SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 295-314, March 2024. Abstract. We study mean-risk optimal portfolio problems where risk is measured by recovery average value at risk, a prominent example in the class of recovery risk measures. We establish existence results in the situation where the joint distribution of portfolio assets is known as well as in the situation where it is uncertain and only assumed to belong to a set of mixtures of benchmark distributions (mixture uncertainty) or to a cloud around a benchmark distribution (box uncertainty). The comparison with the classical average value at risk shows that portfolio selection under its recovery version enables financial institutions to exert better control on the recovery on liabilities while still allowing for tractable computations.
{"title":"Robust Portfolio Selection under Recovery Average Value at Risk","authors":"Cosimo Munari, Justin Plückebaum, Stefan Weber","doi":"10.1137/23m1555491","DOIUrl":"https://doi.org/10.1137/23m1555491","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 295-314, March 2024. <br/> Abstract. We study mean-risk optimal portfolio problems where risk is measured by recovery average value at risk, a prominent example in the class of recovery risk measures. We establish existence results in the situation where the joint distribution of portfolio assets is known as well as in the situation where it is uncertain and only assumed to belong to a set of mixtures of benchmark distributions (mixture uncertainty) or to a cloud around a benchmark distribution (box uncertainty). The comparison with the classical average value at risk shows that portfolio selection under its recovery version enables financial institutions to exert better control on the recovery on liabilities while still allowing for tractable computations.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140322323","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 1, Page 255-294, March 2024. Abstract. The classic optimized certainty equivalent (OCE), proposed by Ben-Tal and Teboulle [Manag. Sci., 11 (1986), pp. 1445–1466], employs the classical expected utility model to evaluate the random risk, in which model each decision maker is characterized by a unique probability measure and only outcome uncertainty is assumed. Due to the lack of information, the distribution ambiguity or Knightian uncertainty prevails in reality. We employ the variational preference of Maccheroni, Marinacci, and Rustichini [Econometrica, 74 (2006), pp. 1447–1498] to address the issue and generalize the concept of OCE. In this paper, we introduce a class of optimized certainty equivalent based on the variational preference, give its dual representation based on [math]-divergence, and study its equivalent characterization of positive homogeneity and coherence. As applications, we investigate the properties of optimized certainty equivalent based on the rank-dependent utility (RDU) model. The dual representation of the RDU-based shortfall risk measure proposed by Mao and Cai [Finance Stoch., 2 (2018), pp. 367–393] is also presented.
{"title":"Generalized Optimized Certainty Equivalent with Applications in the Rank-Dependent Utility Model","authors":"Qinyu Wu, Tiantian Mao, Taizhong Hu","doi":"10.1137/21m1448276","DOIUrl":"https://doi.org/10.1137/21m1448276","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 255-294, March 2024. <br/> Abstract. The classic optimized certainty equivalent (OCE), proposed by Ben-Tal and Teboulle [Manag. Sci., 11 (1986), pp. 1445–1466], employs the classical expected utility model to evaluate the random risk, in which model each decision maker is characterized by a unique probability measure and only outcome uncertainty is assumed. Due to the lack of information, the distribution ambiguity or Knightian uncertainty prevails in reality. We employ the variational preference of Maccheroni, Marinacci, and Rustichini [Econometrica, 74 (2006), pp. 1447–1498] to address the issue and generalize the concept of OCE. In this paper, we introduce a class of optimized certainty equivalent based on the variational preference, give its dual representation based on [math]-divergence, and study its equivalent characterization of positive homogeneity and coherence. As applications, we investigate the properties of optimized certainty equivalent based on the rank-dependent utility (RDU) model. The dual representation of the RDU-based shortfall risk measure proposed by Mao and Cai [Finance Stoch., 2 (2018), pp. 367–393] is also presented.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140312668","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}
Damiano Brigo, Federico Graceffa, Alexander Kalinin
SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 215-254, March 2024. Abstract. We extend the valuation of contingent claims in the presence of default, collateral, and funding to a random functional setting and characterize pre-default value processes by martingales. Pre-default value semimartingales can also be described by BSDEs with random path-dependent coefficients and martingales as drivers. En route, we relax conditions on the available market information and construct a broad class of default times. Moreover, under stochastic volatility, we characterize pre-default value processes via mild solutions to parabolic semilinear PDEs and give sufficient conditions for mild solutions to exist uniquely and to be classical.
{"title":"Mild to Classical Solutions for XVA Equations under Stochastic Volatility","authors":"Damiano Brigo, Federico Graceffa, Alexander Kalinin","doi":"10.1137/22m1506882","DOIUrl":"https://doi.org/10.1137/22m1506882","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 215-254, March 2024. <br/> Abstract. We extend the valuation of contingent claims in the presence of default, collateral, and funding to a random functional setting and characterize pre-default value processes by martingales. Pre-default value semimartingales can also be described by BSDEs with random path-dependent coefficients and martingales as drivers. En route, we relax conditions on the available market information and construct a broad class of default times. Moreover, under stochastic volatility, we characterize pre-default value processes via mild solutions to parabolic semilinear PDEs and give sufficient conditions for mild solutions to exist uniquely and to be classical.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300532","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 1, Page 194-214, March 2024. Abstract. In this work, we study the deep signature algorithms for path-dependent options. We extend the backward scheme in [C. Huré, H. Pham, and X. Warin, Math. Comp., 89 (2020), pp. 1547–1579] for state-dependent FBSDEs with reflections to path-dependent FBSDEs with reflections, by adding the signature layer to the backward scheme. Our algorithm applies to both European and American type option pricing problems, while the payoff function depends on the whole paths of the underlying forward stock process. We prove the convergence analysis of our numerical algorithm with explicit dependence on the truncation order of the signature and the neural network approximation errors. Numerical examples for the algorithm are provided, including Amerasian option under the Black–Scholes model, American option with a path-dependent geometric mean payoff function, and Shiryaev’s optimal stopping problem.
{"title":"Deep Signature Algorithm for Multidimensional Path-Dependent Options","authors":"Erhan Bayraktar, Qi Feng, Zhaoyu Zhang","doi":"10.1137/23m1571563","DOIUrl":"https://doi.org/10.1137/23m1571563","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 194-214, March 2024. <br/> Abstract. In this work, we study the deep signature algorithms for path-dependent options. We extend the backward scheme in [C. Huré, H. Pham, and X. Warin, Math. Comp., 89 (2020), pp. 1547–1579] for state-dependent FBSDEs with reflections to path-dependent FBSDEs with reflections, by adding the signature layer to the backward scheme. Our algorithm applies to both European and American type option pricing problems, while the payoff function depends on the whole paths of the underlying forward stock process. We prove the convergence analysis of our numerical algorithm with explicit dependence on the truncation order of the signature and the neural network approximation errors. Numerical examples for the algorithm are provided, including Amerasian option under the Black–Scholes model, American option with a path-dependent geometric mean payoff function, and Shiryaev’s optimal stopping problem.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140205759","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 1, Page 161-193, March 2024. Abstract. We prove the existence of a continuous-time Radner equilibrium with multiple agents and transaction costs. The agents are incentivized to trade towards a targeted number of shares throughout the trading period and seek to maximize their expected wealth minus a penalty for deviating from their targets. Their wealth is further reduced by transaction costs that are proportional to the number of stock shares traded. The agents’ targeted number of shares are publicly known. In equilibrium, each agent optimally chooses to trade for an initial time interval before stopping trade. Our equilibrium construction and analysis involves identifying the order in which the agents stop trade. The transaction cost level impacts the equilibrium stock price drift. We analyze the equilibrium outcomes and provide numerical examples.
{"title":"A Multi-agent Targeted Trading Equilibrium with Transaction Costs","authors":"Jin Hyuk Choi, Jetlir Duraj, Kim Weston","doi":"10.1137/22m1542982","DOIUrl":"https://doi.org/10.1137/22m1542982","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 161-193, March 2024. <br/> Abstract. We prove the existence of a continuous-time Radner equilibrium with multiple agents and transaction costs. The agents are incentivized to trade towards a targeted number of shares throughout the trading period and seek to maximize their expected wealth minus a penalty for deviating from their targets. Their wealth is further reduced by transaction costs that are proportional to the number of stock shares traded. The agents’ targeted number of shares are publicly known. In equilibrium, each agent optimally chooses to trade for an initial time interval before stopping trade. Our equilibrium construction and analysis involves identifying the order in which the agents stop trade. The transaction cost level impacts the equilibrium stock price drift. We analyze the equilibrium outcomes and provide numerical examples.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196898","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 1, Page 121-160, March 2024. Abstract. This paper studies an optimal consumption problem for a loss-averse agent with reference to past consumption maximum. To account for loss aversion on relative consumption, an S-shaped utility is adopted that measures the difference between the nonnegative consumption rate and a fraction of the historical spending peak. We consider the concave envelope of the utility with respect to consumption, allowing us to focus on an auxiliary HJB variational inequality on the strength of concavification principle and dynamic programming arguments. By applying the dual-transform and smooth-fit conditions, the auxiliary HJB variational inequality is solved in piecewise closed form, and some thresholds of the wealth variable are obtained. The optimal consumption and investment control can be derived in the piecewise feedback form. The rigorous verification proofs on optimality and concavification principle are provided. Some numerical sensitivity analysis and financial implications are also presented.
{"title":"Optimal Consumption with Loss Aversion and Reference to Past Spending Maximum","authors":"Xun Li, Xiang Yu, Qinyi Zhang","doi":"10.1137/22m149212x","DOIUrl":"https://doi.org/10.1137/22m149212x","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 121-160, March 2024. <br/> Abstract. This paper studies an optimal consumption problem for a loss-averse agent with reference to past consumption maximum. To account for loss aversion on relative consumption, an S-shaped utility is adopted that measures the difference between the nonnegative consumption rate and a fraction of the historical spending peak. We consider the concave envelope of the utility with respect to consumption, allowing us to focus on an auxiliary HJB variational inequality on the strength of concavification principle and dynamic programming arguments. By applying the dual-transform and smooth-fit conditions, the auxiliary HJB variational inequality is solved in piecewise closed form, and some thresholds of the wealth variable are obtained. The optimal consumption and investment control can be derived in the piecewise feedback form. The rigorous verification proofs on optimality and concavification principle are provided. Some numerical sensitivity analysis and financial implications are also presented.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140151908","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 1, Page 93-120, March 2024. Abstract. We study the continuous time Kyle–Back model with a risk averse informed trader. We show that in a market with multiple assets and non-Gaussian prices an equilibrium exists. The equilibrium is constructed by considering a Fokker–Planck equation and a system of partial differential equations that are coupled with an optimal transport type constraint at maturity.
{"title":"Multidimensional Kyle–Back Model with a Risk Averse Informed Trader","authors":"Shreya Bose, Ibrahim Ekren","doi":"10.1137/21m1457059","DOIUrl":"https://doi.org/10.1137/21m1457059","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 93-120, March 2024. <br/> Abstract. We study the continuous time Kyle–Back model with a risk averse informed trader. We show that in a market with multiple assets and non-Gaussian prices an equilibrium exists. The equilibrium is constructed by considering a Fokker–Planck equation and a system of partial differential equations that are coupled with an optimal transport type constraint at maturity.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140127413","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}
Jingyi Cao, Dongchen Li, Virginia R. Young, Bin Zou
SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page SC15-SC27, March 2024. Abstract. We find the optimal indemnity to maximize the expected utility of terminal wealth of a buyer of insurance whose preferences are modeled by an exponential utility. The insurance premium is computed by a convex functional. We obtain a necessary condition for the optimal indemnity; then, because the candidate optimal indemnity is given implicitly, we use that necessary condition to develop a numerical algorithm to compute it. We prove that the numerical algorithm converges to a unique indemnity that, indeed, equals the optimal policy. We also illustrate our results with numerical examples.
{"title":"Short Communication: Optimal Insurance to Maximize Exponential Utility When Premium Is Computed by a Convex Functional","authors":"Jingyi Cao, Dongchen Li, Virginia R. Young, Bin Zou","doi":"10.1137/23m1601237","DOIUrl":"https://doi.org/10.1137/23m1601237","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page SC15-SC27, March 2024. <br/> Abstract. We find the optimal indemnity to maximize the expected utility of terminal wealth of a buyer of insurance whose preferences are modeled by an exponential utility. The insurance premium is computed by a convex functional. We obtain a necessary condition for the optimal indemnity; then, because the candidate optimal indemnity is given implicitly, we use that necessary condition to develop a numerical algorithm to compute it. We prove that the numerical algorithm converges to a unique indemnity that, indeed, equals the optimal policy. We also illustrate our results with numerical examples.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071833","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 1, Page 54-92, March 2024. Abstract.A risk measure that is consistent with the second-order stochastic dominance and additive for sums of independent random variables can be represented as a weighted entropic risk measure (WERM). The expected utility maximization problem with risk controlled by WERM and a related risk minimization problem are investigated in this paper. The latter is equivalent to a problem of maximizing a weighted average of constant-absolute-risk-aversion certainty equivalents. The solutions of all the optimization problems are explicitly characterized, and an iterative method is provided to obtain the solutions numerically.
{"title":"Optimal Investment with Risk Controlled by Weighted Entropic Risk Measures","authors":"Jianming Xia","doi":"10.1137/22m152894x","DOIUrl":"https://doi.org/10.1137/22m152894x","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 54-92, March 2024. <br/> Abstract.A risk measure that is consistent with the second-order stochastic dominance and additive for sums of independent random variables can be represented as a weighted entropic risk measure (WERM). The expected utility maximization problem with risk controlled by WERM and a related risk minimization problem are investigated in this paper. The latter is equivalent to a problem of maximizing a weighted average of constant-absolute-risk-aversion certainty equivalents. The solutions of all the optimization problems are explicitly characterized, and an iterative method is provided to obtain the solutions numerically.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140127237","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 1, Page 26-53, March 2024. Abstract. We study optimal control in models with latent factors where the agent controls the distribution over actions, rather than actions themselves, in both discrete and continuous time. To encourage exploration of the state space, we reward exploration with Tsallis entropy and derive the optimal distribution over states—which we prove is [math]-Gaussian distributed with location characterized through the solution of an BS[math]E and BSDE in discrete and continuous time, respectively. We discuss the relation between the solutions of the optimal exploration problems and the standard dynamic optimal control solution. Finally, we develop the optimal policy in a model-agnostic setting along the lines of soft [math]-learning. The approach may be applied in, e.g., developing more robust statistical arbitrage trading strategies.
{"title":"Exploratory Control with Tsallis Entropy for Latent Factor Models","authors":"Ryan Donnelly, Sebastian Jaimungal","doi":"10.1137/22m153505x","DOIUrl":"https://doi.org/10.1137/22m153505x","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 26-53, March 2024. <br/> Abstract. We study optimal control in models with latent factors where the agent controls the distribution over actions, rather than actions themselves, in both discrete and continuous time. To encourage exploration of the state space, we reward exploration with Tsallis entropy and derive the optimal distribution over states—which we prove is [math]-Gaussian distributed with location characterized through the solution of an BS[math]E and BSDE in discrete and continuous time, respectively. We discuss the relation between the solutions of the optimal exploration problems and the standard dynamic optimal control solution. Finally, we develop the optimal policy in a model-agnostic setting along the lines of soft [math]-learning. The approach may be applied in, e.g., developing more robust statistical arbitrage trading strategies.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139751645","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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