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arXiv2017

Incomplete stochastic equilibria for dynamic monetary utility

We study existence and uniqueness of continuous-time stochastic Radner equilibria in an incomplete market model among a group of agents whose preference is characterized by cash invariant time-consistent monetary utilities. An assumption of "smallness" type is shown to be sufficient for existence and uniqueness. In particular, this assumption encapsulates settings with small endowments, small time-horizon, or a large population of weakly heterogeneous agents. Central role in our analysis is played by a fully-coupled nonlinear system of quadratic BSDEs.

math.PR↗
arXiv2016

Optimal Stopping with Random Maturity under Nonlinear Expectations

We analyze an optimal stopping problem with random maturity under a nonlinear expectation with respect to a weakly compact set of mutually singular probabilities $\mathcal{P}$. The maturity is specified as the hitting time to level $0$ of some continuous index process at which the payoff process is even allowed to have a positive jump. When $\mathcal{P}$ is a collection of semimartingale measures, the optimal stopping problem can be viewed as a {\it discretionary} stopping problem for a player who can influence both drift and volatility of the dynamic of underlying stochastic flow.

math.PR↗
arXiv2016

Weak Convergence of Path-Dependent SDEs in Basket CDS Pricing with Contagion Risk

We investigate the computational aspects of the basket CDS pricing with counterparty risk under a credit contagion model of multinames. This model enables us to capture the systematic volatility increases in the market triggered by a particular bankruptcy. The drawback of this problem is its analytical complication due to its path-dependent functional, which bears a potential failure in its convergence of numerical approximation under standing assumptions. In this paper we find sufficient conditions for the desired convergence of the functionals associated with a class of path-dependent stochastic differential equations. The main ingredient is to identify the weak convergence of the approximated solution to the underlying path-dependent stochastic differential equation.

q-fin.MF↗
arXiv2016

Optimal Investment to Minimize the Probability of Drawdown

We determine the optimal investment strategy in a Black-Scholes financial market to minimize the so-called {\it probability of drawdown}, namely, the probability that the value of an investment portfolio reaches some fixed proportion of its maximum value to date. We assume that the portfolio is subject to a payout that is a deterministic function of its value, as might be the case for an endowment fund paying at a specified rate, for example, at a constant rate or at a rate that is proportional to the fund's value.

q-fin.MF↗
arXiv2015

Good deal bounds with convex constraints

We investigate the structure of good deal bounds, which are subintervals of a no-arbitrage pricing bound, for financial market models with convex constraints as an extension of Arai and Fukasawa (2014). The upper and lower bounds of a good deal bound are naturally described by a convex risk measure. We call such a risk measure a good deal valuation; and study its properties. We also discuss superhedging cost and Fundamental Theorem of Asset Pricing for convex constrained markets.

q-fin.MF↗
arXiv2015

New exact Taylor's expansions and simple solutions to PDEs

We provide new exact Taylor's series with fixed coefficients and without the remainder. We demonstrate the usefulness of this contribution by using it to obtain very simple solutions to (non-linear) PDEs. We also apply the method to the portfolio model.

q-fin.MF↗
arXiv2017

An Empirical Approach to Financial Crisis Indicators Based on Random Matrices

The aim of this work is to build financial crisis indicators based on spectral properties of the dynamics of market data. After choosing an optimal size for a rolling window, the historical market data in this window is seen every trading day as a random matrix from which a covariance and a correlation matrix are obtained. The financial crisis indicators that we have built deal with the spectral properties of these covariance and correlation matrices and they are of two kinds. The first one is based on the Hellinger distance, computed between the distribution of the eigenvalues of the empirical covariance matrix and the distribution of the eigenvalues of a reference covariance matrix representing either a calm or agitated market. The idea behind this first type of indicators is that when the empirical distribution of the spectrum of the covariance matrix is deviating from the reference in the sense of Hellinger, then a crisis may be forthcoming. The second type of indicators is based on the study of the spectral radius and the trace of the covariance and correlation matrices as a mean to directly study the volatility and correlations inside the market. The idea behind the second type of indicators is the fact that large eigenvalues are a sign of dynamic instability. The predictive power of the financial crisis indicators in this framework is then demonstrated, in particular by using them as decision-making tools in a protective-put strategy.

q-fin.MF↗
arXiv2015

Local risk-minimization for Barndorff-Nielsen and Shephard models with volatility risk premium

We derive representations of local risk-minimization of call and put options for Barndorff-Nielsen and Shephard models: jump type stochastic volatility models whose squared volatility process is given by a non-Gaussian rnstein-Uhlenbeck process. The general form of Barndorff-Nielsen and Shephard models includes two parameters: volatility risk premium $β$ and leverage effect $ρ$. Arai and Suzuki (2015, arxiv:1503.08589) dealt with the same problem under constraint $β=-\frac{1}{2}$. In this paper, we relax the restriction on $β$; and restrict $ρ$ to $0$ instead. We introduce a Malliavin calculus under the minimal martingale measure to solve the problem.

q-fin.MF↗
arXiv2015

No-Arbitrage Prices of Cash Flows and Forward Contracts as Choquet Representations

In a market of deterministic cash flows, given as an additive, symmetric relation of exchangeability on the finite signed Borel measures on the non-negative real time axis, it is shown that the only arbitrage-free price functional that fulfills some additional mild requirements is the integral of the unit zero-coupon bond prices with respect to the payment measures. For probability measures, this is a Choquet representation, where the Dirac measures, as unit zero-coupon bonds, are the extreme points. Dropping one of the requirements, the Lebesgue decomposition is used to construct counterexamples, where the Choquet price formula does not hold despite of an arbitrage-free market model. The concept is then extended to deterministic streams of assets and currencies in general, yielding a valuation principle for forward markets. Under mild assumptions, it is shown that a foreign cash flow's worth in local currency is identical to the value of the cash flow in local currency for which the Radon-Nikodym derivative with respect to the foreign cash flow is the forward FX rate.

q-fin.MF↗
arXiv2015

Optimal Static Quadratic Hedging

We propose a flexible framework for hedging a contingent claim by holding static positions in vanilla European calls, puts, bonds, and forwards. A model-free expression is derived for the optimal static hedging strategy that minimizes the expected squared hedging error subject to a cost constraint. The optimal hedge involves computing a number of expectations that reflect the dependence among the contingent claim and the hedging assets. We provide a general method for approximating these expectations analytically in a general Markov diffusion market. To illustrate the versatility of our approach, we present several numerical examples, including hedging path-dependent options and options written on a correlated asset.

q-fin.MF↗
arXiv2016

Optimal Skorokhod embedding under finitely-many marginal constraints

The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the optimal Skorokhod embedding problem to the case of finitely-many marginal constraints. Using the classical convex duality approach together with the optimal stopping theory, we obtain the duality results which are formulated by means of probability measures on an enlarged space. We also relate these results to the problem of martingale optimal transport under multiple marginal constraints.

math.PR↗
arXiv2016

Model-free Superhedging Duality

In a model free discrete time financial market, we prove the superhedging duality theorem, where trading is allowed with dynamic and semi-static strategies. We also show that the initial cost of the cheapest portfolio that dominates a contingent claim on every possible path $ω\in Ω$, might be strictly greater than the upper bound of the no-arbitrage prices. We therefore characterize the subset of trajectories on which this duality gap disappears and prove that it is an analytic set.

q-fin.MF↗
arXiv2015

Intrinsic Storage Valuation by Variational Analysis

The mathematical problem concerning intrinsic storage optimisation is formulated and solved by means of variational analysis. The solution, though obtained in implicit form, still sheds light on many important features of the optimal exercise strategy. It is shown how the solution depends on different constraint types including carry cost and cycle constraint. Additionally, the relationship between intrinsic and stochastic solutions is investigated. In particular, we show that the optimal stochastic exercise decision is always close to the intrinsic one.

math.OC↗
arXiv2020

Elicitation Complexity of Statistical Properties

A property, or statistical functional, is said to be elicitable if it minimizes expected loss for some loss function. The study of which properties are elicitable sheds light on the capabilities and limitations of point estimation and empirical risk minimization. While recent work asks which properties are elicitable, we instead advocate for a more nuanced question: how many dimensions are required to indirectly elicit a given property? This number is called the elicitation complexity of the property. We lay the foundation for a general theory of elicitation complexity, including several basic results about how elicitation complexity behaves, and the complexity of standard properties of interest. Building on this foundation, our main result gives tight complexity bounds for the broad class of Bayes risks. We apply these results to several properties of interest, including variance, entropy, norms, and several classes of financial risk measures. We conclude with discussion and open directions.

cs.LG↗
arXiv2016

Martingale property of exponential semimartingales: a note on explicit conditions and applications to financial models

We give a collection of explicit sufficient conditions for the true martingale property of a wide class of exponentials of semimartingales. We express the conditions in terms of semimartingale characteristics. This turns out to be very convenient in financial modeling in general. Especially it allows us to carefully discuss the question of well-definedness of semimartingale Libor models, whose construction crucially relies on a sequence of measure changes.

q-fin.MF↗
arXiv2016

On the Robust Dynkin Game

We study a robust Dynkin game over a set of mutually singular probabilities. We first prove that for the conservative player of the game, her lower and upper value processes coincide (i.e. She has a value process $V $ in the game). Such a result helps people connect the robust Dynkin game with second-order doubly reflected backward stochastic differential equations. Also, we show that the value process $V$ is a submartingale under an appropriately defined nonlinear expectations up to the first time $τ_*$ when $V$ meets the lower payoff process $L$. If the probability set is weakly compact, one can even find an optimal triplet. The mutual singularity of probabilities in causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times. The mutual singularity of probabilities causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times

math.PR↗
arXiv2019

The Long-Term Swap Rate and a General Analysis of Long-Term Interest Rates

We introduce here for the first time the long-term swap rate, characterised as the fair rate of an overnight indexed swap with infinitely many exchanges. Furthermore we analyse the relationship between the long-term swap rate, the long-term yield, see Biagini et al. [2018], Biagini and Härtel [2014], and El Karoui et al. [1997], and the long-term simple rate, considered in Brody and Hughston [2016] as long-term discounting rate. We finally investigate the existence of these long-term rates in two term structure methodologies, the Flesaker-Hughston model and the linear-rational model. A numerical example illustrates how our results can be used to estimate the non-optional component of a CoCo bond.

q-fin.PR↗