from the dual two-person zero-sum game. We provide a solution to the HJBQVI and verify that the solution to the HJBQVI is the value of the dual two-person zero-sum game. We establish the duality result which allows us to derive the optimal strategies and value function of the primal problem from those of the dual problem. We provide examples for a class of utility functions.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142223077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Giulia Di Nunno, Yuliya Mishura, Anton Yurchenko-Tytarenko
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 824-882, September 2024. Abstract.We introduce a new model of financial market with stochastic volatility driven by an arbitrary Hölder continuous Gaussian Volterra process. The distinguishing feature of the model is the form of the volatility equation, which ensures that the solution is “sandwiched” between two arbitrary Hölder continuous functions chosen in advance. We discuss the structure of local martingale measures on this market, investigate integrability and Malliavin differentiability of prices and volatilities, and study absolute continuity of the corresponding probability laws. Additionally, we utilize Malliavin calculus to develop an algorithm of pricing options with discontinuous payoffs.
{"title":"Option Pricing in Sandwiched Volterra Volatility Model","authors":"Giulia Di Nunno, Yuliya Mishura, Anton Yurchenko-Tytarenko","doi":"10.1137/22m1521328","DOIUrl":"https://doi.org/10.1137/22m1521328","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 824-882, September 2024. <br/> Abstract.We introduce a new model of financial market with stochastic volatility driven by an arbitrary Hölder continuous Gaussian Volterra process. The distinguishing feature of the model is the form of the volatility equation, which ensures that the solution is “sandwiched” between two arbitrary Hölder continuous functions chosen in advance. We discuss the structure of local martingale measures on this market, investigate integrability and Malliavin differentiability of prices and volatilities, and study absolute continuity of the corresponding probability laws. Additionally, we utilize Malliavin calculus to develop an algorithm of pricing options with discontinuous payoffs.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142222996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 785-823, September 2024. Abstract.We reconcile rough volatility models and jump models using a class of reversionary Heston models with fast mean reversions and large vol-of-vols. Starting from hyper-rough Heston models with a Hurst index [math]-, we derive a Markovian approximating class of one-dimensional reversionary Heston-type models. Such proxies encode a trade-off between an exploding vol-of-vol and a fast mean-reversion speed controlled by a reversionary timescale [math] and an unconstrained parameter [math]. Sending [math] to 0 yields convergence of the reversionary Heston model toward different explicit asymptotic regimes based on the value of the parameter [math]. In particular, for [math], the reversionary Heston model converges to a class of Lévy jump processes of normal inverse Gaussian type. Numerical illustrations show that the reversionary Heston model is capable of generating at-the-money skews similar to the ones generated by rough, hyper-rough, and jump models.
{"title":"Reconciling Rough Volatility with Jumps","authors":"Eduardo Abi Jaber, Nathan De Carvalho","doi":"10.1137/23m1558847","DOIUrl":"https://doi.org/10.1137/23m1558847","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 785-823, September 2024. <br/> Abstract.We reconcile rough volatility models and jump models using a class of reversionary Heston models with fast mean reversions and large vol-of-vols. Starting from hyper-rough Heston models with a Hurst index [math]-, we derive a Markovian approximating class of one-dimensional reversionary Heston-type models. Such proxies encode a trade-off between an exploding vol-of-vol and a fast mean-reversion speed controlled by a reversionary timescale [math] and an unconstrained parameter [math]. Sending [math] to 0 yields convergence of the reversionary Heston model toward different explicit asymptotic regimes based on the value of the parameter [math]. In particular, for [math], the reversionary Heston model converges to a class of Lévy jump processes of normal inverse Gaussian type. Numerical illustrations show that the reversionary Heston model is capable of generating at-the-money skews similar to the ones generated by rough, hyper-rough, and jump models.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142222997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 734-784, September 2024. Abstract.We derive quantitative error bounds for deep neural networks (DNNs) approximating option prices on a [math]-dimensional risky asset as functions of the underlying model parameters, payoff parameters, and initial conditions. We cover a general class of stochastic volatility models of Markovian nature as well as the rough Bergomi model. In particular, under suitable assumptions we show that option prices can be learned by DNNs up to an arbitrary small error [math] while the network size grows only subpolynomially in the asset vector dimension [math] and the reciprocal [math] of the accuracy. Hence, the approximation does not suffer from the curse of dimensionality. As quantitative approximation results for DNNs applicable in our setting are formulated for functions on compact domains, we first consider the case of the asset price restricted to a compact set, and then we extend these results to the general case by using convergence arguments for the option prices.
{"title":"Approximation Rates for Deep Calibration of (Rough) Stochastic Volatility Models","authors":"Francesca Biagini, Lukas Gonon, Niklas Walter","doi":"10.1137/23m1606769","DOIUrl":"https://doi.org/10.1137/23m1606769","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 734-784, September 2024. <br/> Abstract.We derive quantitative error bounds for deep neural networks (DNNs) approximating option prices on a [math]-dimensional risky asset as functions of the underlying model parameters, payoff parameters, and initial conditions. We cover a general class of stochastic volatility models of Markovian nature as well as the rough Bergomi model. In particular, under suitable assumptions we show that option prices can be learned by DNNs up to an arbitrary small error [math] while the network size grows only subpolynomially in the asset vector dimension [math] and the reciprocal [math] of the accuracy. Hence, the approximation does not suffer from the curse of dimensionality. As quantitative approximation results for DNNs applicable in our setting are formulated for functions on compact domains, we first consider the case of the asset price restricted to a compact set, and then we extend these results to the general case by using convergence arguments for the option prices.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142223013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 700-733, September 2024. Abstract.Systemic risk measures were introduced to capture the global risk and the corresponding contagion effects that are generated by an interconnected system of financial institutions. To this purpose, two approaches were suggested. In the first one, systemic risk measures can be interpreted as the minimal amount of cash needed to secure a system after aggregating individual risks. In the second approach, systemic risk measures can be interpreted as the minimal amount of cash that secures a system by allocating capital to each single institution before aggregating individual risks. Although the theory behind these risk measures has been well investigated by several authors, the numerical part has been neglected so far. In this paper, we use stochastic algorithms schemes in estimating multivariate shortfall risk measure and prove that the resulting estimators are consistent and asymptotically normal. We also test numerically the performance of these algorithms on several examples.
{"title":"Estimation of Systemic Shortfall Risk Measure Using Stochastic Algorithms","authors":"Sarah Kaakaï, Anis Matoussi, Achraf Tamtalini","doi":"10.1137/22m1539344","DOIUrl":"https://doi.org/10.1137/22m1539344","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 700-733, September 2024. <br/> Abstract.Systemic risk measures were introduced to capture the global risk and the corresponding contagion effects that are generated by an interconnected system of financial institutions. To this purpose, two approaches were suggested. In the first one, systemic risk measures can be interpreted as the minimal amount of cash needed to secure a system after aggregating individual risks. In the second approach, systemic risk measures can be interpreted as the minimal amount of cash that secures a system by allocating capital to each single institution before aggregating individual risks. Although the theory behind these risk measures has been well investigated by several authors, the numerical part has been neglected so far. In this paper, we use stochastic algorithms schemes in estimating multivariate shortfall risk measure and prove that the resulting estimators are consistent and asymptotically normal. We also test numerically the performance of these algorithms on several examples.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page SC54-SC67, September 2024. Abstract.When an investor is faced with the option to purchase additional information regarding an asset price, how much should she pay? To address this question, we solve for the indifference price of information in a setting where a trader maximizes her expected utility of terminal wealth over a finite time horizon. If she does not purchase the information, then she solves a partial information stochastic control problem, while if she does purchase the information, then she pays a cost and receives partial information about the asset’s trajectory. We further demonstrate that when the investor can purchase the information at any stopping time prior to the end of the trading horizon, she chooses to do so at a deterministic time.
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page SC54-SC67, September 2024. 摘要.当投资者面临购买资产价格额外信息的选择时，她应该支付多少钱？为了解决这个问题，我们求解了交易者在有限时间跨度内最大化其终端财富预期效用的情况下的信息无差异价格。如果她不购买信息，那么她就解决了一个部分信息随机控制问题，而如果她购买信息，那么她就支付了成本，并获得了关于资产轨迹的部分信息。我们进一步证明，当投资者可以在交易期限结束前的任何停止时间购买信息时，她会选择在确定时间购买。
{"title":"Short Communication: The Price of Information","authors":"Sebastian Jaimungal, Xiaofei Shi","doi":"10.1137/24m1644791","DOIUrl":"https://doi.org/10.1137/24m1644791","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page SC54-SC67, September 2024. <br/> Abstract.When an investor is faced with the option to purchase additional information regarding an asset price, how much should she pay? To address this question, we solve for the indifference price of information in a setting where a trader maximizes her expected utility of terminal wealth over a finite time horizon. If she does not purchase the information, then she solves a partial information stochastic control problem, while if she does purchase the information, then she pays a cost and receives partial information about the asset’s trajectory. We further demonstrate that when the investor can purchase the information at any stopping time prior to the end of the trading horizon, she chooses to do so at a deterministic time.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jonathan Chávez-Casillas, José E. Figueroa-López, Chuyi Yu, Yi Zhang
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 653-699, September 2024. Abstract.A novel high-frequency market making approach in discrete time is proposed that admits closed-form solutions. By taking advantage of demand functions that are linear in the quoted bid and ask spreads with random coefficients, we model the variability of the partial filling of limit orders posted in a limit order book (LOB). As a result, we uncover new patterns as to how the demand’s randomness affects the optimal placement strategy. We also allow the price process to follow general dynamics without any Brownian or martingale assumption as is commonly adopted in the literature. The most important feature of our optimal placement strategy is that it can react or adapt to the behavior of market orders online. Using LOB data, we train our model and reproduce the anticipated final profit and loss of the optimal strategy on a given testing date using the actual flow of orders in the LOB. Our adaptive optimal strategies outperform the nonadaptive strategy and those that quote limit orders at a fixed distance from the midprice.
{"title":"Adaptive Optimal Market Making Strategies with Inventory Liquidation Cost","authors":"Jonathan Chávez-Casillas, José E. Figueroa-López, Chuyi Yu, Yi Zhang","doi":"10.1137/23m1571058","DOIUrl":"https://doi.org/10.1137/23m1571058","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 653-699, September 2024. <br/> Abstract.A novel high-frequency market making approach in discrete time is proposed that admits closed-form solutions. By taking advantage of demand functions that are linear in the quoted bid and ask spreads with random coefficients, we model the variability of the partial filling of limit orders posted in a limit order book (LOB). As a result, we uncover new patterns as to how the demand’s randomness affects the optimal placement strategy. We also allow the price process to follow general dynamics without any Brownian or martingale assumption as is commonly adopted in the literature. The most important feature of our optimal placement strategy is that it can react or adapt to the behavior of market orders online. Using LOB data, we train our model and reproduce the anticipated final profit and loss of the optimal strategy on a given testing date using the actual flow of orders in the LOB. Our adaptive optimal strategies outperform the nonadaptive strategy and those that quote limit orders at a fixed distance from the midprice.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 601-652, September 2024. Abstract.This paper studies the problem of partial hedging within the framework of rough volatility models in an incomplete market setting. We employ a stochastic control problem formulation to minimize the discrepancy between a stochastic target and the terminal value of a hedging portfolio. As rough volatility models are neither Markovian nor semimartingales, stochastic control problems associated with rough models are quite complex to solve. Therefore, we propose a multifactor approximation of the rough volatility model and introduce the associated Markov stochastic control problem. We establish the convergence of the optimal solution for the Markov partial hedging problem to the optimal solution of the original problem as the number of factors tends to infinity. Furthermore, the optimal solution of the Markov problem can be derived by solving a Hamilton–Jacobi–Bellman equation and more precisely a nonlinear partial differential equation (PDE). Due to the inherent complexity of this nonlinear PDE, an explicit formula for the optimal solution is generally unattainable. By introducing the dual solution of the Markov problem and expressing the primal solution as a function of the dual solution, we derive approximate solutions to the Markov problem using a dual control method. This method allows for suboptimal choices of dual control to deduce lower and upper bounds on the optimal solution as well as suboptimal hedging ratios. In particular, explicit formulas for partial hedging strategies in a rough Heston model are derived.
{"title":"Partial Hedging in Rough Volatility Models","authors":"Edouard Motte, Donatien Hainaut","doi":"10.1137/23m1583090","DOIUrl":"https://doi.org/10.1137/23m1583090","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 3, Page 601-652, September 2024. <br/> Abstract.This paper studies the problem of partial hedging within the framework of rough volatility models in an incomplete market setting. We employ a stochastic control problem formulation to minimize the discrepancy between a stochastic target and the terminal value of a hedging portfolio. As rough volatility models are neither Markovian nor semimartingales, stochastic control problems associated with rough models are quite complex to solve. Therefore, we propose a multifactor approximation of the rough volatility model and introduce the associated Markov stochastic control problem. We establish the convergence of the optimal solution for the Markov partial hedging problem to the optimal solution of the original problem as the number of factors tends to infinity. Furthermore, the optimal solution of the Markov problem can be derived by solving a Hamilton–Jacobi–Bellman equation and more precisely a nonlinear partial differential equation (PDE). Due to the inherent complexity of this nonlinear PDE, an explicit formula for the optimal solution is generally unattainable. By introducing the dual solution of the Markov problem and expressing the primal solution as a function of the dual solution, we derive approximate solutions to the Markov problem using a dual control method. This method allows for suboptimal choices of dual control to deduce lower and upper bounds on the optimal solution as well as suboptimal hedging ratios. In particular, explicit formulas for partial hedging strategies in a rough Heston model are derived.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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