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arXiv2013

CVA and FVA to Derivatives Trades Collateralized by Cash

In this article, we combine replication pricing with expectation pricing for derivative trades that are partially collateralized by cash. The derivatives are replicated by underlying assets and cash, using repurchasing agreement (repo) and margining, which incur funding costs. We derive a partial differential equation (PDE) for the derivatives price, obtain and decompose its solution into the risk-free value of the derivative plus credit valuation adjustment (CVA) and funding valuation adjustment (FVA). For most derivatives, as we shall show, CVAs can be evaluated analytically or semi-analytically, while FVAs, as well as the derivatives values, will have to be solved recursively through numerical procedures due to their interdependence. In numerical demonstrations, continuous and discrete margin revisions are considered, respectively, for an equity call option and a vanilla interest-rate swaps.

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arXiv2013

On return rate implied by behavioural present value

The future value of a security is described as a random variable. Distribution of this random variable is the formal image of risk uncertainty. On the other side, any present value is defined as a value equivalent to the given future value. This equivalence relationship is a subjective. Thus follows, that present value is described as a fuzzy number, which is depend on the investor's susceptibility to behavioural factors. All above reasons imply, that return rate is given as a fuzzy probabilistic set. The basic properties of such image of return rate are studied. At the last the set of effective securities is distinguished as a fuzzy set.

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arXiv2013

Inflation-rate Derivatives: From Market Model to Foreign Currency Analogy

In this paper, we establish a market model for the term structure of forward inflation rates based on the risk-neutral dynamics of nominal and real zero-coupon bonds. Under the market model, we can price inflation caplets as well as inflation swaptions with a formula similar to the Black's formula, thus justify the current market practice. We demonstrate how to further extend the market model to cope with volatility smiles. Moreover, we establish a consistency condition on the volatility of real zero-coupon bonds using arbitrage arguments, and with that re-derive the model of Jarrow and Yildirim (2003) with real forward rates based on "foreign currency analogy", and thus interconnect the two modeling paradigms.

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arXiv2013

Efficient Markets, Behavioral Finance and a Statistical Evidence of the Validity of Technical Analysis

This work tried to detect the existence of a relationship between the graphic signals - or patterns - observed day by day in the Brazilian stock market and the trends which happen after these signals, within a period of 8 years, for a number of securities. The results obtained from this study show evidence of the existence of such a relationship, suggesting the validity of the Technical Analysis as an instrument to predict the trend of security prices in the Brazilian stock market within that period.

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arXiv2013

On the Robust superhedging of measurable claims

The problem of robust hedging requires to solve the problem of superhedging under a nondominated family of singular measures. Recent progress was achieved by [9,11]. We show that the dual formulation of this problem is valid in a context suitable for martingale optimal transportation or, more generally, for optimal transportation under controlled stochastic dynamics.

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arXiv2013

Variance optimal hedging for continuous time additive processes and applications

For a large class of vanilla contingent claims, we establish an explicit Föllmer-Schweizer decomposition when the underlying is an exponential of an additive process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electricity market are performed.

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arXiv2013

Technical report : Risk-neutral density recovery via spectral analysis

In this paper, we propose a new method for estimating the conditional risk-neutral density (RND) directly from a cross-section of put option bid-ask quotes. More precisely, we propose to view the RND recovery problem as an inverse problem. We first show that it is possible to define restricted put and call operators that admit a singular value decomposition (SVD), which we compute explicitly. We subsequently show that this new framework allows us to devise a simple and fast quadratic programming method to recover the smoothest RND whose corresponding put prices lie inside the bid-ask quotes. This method is termed the spectral recovery method (SRM). Interestingly, the SVD of the restricted put and call operators sheds some new light on the RND recovery problem. The SRM improves on other RND recovery methods in the sense that: - it is fast and simple to implement since it requires solution of a single quadratic program, while being fully nonparametric; - it takes the bid ask quotes as sole input and does not require any sort of calibration, smoothing or preprocessing of the data; - it is robust to the paucity of price quotes; - it returns the smoothest density giving rise to prices that lie inside the bid ask quotes. The estimated RND is therefore as well-behaved as can be; - it returns a closed form estimate of the RND on the interval [0,B] of the positive real line, where B is a positive constant that can be chosen arbitrarily. We thus obtain both the middle part of the RND together with its full left tail and part of its right tail. We confront this method to both real and simulated data and observe that it fares well in practice. The SRM is thus found to be a promising alternative to other RND recovery methods.

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arXiv2013

The Pricing of Multiple-Expiry Exotics

In this paper we extend Buchen's method to develop a new technique for pricing of some exotic options with several expiry dates(more than 3 expiry dates) using a concept of higher order binary option. At first we introduce the concept of higher order binary option and then provide the pricing formulae of $n$-th order binaries using PDE method. After that, we apply them to pricing of some multiple-expiry exotic options such as Bermudan option, multi time extendable option, multi shout option and etc. Here, when calculating the price of concrete multiple-expiry exotic options, we do not try to get the formal solution to corresponding initial-boundary problem of the Black-Scholes equation, but explain how to express the expiry payoffs of the exotic options as a combination of the payoffs of some class of higher order binary options. Once the expiry payoffs are expressed as a linear combination of the payoffs of some class of higher order binary options, in order to avoid arbitrage, the exotic option prices are obtained by static replication with respect to this family of higher order binaries.

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arXiv2013

Pricing Corporate Defaultable Bond using Declared Firm Value

We study the pricing problem for corporate defaultable bond from the viewpoint of the investors outside the firm that could not exactly know about the information of the firm. We consider the problem for pricing of corporate defaultable bond in the case when the firm value is only declared in some fixed discrete time and unexpected default intensity is determined by the declared firm value. Here we provide a partial differential equation model for such a defaultable bond and give its pricing formula. Our pricing model is derived to solving problems of partial differential equations with random constants (de- fault intensity) and terminal values of binary types. Our main method is to use the solving method of a partial differential equation with a random constant in every subinterval and to take expectation to remove the random constants.

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arXiv2013

Pricing Step Options under the CEV and other Solvable Diffusion Models

We consider a special family of occupation-time derivatives, namely proportional step options introduced by Linetsky in [Math. Finance, 9, 55--96 (1999)]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.

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arXiv2013

A Class of Solvable Optimal Stopping Problems of Spectrally Negative Jump Diffusions

We consider the optimal stopping of a class of spectrally negative jump diffusions. We state a set of conditions under which the value is shown to have a representation in terms of an ordinary nonlinear programming problem. We establish a connection between the considered problem and a stopping problem of an associated continuous diffusion process and demonstrate how this connection may be applied for characterizing the stopping policy and its value. We also establish a set of typically satisfied conditions under which increased volatility as well as higher jump-intensity decelerates rational exercise by increasing the value and expanding the continuation region.

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arXiv2013

How to make Dupire's local volatility work with jumps

There are several (mathematical) reasons why Dupire's formula fails in the non-diffusion setting. And yet, in practice, ad-hoc preconditioning of the option data works reasonably well. In this note we attempt to explain why. In particular, we propose a regularization procedure of the option data so that Dupire's local vol diffusion process recreates the correct option prices, even in manifest presence of jumps.

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arXiv2013

Optimal dividends problem with a terminal value for spectrally positive Levy processes

In this paper we consider a modified version of the classical optimal dividends problem of de Finetti in which the dividend payments subject to a penalty at ruin. We assume that the risk process is modeled by a general spectrally positive Levy process before dividends are deducted. Using the fluctuation theory of spectrally positive Levy processes we give an explicit expression of the value function of a barrier strategy. Subsequently we show that a barrier strategy is the optimal strategy among all admissible ones. Our work is motivated by the recent work of Bayraktar, Kyprianou and Yamazaki (2013).

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arXiv2013

An Optimal Pairs-Trading Rule

This paper is concerned with a pairs trading rule. The idea is to monitor two historically correlated securities. When divergence is underway, i.e., one stock moves up while the other moves down, a pairs trade is entered which consists of a pair to short the outperforming stock and to long the underperforming one. Such a strategy bets the "spread" between the two would eventually converge. In this paper, a difference of the pair is governed by a mean-reverting model. The objective is to trade the pair so as to maximize an overall return. A fixed commission cost is charged with each transaction. In addition, a stop-loss limit is imposed as a state constraint. The associated HJB equations (quasi-variational inequalities) are used to characterize the value functions. It is shown that the solution to the optimal stopping problem can be obtained by solving a number of quasi-algebraic equations. We provide a set of sufficient conditions in terms of a verification theorem. Numerical examples are reported to demonstrate the results.

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arXiv2013

Swing options in commodity markets: A multidimensional Lévy diffusion model

We study valuation of swing options on commodity markets when the commodity prices are driven by multiple factors. The factors are modeled as diffusion processes driven by a multidimensional Lévy process. We set up a valuation model in terms of a dynamic programming problem where the option can be exercised continuously in time. Here, the number of swing rights is given by a total volume constraint. We analyze some general properties of the model and study the solution by analyzing the associated HJB-equation. Furthermore, we discuss the issues caused by the multi-dimensionality of the commodity price model. The results are illustrated numerically with three explicit examples.

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arXiv2014

Asymptotic arbitrage in the Heston model

In the context of the Heston model, we establish a precise link between the set of equivalent martingale measures, the ergodicity of the underlying variance process and the concept of asymptotic arbitrage proposed in Kabanov-Kramkov and in Follmer-Schachermayer.

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arXiv2013

CoCo Bonds Valuation with Equity- and Credit-Calibrated First Passage Structural Models

After the beginning of the credit and liquidity crisis, financial institutions have been considering creating a convertible-bond type contract focusing on Capital. Under the terms of this contract, a bond is converted into equity if the authorities deem the institution to be under-capitalized. This paper discusses this Contingent Capital (or Coco) bond instrument and presents a pricing methodology based on firm value models. The model is calibrated to readily available market data. A stress test of model parameters is illustrated to account for potential model risk. Finally, a brief overview of how the instrument performs is presented.

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arXiv2014

The arbitrage-free Multivariate Mixture Dynamics Model: Consistent single-assets and index volatility smiles

We introduce a multivariate diffusion model that is able to price derivative securities featuring multiple underlying assets. Each asset volatility smile is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allows to reconcile single name and index/basket volatility smiles in a consistent framework. Our approach could be dubbed a multidimensional local volatility approach with vector-state dependent diffusion matrix. The model is quite tractable, leading to a complete market and not requiring Fourier techniques for calibration and dependence measures, contrary to multivariate stochastic volatility models such as Wishart. We prove existence and uniqueness of solutions for the model stochastic differential equations, provide formulas for a number of basket options, and analyze the dependence structure of the model in detail by deriving a number of results on covariances, its copula function and rank correlation measures and volatilities-assets correlations. A comparison with sampling simply-correlated suitably discretized one-dimensional mixture dynamical paths is made, both in terms of option pricing and of dependence, and first order expansion relationships between the two models' local covariances are derived. We also show existence of a multivariate uncertain volatility model of which our multivariate local volatilities model is a Markovian projection, highlighting that the projected model is smoother and avoids a number of drawbacks of the uncertain volatility version. We also show a consistency result where the Markovian projection of a geometric basket in the multivariate model is a univariate mixture dynamics model. A few numerical examples on basket and spread options pricing conclude the paper.

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