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arXiv2015

Regularity properties in a state-constrained expected utility maximization problem

We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the initial condition fulfilled by the corresponding value function and show its first regularity property. Moreover, we can prove the existence and uniqueness of optimal strategies under rather mild model assumptions. On the one hand, this result is of independent interest. On the other hand, it will then allow us to derive further regularity properties of the corresponding value function, in particular its continuity and partial differentiability. As a consequence of the continuity of the value function, we will prove the dynamic programming principle without appealing to the classical measurable selection arguments.

q-fin.MF↗

arXiv2018

Asymptotic Expansion for Forward-Backward SDEs with Jumps

This work provides a semi-analytic approximation method for decoupled forwardbackward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with σ-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications.

q-fin.CP↗

arXiv2016

Hedging with Temporary Price Impact

We consider the problem of hedging a European contingent claim in a Bachelier model with transient price impact as proposed by Almgren and Chriss. Following the approach of Rogers and Singh and Naujokat and Westray, the hedging problem can be regarded as a cost optimal tracking problem of the frictionless hedging strategy. We solve this problem explicitly for general predictable target hedging strategies. It turns out that, rather than towards the current target position, the optimal policy trades towards a weighted average of expected future target positions. This generalizes an observation of Garleanu and Pedersen from their homogenous Markovian optimal investment problem to a general hedging problem. Our findings complement a number of previous studies in the literature on optimal strategies in illiquid markets where the frictionless strategy is confined to diffusions. The consideration of general predictable reference strategies is made possible by the use of a convex analysis approach instead of the more common dynamic programming methods.

q-fin.MF↗

arXiv2015

Viscosity properties with singularities in a state-constrained expected utility maximization problem

We consider the value function originating from an expected utility maximization problem with finite fuel constraint and show its close relation to a nonlinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity. On one hand, we give a so-called verification argument based on the dynamic programming principle, which allows us to derive conditions under which a classical solution of the HJB equation coincides with our value function (provided that it is smooth enough). On the other hand, we establish a comparison principle, which allows us to characterize our value function as the unique viscosity solution of the HJB equation.

q-fin.MF↗

arXiv2015

Performance analysis of the optimal strategy under partial information

The question addressed in this paper is the performance of the optimal strategy, and the impact of partial information. The setting we consider is that of a stochastic asset price model where the trend follows an unobservable Ornstein-Uhlenbeck process. We focus on the optimal strategy with a logarithmic utility function under full or partial information. For both cases, we provide the asymptotic expectation and variance of the logarithmic return as functions of the signal-to-noise ratio and of the trend mean reversion speed. Finally, we compare the asymptotic Sharpe ratios of these strategies in order to quantify the loss of performance due to partial information.

q-fin.PM↗

arXiv2018

Affine representations of fractional processes with applications in mathematical finance

Fractional processes have gained popularity in financial modeling due to the dependence structure of their increments and the roughness of their sample paths. The non-Markovianity of these processes gives, however, rise to conceptual and practical difficulties in computation and calibration. To address these issues, we show that a certain class of fractional processes can be represented as linear functionals of an infinite dimensional affine process. This can be derived from integral representations similar to those of Carmona, Coutin, Montseny, and Muravlev. We demonstrate by means of several examples that this allows one to construct tractable financial models with fractional features.

q-fin.MF↗

arXiv2015

Asymptotic Lower Bounds for Optimal Tracking: a Linear Programming Approach

We consider the problem of tracking a target whose dynamics is modeled by a continuous Itō semi-martingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds for this problem, depending on the cost structure. These lower bounds can be related to the time-average control of Brownian motion, which is characterized as a deterministic linear programming problem. A comprehensive list of examples with explicit expressions for the lower bounds is provided.

math.PR↗

arXiv2015

Extending the Black-Scholes Option Pricing Theory to Account for an Option Market Maker's Funding Costs

An option market maker incurs funding costs when carrying and hedging inventory. To hedge a net long delta inventory, for example, she pays a fee to borrow stock from the securities lending market. Because of haircuts, she posts additional cash margin to the lender which needs to be financed at her unsecured debt rate. This paper incorporates funding asymmetry (borrowed cash and invested cash earning different interest rates) and realistic stock financing cost into the classic option pricing theory. It is shown that an option position can be dynamically replicated and self financed in the presence of these funding costs. Noting that the funding amounts and costs are different for long and short positions, we extend Black-Scholes partial differential equations (PDE) per position side. The PDE's nonlinear funding cost terms create a free funding boundary and would result in the bid price for a long position on an option lower than the ask price for a short position. An iterative Crank-Nicholson finite difference method is developed to compute European and American vanilla option prices. Numerical results show that reasonable funding cost parameters can easily produce same magnitude of bid/ask spread of less liquid, longer term options as observed in the market place. Portfolio level pricing examples show the netting effect of hedges, which could moderate impact of funding costs.

q-fin.PR↗

arXiv2015

Nonlinear PDEs risen when solving some optimization problems in finance, and their solutions

We consider a specific type of nonlinear partial differential equations (PDE) that appear in mathematical finance as the result of solving some optimization problems. We review some existing in the literature examples of such problems, and discuss the properties of these PDEs. We also demonstrate how to solve them numerically in a general case, and analytically in some particular case.

q-fin.CP↗

arXiv2018

A martingale representation theorem and valuation of defaultable securities

We consider a market model where there are two levels of information. The public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm for credit risk, and the death time of an insured for life insurance. By using the expansion of filtration, the random time uncertainty and its entailed risk are fully considered without any mathematical restriction. In this context with no model's specification for the random time, the main challenge lies in finding the dynamics and the structures for the value processes of defaultable or mortality and/or longevity securities which are vital for the insurance securitization. To overcome this obstacle, we elaborate our optional martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into precise and unique orthogonal local martingales (i.e. local martingales whose product remains a local martingale). This constitutes our first and probably the principal contribution. Even though the driving motivation for this representation resides in credit risk theory, our results are applicable to several other financial and economics contexts, such as life insurance and financial markets with random horizon. Thanks to this optional representation, we decompose any defaultable or mortality and/or longevity liability into the sum of "non-correlated" risks using a risk basis. This constitutes our second contribution.

q-fin.MF↗

arXiv2016

An elementary approach to the option pricing problem

Our goal here is to discuss the pricing problem of European and American options in discrete time using elementary calculus so as to be an easy reference for first year undergraduate students. Using the binomial model we compute the fair price of European and American options. We explain the notion of Arbitrage and the notion of the fair price of an option using common sense. We give a criterion that the holder can use to decide when it is appropriate to exercise the option. We prove the put-call parity formulas for both European and American options and we discuss the relation between American and European options. We give also the bounds for European and American options. We also discuss the portfolio's optimization problem and the fair value in the case where the holder can not produce the opposite portfolio.

q-fin.MF↗

arXiv2015

Coherent CVA and FVA with Liability Side Pricing of Derivatives

This article presents FVA and CVA of a bilateral derivative in a coherent manner, based on recent developments in fair value accounting and ISDA standards. We argue that a derivative liability, after primary risk factors being hedged, resembles in economics an issued variable funding note, and should be priced at the market rate of the issuer's debt. For the purpose of determining the fair value, the party on the liability side is economically neutral to make a deposit to the other party, which earns his current debt rate and effectively provides funding and hedging for the party holding the derivative asset. The newly derived partial differential equation for an option discounts the derivative's receivable part with counterparty's curve and payable part with own financing curve. The price difference from the counterparty risk free price, or total counterparty risk adjustment, is precisely defined by discounting the product of the risk free price and the credit spread at the local liability curve. Subsequently the adjustment can be broken into a default risk component -- CVA and a funding component -- FVA, consistent with a simple note's fair value treatment and in accordance with the usual understanding of a bond's credit spread consisting of a CDS spread and a basis. As for FVA, we define a cost -- credit funding adjustment (CFA) and a benefit -- debit funding adjustment (DFA), in parallel to CVA and DVA and attributed to counterparty's and own funding basis. This resolves a number of outstanding FVA debate issues, such as double counting, violation of the law of one price, misuse of cash flow discounting, and controversial hedging of own default risk. It also allows an integrated implementation strategy and reuse of existing CVA infrastructure.

q-fin.PR↗

arXiv2015

Pricing of high-dimensional options

Pricing of high-dimensional options is one of the most important problems in Mathematical Finance. The objective of this manuscript is to present an original self-contained treatment of the multidimensional pricing. During the past decades the Black-Scholes this model, which essentially is based on the log-normal assumption, has been increasingly criticised. In particular, it was noticed by Mandelbrot that empirical log-returns distributions are more concentrated around the origin and have considerably heavier tails. This suggests to adjust the Black-Scholes model by the introduction of the Levy processes instead of Brownian ones. This approach has been extensively studied in a univariate setup since the nineties. In the multivariate settings the theory is not so advanced. We present a general method of high-dimensional option pricing based on a wide range of jump-diffusion models. Namely, we construct approximation formulas for the price of spread options. It is important to get an efficient approximation for the respective density function, since the reward function has usually a simple structure. Instead of a commonly used tabulation approach, we use the respective m-widths to compare a wide range of numerical methods. We give an algorithm of almost optimal, in the sense of the respective m-widths, reconstruction of density functions. To demonstrate the power of our approach we consider in details a concrete class of Levy driven processes and present the respective rates of convergence of approximation formulas. The interrelationship between the theory and tools reflects the richness and deep connections in Financial Mathematics, Stochastic Processes, Theory of Martingales, Functional Analysis, Topology and Harmonic Analysis.

q-fin.MF↗

arXiv2015

Modern Monetary Circuit Theory, Stability of Interconnected Banking Network, and Balance Sheet Optimization for Individual Banks

A modern version of Monetary Circuit Theory with a particular emphasis on stochastic underpinning mechanisms is developed. It is explained how money is created by the banking system as a whole and by individual banks. The role of central banks as system stabilizers and liquidity providers is elucidated. It is shown how in the process of money creation banks become naturally interconnected. A novel Extended Structural Default Model describing the stability of the Interconnected Banking Network is proposed. The purpose of banks' capital and liquidity is explained. Multi-period constrained optimization problem for banks's balance sheet is formulated and solved in a simple case. Both theoretical and practical aspects are covered.

econ.GN↗

arXiv2017

Stochastic control for a class of nonlinear kernels and applications

We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimisation, over a set of possibly non-dominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalisations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semi-martingale characterisation of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in [86]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a nonlinear optional decomposition in a robust setting, extending recent results of [71], which we then use to obtain a super-hedging duality in uncertain, incomplete and nonlinear financial markets. Finally, we relate, under additional regularity assumptions, the value function to a viscosity solution of an appropriate path-dependent partial differential equation (PPDE).

math.PR↗

arXiv2016

Pathwise no-arbitrage in a class of Delta hedging strategies

We consider a strictly pathwise setting for Delta hedging exotic options, based on Föllmer's pathwise Itō calculus. Price trajectories are $d$-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix. The existence of Delta hedging strategies in this pathwise setting is established via existence results for recursive schemes of parabolic Cauchy problems and via the existence of functional Cauchy problems on path space. Our main results establish the nonexistence of pathwise arbitrage opportunities in classes of strategies containing these Delta hedging strategies and under relatively mild conditions on the local volatility matrix.

q-fin.MF↗

arXiv2015

Real Options and Threshold Strategies

The paper considers an investment timing problem appearing in real options theory. Present values from an investment project are modeled by general diffusion process. We prove necessary and sufficient conditions under which an optimal investment time is induced by threshold strategy. We study also the conditions of optimality of threshold strategy (over all threshold strategies) and discuss the connection between solutions to investment timing problem and to free-boundary problem.

q-fin.MF↗

arXiv2016

Trajectory based models. Evaluation of minmax pricing bounds

The paper studies sub and super-replication price bounds for contingent claims defined on general trajectory based market models. No prior probabilistic or topological assumptions are placed on the trajectory space, trading is assumed to take place at a finite number of occasions but not bounded in number nor necessarily equally spaced in time. For a given option, there exists an interval bounding the set of possible fair prices; such interval exists under more general conditions than the usual no-arbitrage requirement. The paper develops a backward recursive method to evaluate the option bounds; the global minmax optimization, defining the price interval, is reduced to a local minmax optimization via dynamic programming. Trajectory sets are introduced for which existing non-probabilistic markets models are nested as a particular case. Several examples are presented, the effect of the presence of arbitrage on the price bounds is illustrated.

q-fin.MF↗