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Similar Journals
 SIAM Journal on Financial MathematicsJournal Prestige (SJR): 1.222 Citation Impact (citeScore): 1Number of Followers: 3      Hybrid journal (It can contain Open Access articles) ISSN (Online) 1945-497X Published by Society for Industrial and Applied Mathematics  [17 journals]
• Short Communication: Minimal Quantile Functions Subject to Stochastic
Dominance Constraints

Authors: Xiangyu Wang, Jianming Xia, Zuo Quan Xu, Zhou Yang
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page SC87-SC98, September 2022.
We consider a problem of finding a second-order stochastic dominance (SSD)--minimal quantile function subject to the mixture of first-order stochastic dominance (FSD) and SSD constraints. The SSD-minimal solution is explicitly worked out and has a close relation to the Skorokhod problem. This result is then applied to explicitly solve a risk minimizing problem in financial economics.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-09-08T07:00:00Z
DOI: 10.1137/22M1488557
Issue No: Vol. 13, No. 3 (2022)

• Optimal Ratcheting of Dividends in a Brownian Risk Model

Authors: Hansjörg Albrecher, Pablo Azcue, Nora Muler
Pages: 657 - 701
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 657-701, September 2022.
We study the problem of optimal dividend payout from a surplus process governed by Brownian motion with drift under the additional constraint of ratcheting, i.e., the dividend rate can never decrease. We solve the resulting two-dimensional optimal control problem, identifying the value function to be the unique viscosity solution of the corresponding Hamilton--Jacobi--Bellman equation. For finitely many admissible dividend rates we prove that threshold strategies are optimal, and for any closed interval of admissible dividend rates we establish the $\varepsilon$-optimality of curve strategies. This work is a counterpart of [H. Albrecher, P. Azcue, and N. Muler, SIAM J. Control Optim., 58 (2020) pp. 1822--1845], where the ratcheting problem was studied for a compound Poisson surplus process with drift. In the present Brownian setup, calculus of variation techniques allow us to obtain a much more explicit analysis and description of the optimal dividend strategies. We also give some numerical illustrations of the optimality results.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-07-05T07:00:00Z
DOI: 10.1137/20M1387171
Issue No: Vol. 13, No. 3 (2022)

• Principal Eigenportfolios for U.S. Equities

Authors: Marco Avellaneda, Brian Healy, Andrew Papanicolaou, George Papanicolaou
Pages: 702 - 744
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 702-744, September 2022.
We analyze portfolios constructed from the principal eigenvector of the equity returns' correlation matrix and compare these portfolios with the capitalization weighted market portfolio. It is well known empirically that principal eigenportfolios are a good proxy for the market portfolio. We quantify this property through the large-dimensional asymptotic analysis of a spike model with diverging top eigenvalue, comprising a rank-one matrix and a random matrix. We show that, in this limit, the top eigenvector of the correlation matrix is close to the vector of market betas divided componentwise by returns standard deviation. Historical returns data are generally consistent with this analysis of the correspondence between the top eigenportfolio and the market portfolio. We further examine this correspondence using eigenvectors obtained from hierarchically constructed tensors where stocks are separated into their respective industry sectors. This hierarchical approach results in a principal factor whose portfolio weights are all positive for a greater percentage of time compared to the weights of the vanilla eigenportfolio computed from the correlation matrix. Returns from hierarchical construction are also more robust with respect to the duration of the time window used for estimation. All principal eigenportfolios that we observe have returns that exceed those of the market portfolio between 1994 and 2020. We attribute these excess returns to the brief periods where short holdings are more than a small percentage of portfolio weight.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-07-18T07:00:00Z
DOI: 10.1137/20M1383501
Issue No: Vol. 13, No. 3 (2022)

• Stochastic Control of Optimized Certainty Equivalents

Authors: Julio Backhoff Veraguas, A. Max Reppen, Ludovic Tangpi
Pages: 745 - 772
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 745-772, September 2022.
Optimized certainty equivalents (OCEs) are a family of risk measures widely used by both practitioners and academics. This is mostly due to its tractability and the fact that it encompasses important examples, including entropic risk measures and average value-at-risk. In this work we consider stochastic optimal control problems where the objective criterion is given by an OCE risk measure or, in other words, a risk minimization problem for controlled diffusions. A major difficulty arises since OCEs are often time-inconsistent. Nevertheless, via an enlargement of state space we achieve a substitute of sorts for time-consistency in fair generality. This allows us to derive a dynamic programming principle and thus recover central results of (risk-neutral) stochastic control theory. In particular, we show that the value of our risk minimization problem can be characterized as a viscosity solution of a Hamilton--Jacobi--Bellman--Isaacs equation. We further establish a comparison principle and uniqueness of the latter under suitable technical conditions.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-07-28T07:00:00Z
DOI: 10.1137/21M1407732
Issue No: Vol. 13, No. 3 (2022)

• Perpetual American Standard and Lookback Options with Event Risk and
Asymmetric Information

Authors: Pavel V. Gapeev, Libo Li
Pages: 773 - 801
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 773-801, September 2022.
We derive closed-form solutions to the perpetual American standard and floating-strike lookback put and call options in an extension of the Black--Merton--Scholes model with event risk and asymmetric information. It is assumed that the contracts are terminated by their writers with linear or fractional recoveries at the last hitting times for the underlying asset price process of its ultimate maximum or minimum over the infinite time interval which are not stopping times with respect to the reference filtration. We show that the optimal exercise times for the holders are the first times at which the asset price reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. The optimal exercise boundaries are proven to be the maximal or minimal solutions of some first-order nonlinear ordinary differential equations.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-08-01T07:00:00Z
DOI: 10.1137/21M1396848
Issue No: Vol. 13, No. 3 (2022)

• Robust Consumption-Investment with Return Ambiguity: A Dual Approach with
Volatility Ambiguity

Authors: Kyunghyun Park, Hoi Ying Wong
Pages: 802 - 843
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 802-843, September 2022.
Consider a robust consumption-investment problem for a risk- and ambiguity-averse investor who is concerned about return ambiguity in risky asset prices. When the investor aims to maximize the worst-case scenario of his/her consumption-investment objective, we propose a dual approach to the robust optimization problem in a dual economy with volatility ambiguity. Using the $G$-expectation framework, we establish the duality theorem to bridge between the primal problem with return ambiguity and the dual problem with volatility ambiguity, and hence characterize the robust strategy for a general class of utility functions subject to the nonnegative consumption rate and wealth constraints. The volatility ambiguity in the dual problem induces correlation ambiguity when the primal economy comprises multiple risky assets with return ambiguity. By analyzing the dual economy, we show that the robust investment strategy favors a sparse portfolio, in addition to its usual feature---having the least exposure to ambiguity.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-08-01T07:00:00Z
DOI: 10.1137/21M1440189
Issue No: Vol. 13, No. 3 (2022)

• Forward Utility and Market Adjustments in Relative Investment-Consumption
Games of Many Players

Authors: Gonçalo dos Reis, Vadim Platonov
Pages: 844 - 876
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 844-876, September 2022.
We study a portfolio management problem featuring many-player and mean field competition, investment and consumption, and relative performance concerns under the forward performance processes (FPP) framework. We focus on agents using power Constant Relative Risk Aversion type FPPs for their investment-consumption optimization problem under a common noise Merton market model. We solve both the many-player and mean field game providing closed-form expressions for the solutions where the limit of the former yields the latter. In our case, the FPP framework yields a continuum of solutions for the consumption component as indexed to a market parameter we coin “market-risk relative consumption preference.” The parameter permits the agent to set a preference for their consumption going forward in time that, in the competition case, reflects a common market behavior. We show the FPP framework, under both competition and no-competition, allows the agent to disentangle her risk-tolerance and elasticity of intertemporal substitution (EIS) just like Epstein--Zin preferences under a recursive utility framework and unlike the classical utility theory one. This, in turn, allows a finer analysis on the agent's consumption “income” and “substitution” regimes, and, of independent interest, motivates a new strand of economics research on EIS under the FPP framework. We find that competition rescales the agent's perception of consumption in a nontrivial manner. We provide numerical illustrations of our results.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-08-04T07:00:00Z
DOI: 10.1137/20M138421X
Issue No: Vol. 13, No. 3 (2022)

• Insiders and Their Free Lunches: The Role of Short Positions

Pages: 877 - 902
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 877-902, September 2022.
Given a stock price process, we analyze the potential of arbitrage in a context of short-selling prohibitions. We introduce the notion of minimal supermartingale measure, and we analyze its properties in connection with the minimal martingale measure. This question is more specifically analyzed in the case of an investor having additional, inside information. In particular, we establish conditions when minimal martingale and supermartingale measures both fail to exist. These correspond to the case when the insider information includes some nonnull events that are perceived as having null probabilities by the uninformed market investors, even as they cannot observe them. The results may have different applications, such as in problems related to the local risk minimization for insiders whenever strategies are implemented without short selling.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-08-08T07:00:00Z
DOI: 10.1137/20M1375826
Issue No: Vol. 13, No. 3 (2022)

• Optimal Dynamic Reinsurance Under Heterogeneous Beliefs and CARA Utility

Authors: Hui Meng, Pengyu Wei, Wanlu Zhang, Sheng Chao Zhuang
Pages: 903 - 943
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 903-943, September 2022.
This paper examines the optimal dynamic reinsurance policy for an insurance company under belief heterogeneity. We assume the reinsurance premium is calculated according to the mean-conditional value-at-risk principle and impose the incentive compatibility constraint to rule out moral hazard. Under the objective of maximizing the exponential utility function, we obtain the optimal strategies in closed form via a “relaxation and modification” approach. The optimal contracts have more complicated structures than the standard proportional and excess-of-loss reinsurance widely investigated in the literature. In particular, we demonstrate that the insurer may optimally choose to purchase proportional reinsurance in different layers when the reinsurer is more pessimistic about the underlying loss. Our model lends support to the observation that reinsurance contracts in practice often involve proportional reinsurance in multiple layers. We also demonstrate that belief heterogeneity can explain the inverse relationship between the purchase of reinsurance and the size of the insurer's losses observed in the reinsurance market.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-08-11T07:00:00Z
DOI: 10.1137/21M1411093
Issue No: Vol. 13, No. 3 (2022)

• Optimal Trading with Signals and Stochastic Price Impact

Authors: Jean-Pierre Fouque, Sebastian Jaimungal, Yuri F. Saporito
Pages: 944 - 968
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 944-968, September 2022.
Trading frictions are stochastic. They are, moreover, in many instances fast mean-reverting. Here, we study how to optimally trade in a market with stochastic price impact and study approximations to the resulting optimal control problem using singular perturbation methods. We prove, by constructing sub- and supersolutions, that the approximations are accurate to the specified order. Finally, we perform some numerical experiments to illustrate the effect that stochastic trading frictions have on optimal trading.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-08-11T07:00:00Z
DOI: 10.1137/21M1394473
Issue No: Vol. 13, No. 3 (2022)

• Optimal Investment with Time-Varying Stochastic Endowments

Authors: Christoph Belak, An Chen, Carla Mereu, Robert Stelzer
Pages: 969 - 1003
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 969-1003, September 2022.
This paper considers a utility maximization and optimal asset allocation problem in the presence of a stochastic endowment that cannot be fully hedged through trading in the financial market. After studying continuity properties of the value function for general utility functions, we rely on the dynamic programming approach to solve the optimization problem for power utility investors including the empirically relevant and mathematically challenging case of relative risk aversion larger than one. For this, we argue that the value function is the unique viscosity solution of the Hamilton--Jacobi--Bellman (HJB) equation. The homogeneity of the value function is then used to reduce the HJB equation by one dimension, which allows us to prove that the value function is even a classical solution thereof. Using this, an optimal strategy is derived and its asymptotic behavior in the large wealth regime is discussed.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-08-16T07:00:00Z
DOI: 10.1137/21M1453402
Issue No: Vol. 13, No. 3 (2022)

• Robust Portfolio Choice with Sticky Wages

Authors: Sara Biagini, Fausto Gozzi, Margherita Zanella
Pages: 1004 - 1039
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 1004-1039, September 2022.
We present a robust version of the life-cycle optimal portfolio choice problem in the presence of labor income, as introduced in Biffis, Gozzi, and Prosdocimi [SIAM J. Control Optim., 58 (2020), pp. 1906--1938] and Dybvig and Liu [J. Econom. Theory, 145 (2010), pp. 885--907]. In particular, in Biffis, Gozzi, and Prosdocimi the influence of past wages on the future ones is modeled linearly in the evolution equation of labor income, through a given weight function. The optimization relies on the resolution of an infinite dimensional HJB equation. We improve the state of art in three ways. First, we allow the weight to be a Radon measure. This accommodates for more realistic weighting of the sticky wages, e.g., on a discrete temporal grid according to some periodic income. Second, there is a general correlation structure between labor income and stock markets. This naturally affects the optimal hedging demand, which may increase or decrease according to the correlation sign. Third, we allow the weight to change with time, possibly lacking perfect identification. The uncertainty is specified by a given set of Radon measures K, in which the weight process takes values. This renders the inevitable uncertainty on how the past affects the future, and includes the standard case of error bounds on a specific estimate for the weight. Under uncertainty averse preferences, the decision maker takes a maxmin approach to the problem. Our analysis confirms the intuition: in the infinite dimensional setting, the optimal policy remains the best investment strategy under the worst case weight.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-08-16T07:00:00Z
DOI: 10.1137/21M1429722
Issue No: Vol. 13, No. 3 (2022)

• Power Mixture Forward Performance Processes

Authors: Levon Avanesyan, Ronnie Sircar
Pages: 1040 - 1062
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 1040-1062, September 2022.
We consider the forward investment problem in market models where the stock prices are continuous semimartingales adapted to a Brownian filtration. We construct a broad class of forward performance processes with initial conditions of power mixture type, $u(x) = \int_{\mathbb{I}} \frac{x^{1-\gamma}}{1-\gamma }\nu(d \gamma)$. We proceed to define and fully characterize two-power mixture forward performance processes with constant risk aversion coefficients in the interval $(0,1)$, and derive properties of two-power mixture forward performance processes when the risk aversion coefficients are continuous stochastic processes. Finally, we discuss the problem of managing an investment pool of two investors, whose respective preferences evolve as power forward performance processes.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-08-18T07:00:00Z
DOI: 10.1137/20M1385500
Issue No: Vol. 13, No. 3 (2022)

• Realization Utility with Path-Dependent Reference Points

Authors: Linghui Kong, Cong Qin, Xingye Yue
Pages: 1063 - 1111
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 1063-1111, September 2022.
In this paper, we propose a model to study the impact of path-dependent reference points on optimal trading strategies of a realization utility investor, where the reference points are initial purchase prices adjusted dynamically by subsequent paper gains and losses. By homogeneity, this optimal trading problem can be reformulated as a new one with a unique state variable $x$ (wealth-reference ratio), which can be solved analytically. Importantly, we find that the introduction of path-dependent reference points can generate two interesting effects: (a) a discount effect, i.e., a constant subjective discount factor in the original problem becomes stochastic in the new problem; and (b) a mean-reverting effect, i.e., the state variable $x$ follows a mean-reverting process. These two effects offset each other in the state of paper gain ($x>1$) while getting reinforced in the state of paper loss ($x Citation: SIAM Journal on Financial Mathematics PubDate: 2022-08-23T07:00:00Z DOI: 10.1137/21M1411457 Issue No: Vol. 13, No. 3 (2022) • Multiple Anchor Point Shrinkage for the Sample Covariance Matrix • Free pre-print version: Loading... Authors: Hubeyb Gurdogan, Alec Kercheval Pages: 1112 - 1143 Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 1112-1143, September 2022. Estimation of the covariance of a high-dimensional returns vector is well-known to be impeded by the lack of long data history. We extend the work of Goldberg, Papanicolaou, and Shkolnik [SIAM J. Financial Math., 13 (2022), pp. 521--550] on shrinkage estimates for the leading eigenvector of the covariance matrix in the high-dimensional, low sample size regime, which has immediate application to estimating minimum variance portfolios. We introduce a more general framework of shrinkage targets---multiple anchor point shrinkage---that allows the practitioner to incorporate additional information---such as sector separation of equity betas, or prior beta estimates from the recent past---to the estimation. We prove some asymptotic statements and illustrate our results with some numerical experiments. Citation: SIAM Journal on Financial Mathematics PubDate: 2022-08-24T07:00:00Z DOI: 10.1137/21M1446411 Issue No: Vol. 13, No. 3 (2022) • Analysis of Markov Chain Approximation for Diffusion Models with Nonsmooth Coefficients • Free pre-print version: Loading... Authors: Gongqiu Zhang, Lingfei Li Pages: 1144 - 1190 Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 1144-1190, September 2022. In this study, we analyze the convergence of continuous-time Markov chain approximation for one-dimensional diffusions with nonsmooth coefficients. We obtain a sharp estimate of the convergence rate for the value function and its first and second derivatives, which is generally first order. To improve it to second order, we propose two methods: applying the midpoint rule that places all nonsmooth points midway between two neighboring grid points or applying harmonic averaging to smooth the model coefficients. We conduct numerical experiments for various financial applications to confirm the theoretical estimates. We also show that the midpoint rule can be applied to achieve second-order convergence for some jump-diffusion and two-factor short rate models. Citation: SIAM Journal on Financial Mathematics PubDate: 2022-09-15T07:00:00Z DOI: 10.1137/21M1440098 Issue No: Vol. 13, No. 3 (2022) • Escrow and Clawback • Free pre-print version: Loading... Authors: Steven Shreve, Jing Wang Pages: 1191 - 1229 Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 3, Page 1191-1229, September 2022. Since the financial crisis of 2008, clawback provisions have been implemented by several high-profile banks and are also required by regulators in order to mitigate the cost of financial failures and to deter excessive risk taking. We construct a model to investigate the long-term effect on the bank's revenue of deferring (escrowing) a trader's bonuses to facilitate clawback. We formulate the question by setting up an infinite-horizon dynamic programming model. Within this model, the trader's optimal investment and consumption strategy, with and without bonus escrow, can be expressed by explicit analytic formulas. These formulas enable calculation and comparison of the bank's total expected revenue under the two bonus payout schemes. The results of the comparison depend on the parameters describing the trader's risk appetite, the discount factor, and the bank's level of patience, in addition to the market parameters. In particular, when the bank's total expected discounted revenue is finite under both types of bonus payment schemes and the bank is sufficiently patient, the bank benefits by escrowing the trader's bonus, although not escrowing the trader's bonus brings better short-term revenue. Citation: SIAM Journal on Financial Mathematics PubDate: 2022-09-15T07:00:00Z DOI: 10.1137/21M1455619 Issue No: Vol. 13, No. 3 (2022) • Short Communication: Projection of Functionals and Fast Pricing of Exotic Options • Free pre-print version: Loading... Authors: Valentin Tissot-Daguette Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page SC74-SC86, June 2022. We investigate the approximation of path functionals. In particular, we advocate the use of the Karhunen--Loève expansion, the continuous analogue of principal component analysis, to extract relevant information from the image of a functional. Having an accurate estimate of functionals is of paramount importance in the context of exotic derivatives pricing, as presented in the practical applications. Specifically, we show how a simulation-based procedure, which we call the Karhunen--Loève Monte Carlo (KLMC) algorithm, allows fast and efficient computation of the price of path-dependent options. We also explore the path signature as an alternative tool to project both paths and functionals. Citation: SIAM Journal on Financial Mathematics PubDate: 2022-06-27T07:00:00Z DOI: 10.1137/21M1451439 Issue No: Vol. 13, No. 2 (2022) • Short Communication: On the Weak Convergence Rate in the Discretization of Rough Volatility Models • Free pre-print version: Loading... Authors: Christian Bayer, Masaaki Fukasawa, Shonosuke Nakahara Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page SC66-SC73, June 2022. We study the weak convergence rate in the discretization of rough volatility models. After showing a lower bound$2H$under a general model, where$H$is the Hurst index of the volatility process, we give a sharper bound$H + 1/2\$ under a linear model.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-06-13T07:00:00Z
DOI: 10.1137/22M1482871
Issue No: Vol. 13, No. 2 (2022)

• Short Communication: Super-Replication Prices with Multiple Priors in
Discrete Time

Authors: Romain Blanchard, Laurence Carassus
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page SC53-SC65, June 2022.
In the frictionless discrete time financial market of Bouchard and Nutz (2015), we propose a full characterization of the quasi-sure super-replication price, as the supremum of the mono-prior super-replication prices, through an extreme prior and through martingale measures.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-16T07:00:00Z
DOI: 10.1137/22M1470013
Issue No: Vol. 13, No. 2 (2022)

• Short Communication: Chances for the Honest in Honest versus Insider

Authors: Mauricio Elizalde, Carlos Escudero
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page SC39-SC52, June 2022.
We study a Black--Scholes market with a finite time horizon and two investors: an honest and an insider trader. We analyze it with anticipating stochastic calculus in two steps. First, we recover the classical result on portfolio optimization that shows that the expected logarithmic utility of the insider is strictly greater than that of the honest trader. Then, we prove that whenever the market is viable, the honest trader can get a higher logarithmic utility, and therefore more wealth, than the insider with a strictly positive probability. Our proof relies on the analysis of a sort of forward integral variant of the Doléans--Dade exponential process. The main financial conclusion is that the logarithmic utility is perhaps too conservative for some insiders.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-04-28T07:00:00Z
DOI: 10.1137/21M1439547
Issue No: Vol. 13, No. 2 (2022)

• Mean-Variance Portfolio Selection in Contagious Markets

Authors: Yang Shen, Bin Zou
Pages: 391 - 425
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 391-425, June 2022.
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 the problem and obtain the efficient strategy and efficient frontier in semiclosed form, subject to a nonlocal partial differential equation. Numerical examples are provided to illustrate our results.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-04-07T07:00:00Z
DOI: 10.1137/20M1320560
Issue No: Vol. 13, No. 2 (2022)

• American Options in the Volterra Heston Model

Authors: Etienne Chevalier, Sergio Pulido, Elizabeth Zún͂iga
Pages: 426 - 458
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 426-458, June 2022.
We price American options using kernel-based approximations of the Volterra Heston model. We choose these approximations because they allow simulation-based techniques for pricing. We prove the convergence of American option prices in the approximating sequence of models towards the prices in the Volterra Heston model. A crucial step in the proof is to exploit the affine structure of the model in order to establish explicit formulas and convergence results for the conditional Fourier--Laplace transform of the log price and an adjusted version of the forward variance. We illustrate with numerical examples our convergence result and the behavior of American option prices with respect to certain parameters of the model.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-04-27T07:00:00Z
DOI: 10.1137/21M140674X
Issue No: Vol. 13, No. 2 (2022)

• Strong Convergence to the Mean Field Limit of a Finite Agent Equilibrium

Authors: Masaaki Fujii, Akihiko Takahashi
Pages: 459 - 490
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 459-490, June 2022.
We study an equilibrium-based continuous asset pricing problem for the securities market. In the previous work [M. Fujii and A. Takahashi (2022), SIAM J. Control Optim., 60, pp. 259--279], we have shown that a certain price process, which is given by the solution to a forward-backward stochastic differential equation of conditional McKean--Vlasov type, asymptotically clears the market in the large population limit. In the current work, under suitable conditions, we show the existence of a finite agent equilibrium and its strong convergence to the corresponding mean-field limit given in [M. Fujii and A. Takahashi (2022), SIAM J. Control Optim., 60, pp. 259--279]. As an important byproduct, we get the direct estimate on the difference of the equilibrium price between the two markets: the one consisting of heterogeneous agents of finite population size and the other of homogeneous agents of infinite population size.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-04-28T07:00:00Z
DOI: 10.1137/21M1441055
Issue No: Vol. 13, No. 2 (2022)

• Exact Solutions and Approximations for Optimal Investment Strategies and
Indifference Prices

Authors: Michel Vellekoop, Marcellino Gaudenzi
Pages: 491 - 520
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 491-520, June 2022.
In this paper a new approach is proposed to determine the optimal strategy for investment in risky assets by a risk averse investor. To generate approximations for such problems in continuous time, we define a sequence of models in discrete time with a finite state space and a restricted class of utility functions for which the exact optimal strategy can be found. We prove that the graphs of optimal policies form a connected subset of two bundles of parallel lines in the plane and that the optimization problem can be reduced to a sequence of simple binary decisions. This allows us to avoid the search over real numbers that is required for every possible value of the state when finite difference schemes for the Hamilton--Jacobi--Bellman equations in continuous time are used. A very efficient calculation scheme is defined which generates the exact solutions for our discrete time approximations, and we use known results from the theory of viscosity solutions to give conditions which guarantee that a sequence of such approximations for a given problem in continuous time converges to the correct limit. We show in a number of examples how the method can be used to find indifference prices in incomplete markets and that our approach can outperform alternative methods that are based on finite difference schemes.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-02T07:00:00Z
DOI: 10.1137/21M1393303
Issue No: Vol. 13, No. 2 (2022)

• The Dispersion Bias

Authors: Lisa R. Goldberg, Alex Papanicolaou, Alex Shkolnik
Pages: 521 - 550
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 521-550, June 2022.
We identify and correct excess dispersion in the leading eigenvector of a sample covariance matrix when the number of variables vastly exceeds the number of observations. Our correction is data-driven, and it materially diminishes the substantial impact of estimation error on weights and risk forecasts of minimum variance portfolios. We quantify that impact with a novel metric, the optimization bias, which has a positive lower bound prior to correction and tends to zero almost surely after correction. Our analysis sheds light on aspects of how estimation error corrupts an estimated covariance matrix and is transmitted to portfolios via quadratic optimization.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-03T07:00:00Z
DOI: 10.1137/21M144058X
Issue No: Vol. 13, No. 2 (2022)

Authors: Eyal Neuman, Moritz Voß
Pages: 551 - 575
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 551-575, June 2022.
We study optimal liquidation in the presence of linear temporary and transient price impact along with taking into account a general price predicting finite-variation signal. We formulate this problem as minimization of a cost-risk functional over a class of absolutely continuous and signal-adaptive strategies. The stochastic control problem is solved by following a probabilistic and convex analytic approach. We show that the optimal trading strategy is given by a system of four coupled forward-backward SDEs, which can be solved explicitly. Our results reveal how the induced transient price distortion provides together with the predictive signal an additional predictor about future price changes. As a consequence, the optimal signal-adaptive trading rate trades off exploiting the predictive signal against incurring the transient displacement of the execution price from its unaffected level. This answers an open question from [C. A. Lehalle and E. Neuman, Finance Stoch., 23 (2019), pp. 275--311] as we show how to derive the unique optimal signal-adaptive liquidation strategy when price impact is not only temporary but also transient.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-05T07:00:00Z
DOI: 10.1137/20M1375486
Issue No: Vol. 13, No. 2 (2022)

• Functional Portfolio Optimization in Stochastic Portfolio Theory

Authors: Steven Campbell, Ting-Kam Leonard Wong
Pages: 576 - 618
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 576-618, June 2022.
In this paper we develop a concrete and fully implementable approach to the optimization of functionally generated portfolios in stochastic portfolio theory. The main idea is to optimize over a family of rank-based portfolios parameterized by an exponentially concave function on the unit interval. This choice can be motivated by the long term stability of the capital distribution observed in large equity markets and allows us to circumvent the curse of dimensionality. The resulting optimization problem, which is convex, allows for various regularizations and constraints to be imposed on the generating function. We prove an existence and uniqueness result for our optimization problem and provide a stability estimate in terms of a Wasserstein metric of the input measure. Then we formulate a discretization which can be implemented numerically using available software packages and analyze its approximation error. Finally, we present empirical examples using CRSP data from the U.S. stock market, including the performance of the portfolios allowing for dividends, defaults, and transaction costs.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-05T07:00:00Z
DOI: 10.1137/21M1417715
Issue No: Vol. 13, No. 2 (2022)

• Performance Fees with Stochastic Benchmark

Authors: Gu Wang
Pages: 619 - 652
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 619-652, June 2022.
A hedge fund manager invests the fund in a constant investment opportunity, and receives performance fees when the fund reaches a new maximum relative to a stochastic benchmark, aiming to maximize the expected power utility from fees in the long run. The manager's optimal portfolio includes a Merton component with an effective risk aversion parameter shifted from his/her own risk aversion toward one, and an extra component which hedges the risks in the benchmark. The effective risk aversion and the hedging component depend on how the fund investment opportunity compares to the benchmark, which allows investors to regulate the manager's risk taking with a carefully chosen benchmark.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-05-19T07:00:00Z
DOI: 10.1137/21M1401826
Issue No: Vol. 13, No. 2 (2022)

• Erratum: The Robust Superreplication Problem: A Dynamic Approach

Authors: Laurence Carassus, Jan Obłój, Johannes Wiesel
Pages: 653 - 655
Abstract: SIAM Journal on Financial Mathematics, Volume 13, Issue 2, Page 653-655, June 2022.
The assertions of Proposition 3.7 in our paper “The robust superreplication problem: A dynamic approach” [L. Carassus, J. Obłój, and J. Wiesel, SIAM J. Financial Math., 10 (2019), pp. 907--941] may fail to hold without an additional assumption, which we detail in this erratum.
Citation: SIAM Journal on Financial Mathematics
PubDate: 2022-06-28T07:00:00Z
DOI: 10.1137/21M1447040
Issue No: Vol. 13, No. 2 (2022)

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