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arXiv2013

An alternative proof of a result of Takaoka

In Karatzas and Kardaras's paper on semimartingale financial models, it is proved that the NUPBR condition is a property of the local characteristic of the asset process alone. In Takaoka's paper on NUPBR, it is proved that the NUPBR condition is equivalent to the existence of a simga-martingale deflator. However, Takaoka's paper founds its proof on Delbaen and Schachermayer's fundamental asset pricing theorem, i.e. the NFLVR condition, which is not a pure property of the local characteristic of the asset process. In this paper we give an alternative proof of the result of Takaoka, which makes use only the properties of the local characteristic of the asset process.

math.PR↗

arXiv2013

Fractional G-White Noise Theory, Wavelet Decomposition for Fractional G-Brownian Motion, and Bid-Ask Pricing Application to Finance Under Uncertainty

G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index $H\in (0,1)$. This process has stationary increments, self-similarity, and long rang dependence properties in the sense of sub-linearity. These properties make the fractional G-Brownian motion a suitable driven process in mathematical finance. We construct wavelet decomposition of the fGBm by wavelet with compactly support. We develop fractional G-white noise theory, define G-Itô-Wick stochastic integral, establish the fractional G-Itô formula and the fractional G-Clark-Ocone formula, and derive the G-Girsanov's Theorem. For application the G-white noise theory, we consider the financial market modelled by G-Wick-Itô type of SDE driven by fGBm. The financial asset price modelled by fGBm has volatility uncertainty, using G-Girsanov's Theorem and G-Clark-Ocone Theorem, we derive that sublinear expectation of the discounted European contingent claim is the bid-ask price of the claim.

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arXiv2013

Valuation and hedging of OTC contracts with funding costs, collateralization and counterparty credit risk: Part 1

The research presented in this work is motivated by some recent papers regarding hedging and valuation of financial securities subject to funding costs, collateralization and counterparty credit risk. Our goal is to provide a sound theoretical underpinning for some results presented in these papers by developing a unified martingale framework for the non-linear approach to hedging and pricing of OTC financial contracts. The impact that various funding bases and margin covenants exert on the values and hedging strategies for OTC contracts is examined.

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arXiv2013

Non-Equivalent Beliefs and Subjective Equilibrium Bubbles

This paper develops a dynamic equilibrium model where agents exhibit a strong form of belief heterogeneity: they disagree about zero probability events. It is shown that, somewhat surprisingly, equilibrium exists in this setting, and that the disagreement about nullsets naturally leads to equilibrium asset pricing bubbles. The bubbles are subjective in the sense that they are perceived by some but not necessarily all agents. In contrast to existing models, bubbles arise with no restrictions on trade beyond a standard solvency constraint.

q-fin.GN↗

arXiv2015

Social Discounting and the Long Rate of Interest

The well-known theorem of Dybvig, Ingersoll and Ross shows that the long zero-coupon rate can never fall. This result, which, although undoubtedly correct, has been regarded by many as surprising, stems from the implicit assumption that the long-term discount function has an exponential tail. We revisit the problem in the setting of modern interest rate theory, and show that if the long "simple" interest rate (or Libor rate) is finite, then this rate (unlike the zero-coupon rate) acts viably as a state variable, the value of which can fluctuate randomly in line with other economic indicators. New interest rate models are constructed, under this hypothesis and certain generalizations thereof, that illustrate explicitly the good asymptotic behaviour of the resulting discount bond systems. The conditions necessary for the existence of such "hyperbolic" and "generalized hyperbolic" long rates are those of so-called social discounting, which allow for long-term cash flows to be treated as broadly "just as important" as those of the short or medium term. As a consequence, we are able to provide a consistent arbitrage-free valuation framework for the cost-benefit analysis and risk management of long-term social projects, such as those associated with sustainable energy, resource conservation, and climate change.

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arXiv2013

Dynamic Term Structure Modelling with Default and Mortality Risk: New Results on Existence and Monotonicity

This paper considers general term structure models like the ones appearing in portfolio credit risk modelling or life insurance. We give a general model starting from families of forward rates driven by infinitely many Brownian motions and an integer-valued random measure, generalizing existing approaches in the literature. Then we derive drift conditions which are equivalent to no asymptotic free lunch on the considered market. Existence results are also given. In practice, models possessing a certain monotonicity are favorable and we study general conditions which guarantee this. The setup is illustrated with some examples.

q-fin.PR↗

arXiv2013

Optimal exercise of swing contracts in energy markets: an integral constrained stochastic optimal control problem

We characterize the value of swing contracts in continuous time as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation with suitable boundary conditions. The case of contracts with penalties is straightforward, and in that case only a terminal condition is needed. Conversely, the case of contracts with strict constraints gives rise to a stochastic control problem with a nonstandard state constraint. We approach this problem by a penalty method: we consider a general constrained problem and approximate the value function with a sequence of value functions of appropriate unconstrained problems with a penalization term in the objective functional. Coming back to the case of swing contracts with strict constraints, we finally characterize the value function as the unique viscosity solution with polynomial growth of the Hamilton-Jacobi-Bellman equation subject to appropriate boundary conditions.

math.OC↗

arXiv2013

Importance sampling for jump processes and applications to finance

Adaptive importance sampling techniques are widely known for the Gaussian setting of Brownian driven diffusions. In this work, we want to extend them to jump processes. Our approach relies on a change of the jump intensity combined with the standard exponential tilting for the Brownian motion. The free parameters of our framework are optimized using sample average approximation techniques. We illustrate the efficiency of our method on the valuation of financial derivatives in several jump models.

math.PR↗

arXiv2014

Strict Local Martingales with Jumps

A strict local martingale is a local martingale which is not a martingale. There are few explicit examples of "naturally occurring" strict local martingales with jumps available in the literature. The purpose of this paper is to provide such examples, and to illustrate how they might arise via filtration shrinkage, a phenomenon we would contend is common in applications such as filtering, control, and especially in mathematical finance. We give a method for constructing such examples and analyze one particular method in detail.

math.PR↗

arXiv2013

On model-independent pricing/hedging using shortfall risk and quantiles

We consider the pricing and hedging of exotic options in a model-independent set-up using \emph{shortfall risk and quantiles}. We assume that the marginal distributions at certain times are given. This is tantamount to calibrating the model to call options with discrete set of maturities but a continuum of strikes. In the case of pricing with shortfall risk, we prove that the minimum initial amount is equal to the super-hedging price plus the inverse of the utility at the given shortfall level. In the second result, we show that the quantile hedging problem is equivalent to super-hedging problems for knockout options. These results generalize the duality results of [5,6] to the model independent setting of [1].

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arXiv2013

Utility indifference valuation for non-smooth payoffs with an application to power derivatives

We consider the problem of exponential utility indifference valuation under the simplified framework where traded and nontraded assets are uncorrelated but where the claim to be priced possibly depends on both. Traded asset prices follow a multivariate Black and Scholes model, while nontraded asset prices evolve as generalized Ornstein-Uhlenbeck processes. We provide a BSDE characterization of the utility indifference price (UIP) for a large class of non-smooth, possibly unbounded, payoffs depending simultaneously on both classes of assets. Focusing then on European claims and using the Gaussian structure of the model allows us to employ some BSDE techniques (in particular, a Malliavin-type representation theorem due to Ma (2002)) to prove the regularity of Z and to characterize the UIP for possibly discontinuous European payoffs as a viscosity solution of a suitable PDE with continuous space derivatives. The optimal hedging strategy is also identified essentially as the delta hedging strategy corresponding to the UIP. Since there are no closed-form formulas in general, we also obtain asymptotic expansions for prices and hedging strategies when the risk aversion parameter is small. Finally, our results are applied to pricing and hedging power derivatives in various structural models for energy markets.

q-fin.PR↗

arXiv2021

On the Pricing of Storable Commodities

This paper introduces an information-based model for the pricing of storable commodities such as crude oil and natural gas. The model uses the concept of market information about future supply and demand as a basis for valuation. Physical ownership of a commodity is taken to provide a stream of convenience dividends equivalent to a continuous cash flow. The market filtration is assumed to be generated jointly by (i) current and past levels of the dividend rate, and (ii) partial information concerning the future of the dividend flow. The price of a commodity is the expectation under a suitable pricing measure of the totality of the discounted risk-adjusted future convenience dividend, conditional on the information provided by the market filtration. In the situation where the dividend rate is modelled by an Ornstein-Uhlenbeck process, the prices of options on commodities can be derived in closed form. The approach that we present can be applied to other assets that yield potentially negative effective cash flows, such as real estate, factories, refineries, mines, and power generating plants.

q-fin.PR↗

arXiv2013

When terminal facelift enforces Delta constraints

This paper deals with the super-replication of non path-dependent European claims under additional convex constraints on the number of shares held in the portfolio. The corresponding super-replication price of a given claim has been widely studied in the literature and its terminal value, which dominates the claim of interest, is the so-called facelift transform of the claim. We investigate under which conditions the super-replication price and strategy of a large class of claims coincide with the exact replication price and strategy of the facelift transform of this claim. In one dimension, we observe that this property is satisfied for any local volatility model. In any dimension, we exhibit an analytical necessary and sufficient condition for this property, which combines the dynamics of the stock together with the characteristics of the closed convex set of constraints. To obtain this condition, we introduce the notion of first order viability property for linear parabolic PDEs. We investigate in details several practical cases of interest: multidimensional Black Scholes model, non-tradable assets or short selling restrictions.

q-fin.PR↗

arXiv2013

CVA for Bilateral Counterparty Risk under Alternative Settlement Conventions

We depart from the usual methods for pricing contracts with the counterparty credit risk found in most of the existing literature. In effect, typically, these models do not account for either systemic effects or at-first-default contagion and postulate that the contract value at default equals either the risk-free value or the pre-default value. We propose instead a fairly general framework, which allows us to perform effective Credit Value Adjustment (CVA) computations for a contract with bilateral counterparty risk in the presence of systemic and wrong or right way risks. Our general methodology focuses on the role of alternative settlement clauses, but it is also aimed to cover various features of margin agreements. A comparative analysis of numerical results reported in the final section supports our initial conjecture that alternative specifications of settlement values have a non-negligible impact on the CVA computation for contracts with bilateral counterparty risk. This emphasizes the practical importance of more sophisticated models that are capable of fully reflecting the actual features of financial contracts, as well as the influence of the market environment.

q-fin.PR↗

arXiv2013

Quantum Tunneling of Stock Price in Range Bound Market Conditions

Applications of Quantum Tunneling effect have long gone beyond the traditional physical meaning. Initially created by Gamow to explain α-decay of nuclear particles, along the time, quantum tunneling found fertile domain of research in chemistry and recently in biology, where the new discipline of Quantum Biology emerges. The present paper extends the applicability of quantum tunneling to financial markets. In a recent paper [1] a time-independent equation for pricing the options having the underlying stock in a range bound markets is found. The equation is identical with a time-independent Schrodinger equation but incorporates elements of finance. The financial time-independent equation for option pricing is solved to explain a particular explosive violent movement of stock price in range bound markets. The aforementioned particular stock price movement is assimilated with a quantum tunneling effect. The probability of stock price to quantum tunneling out of the bounded region, known as transmission coefficient, is deduced. Quantum aspects of tunneling effect in financial markets are discussed. Recent evidences of price quantum tunneling in stock market are also shown.

q-fin.PR↗

arXiv2013

An identity of hitting times and its application to the valuation of guaranteed minimum withdrawal benefit

In this paper we explore an identity in distribution of hitting times of a finite variation process (Yor's process) and a diffusion process (geometric Brownian motion with affine drift), which arise from various applications in financial mathematics. As a result, we provide analytical solutions to the fair charge of variable annuity guaranteed minimum withdrawal benefit (GMWB) from a policyholder's point of view, which was only previously obtained in the literature by numerical methods. We also use complex inversion methods to derive analytical solutions to the fair charge of the GMWB from an insurer's point of view, which is used in the market practice, however, based on Monte Carlo simulations. Despite of their seemingly different formulations, we can prove under certain assumptions the two pricing approaches are equivalent.

q-fin.CP↗

arXiv2013

Efficient valuation method for the SABR model

In this article, we show how the scaling symmetry of the SABR model can be utilized to efficiently price European options. For special kinds of payoffs, the complexity of the problem is reduced by one dimension. For more generic payoffs, instead of solving the 1+2 dimensional SABR PDE, it is sufficient to solve $N_V$ uncoupled 1+1 dimensional PDE's, where $N_V$ is the number of points used to discretize one dimension. Furthermore, the symmetry argument enables us to obtain prices of multiple options, whose payoffs are related to each other by convolutions, by valuing one of them. The results of the method are compared with the Monte Carlo simulation.

q-fin.CP↗

arXiv2014

Admissible Trading Strategies under Transaction Costs

A well known result in stochastic analysis reads as follows: for an $\mathbb{R}$-valued super-martingale $X = (X_t)_{0\leq t \leq T}$ such that the terminal value $X_T$ is non-negative, we have that the entire process $X$ is non-negative. An analogous result holds true in the no arbitrage theory of mathematical finance: under the assumption of no arbitrage, a portfolio process $x+(H\cdot S)$ verifying $x+(H\cdot S)_T\geq 0$ also satisfies $x+(H\cdot S)_t\geq 0,$ for all $0 \leq t \leq T$. In the present paper we derive an analogous result in the presence of transaction costs. A counter-example reveals that the consideration of transaction costs makes things more delicate than in the frictionless setting.

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