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At least 415 records · Page 23Linked to original sources
arXiv2012

Time-Changed Ornstein-Uhlenbeck Processes And Their Applications In Commodity Derivative Models

This paper studies subordinate Ornstein-Uhlenbeck (OU) processes, i.e., OU diffusions time changed by Lévy subordinators. We construct their sample path decomposition, show that they possess mean-reverting jumps, study their equivalent measure transformations, and the spectral representation of their transition semigroups in terms of Hermite expansions. As an application, we propose a new class of commodity models with mean-reverting jumps based on subordinate OU process. Further time changing by the integral of a CIR process plus a deterministic function of time, we induce stochastic volatility and time inhomogeneity, such as seasonality, in the models. We obtain analytical solutions for commodity futures options in terms of Hermite expansions. The models are consistent with the initial futures curve, exhibit Samuelson's maturity effect, and are flexible enough to capture a variety of implied volatility smile patterns observed in commodities futures options.

q-fin.PR↗
arXiv2012

On Pricing Basket Credit Default Swaps

In this paper we propose a simple and efficient method to compute the ordered default time distributions in both the homogeneous case and the two-group heterogeneous case under the interacting intensity default contagion model. We give the analytical expressions for the ordered default time distributions with recursive formulas for the coefficients, which makes the calculation fast and efficient in finding rates of basket CDSs. In the homogeneous case, we explore the ordered default time in limiting case and further include the exponential decay and the multistate stochastic intensity process. The numerical study indicates that, in the valuation of the swap rates and their sensitivities with respect to underlying parameters, our proposed model outperforms the Monte Carlo method.

q-fin.PR↗
arXiv2012

Yield to maturity modelling and a Monte Carlo Technique for pricing Derivatives on Constant Maturity Treasury (CMT) and Derivatives on forward Bonds

This paper proposes a Monte Carlo technique for pricing the forward yield to maturity, when the volatility of the zero-coupon bond is known. We make the assumption of deterministic default intensity (Hazard Rate Function). We make no assumption on the volatility of the yield. We actually calculate the initial value of the forward yield, we calculate the volatility of the yield, and we write the diffusion of the yield. As direct application we price options on Constant Maturity Treasury (CMT) in the Hull and White Model for the short interest rate. Tests results with Caps and Floors on 10 years constant maturity treasury (CMT10) are satisfactory. This work can also be used for pricing options on bonds or forward bonds.

q-fin.CP↗
arXiv2012

Libor model with expiry-wise stochastic volatility and displacement

We develop a multi-factor stochastic volatility Libor model with displacement, where each individual forward Libor is driven by its own square-root stochastic volatility process. The main advantage of this approach is that, maturity-wise, each square-root process can be calibrated to the corresponding cap(let)vola-strike panel at the market. However, since even after freezing the Libors in the drift of this model, the Libor dynamics are not affine, new affine approximations have to be developed in order to obtain Fourier based (approximate) pricing procedures for caps and swaptions. As a result, we end up with a Libor modeling package that allows for efficient calibration to a complete system of cap/swaption market quotes that performs well even in crises times, where structural breaks in vola-strike-maturity panels are typically observed.

q-fin.PR↗
arXiv2012

On absolutely continuous compensators and nonlinear filtering equations in default risk models

We discuss the pricing of defaultable assets in an incomplete information model where the default time is given by a first hitting time of an unobservable process. We show that in a fairly general Markov setting, the indicator function of the default has an absolutely continuous compensator. Given this compensator we then discuss the optional projection of a class of semimartingales onto the filtration generated by the observation process and the default indicator process. Available formulas for the pricing of defaultable assets are analyzed in this setting and some alternative formulas are suggested.

math.PR↗
arXiv2012

Central Counterparty Risk

A clearing member of a Central Counterparty (CCP) is exposed to losses on their default fund and initial margin contributions. Such losses can be incurred whenever the CCP has insufficient funds to unwind the portfolio of a defaulting clearing member. This does not necessarily require the default of the CCP itself. In this note we aim to quantify the risk a financial institution has when facing a CCP. We show that a clearing member's CCP risk is given by a sum of exposures to each of the other clearing members. This arises because of the implicit default insurance that each member has provided in the form of mutualised, loss sharing collateral. We calculate the exposures by explicitly modeling the capital structure of a CCP as well as the loss distributions of the individual member portfolios. An important consideration in designing the model is the limited transparency with respect to the portfolio composition and collateral levels of individual clearing members. To overcome this we leverage the fact that, for a typical CCP, margin levels are risk-based. In particular, we parameterise the portfolio loss tail as a Pareto distribution and we calibrate this to the CCP defined probability of losses exceeding the posted initial margin levels. A key aspect of the model is that we explicitly take into account wrong-way risk, i.e. the fact that member defaults are more likely to occur in stressed market conditions, as well as potential contagion between a member's default and the losses on his portfolio.

q-fin.RM↗
arXiv2013

Bilateral Credit Valuation Adjustment of an Optional Early Termination Clause

Is an option to early terminate a swap at its market value worth zero? At first sight it is, but in presence of counterparty risk it depends on the criteria used to determine such market value. In case of a single uncollateralised swap transaction under ISDA between two defaultable counterparties, the additional unilateral option to early terminate the swap at predefined dates requires a Bermudan credit valuation adjustment. We give a general pricing formula assuming a default-free close-out amount, and apply it in a simplified setting with deterministic intensity and one single date of optional early termination, showing that the impact on the fair value of the transaction at inception might be non negligible.

q-fin.PR↗
arXiv2013

Exponential Lévy-type models with stochastic volatility and stochastic jump-intensity

We consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential Lévy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time through common driving factors -- one fast-varying and one slow-varying. Using Fourier analysis we derive an explicit formula for the approximate price of any European-style derivative whose payoff has a generalized Fourier transform; in particular, this includes European calls and puts. From a theoretical perspective, our results extend the class of multiscale stochastic volatility models of \citet*{fpss} to models of the exponential Lévy type. From a financial perspective, the inclusion of jumps and stochastic volatility allow us to capture the term-structure of implied volatility. To illustrate the flexibility of our modeling framework we extend five exponential Lévy processes to include stochastic volatility and jump-intensity. For each of the extended models, using a single fast-varying factor of volatility and jump-intensity, we perform a calibration to the S&P500 implied volatility surface. Our results show decisively that the extended framework provides a significantly better fit to implied volatility than both the traditional exponential Lévy models and the fast mean-reverting stochastic volatility models of \citet{fpss}.

q-fin.PR↗
arXiv2012

Pricing options on illiquid assets with liquid proxies using utility indifference and dynamic-static hedging

This work addresses the problem of optimal pricing and hedging of a European option on an illiquid asset Z using two proxies: a liquid asset S and a liquid European option on another liquid asset Y. We assume that the S-hedge is dynamic while the Y-hedge is static. Using the indifference pricing approach we derive a HJB equation for the value function, and solve it analytically (in quadratures) using an asymptotic expansion around the limit of the perfect correlation between assets Y and Z. While in this paper we apply our framework to an incomplete market version of the credit-equity Merton's model, the same approach can be used for other asset classes (equity, commodity, FX, etc.), e.g. for pricing and hedging options with illiquid strikes or illiquid exotic options.

q-fin.PR↗
arXiv2012

New solvable stochastic volatility models for pricing volatility derivatives

Classical solvable stochastic volatility models (SVM) use a CEV process for instantaneous variance where the CEV parameter $γ$ takes just few values: 0 - the Ornstein-Uhlenbeck process, 1/2 - the Heston (or square root) process, 1- GARCH, and 3/2 - the 3/2 model. Some other models were discovered in \cite{Labordere2009} by making connection between stochastic volatility and solvable diffusion processes in quantum mechanics. In particular, he used to build a bridge between solvable (super)potentials (the Natanzon (super)potentials, which allow reduction of a Schrödinger equation to a Gauss confluent hypergeometric equation) and existing SVM. In this paper we discuss another approach to extend the class of solvable SVM in terms of hypergeometric functions. Thus obtained new models could be useful for pricing volatility derivatives (variance and volatility swaps, moment swaps).

q-fin.PR↗
arXiv2012

Valuation and hedging of the ruin-contingent life annuity (RCLA)

This paper analyzes a novel type of mortality contingent-claim called a ruin-contingent life annuity (RCLA). This product fuses together a path-dependent equity put option with a "personal longevity" call option. The annuitant's (i.e. long position) payoff from a generic RCLA is \$1 of income per year for life, akin to a defined benefit pension, but deferred until a pre-specified financial diffusion process hits zero. We derive the PDE and relevant boundary conditions satisfied by the RCLA value (i.e. the hedging cost) assuming a complete market where No Arbitrage is possible. We then describe some efficient numerical techniques and provide estimates of a typical RCLA under a variety of realistic parameters. The motivation for studying the RCLA on a stand-alone basis is two-fold. First, it is implicitly embedded in approximately \$1 trillion worth of U.S. variable annuity (VA) policies; which have recently attracted scrutiny from financial analysts and regulators. Second, the U.S. administration - both Treasury and Department of Labor - have been encouraging Defined Contribution (401k) plans to offer stand-alone longevity insurance to participants, and we believe the RCLA would be an ideal and cost effective candidate for that job.

q-fin.PR↗
arXiv2012

Variance Optimal Hedging for discrete time processes with independent increments. Application to Electricity Markets

We consider the discretized version of a (continuous-time) two-factor model introduced by Benth and coauthors for the electricity markets. For this model, the underlying is the exponent of a sum of independent random variables. We provide and test an algorithm, which is based on the celebrated Foellmer-Schweizer decomposition for solving the mean-variance hedging problem. In particular, we establish that decomposition explicitely, for a large class of vanilla contingent claims. Interest is devoted in the choice of rebalancing dates and its impact on the hedging error, regarding the payoff regularity and the non stationarity of the log-price process.

q-fin.PR↗
arXiv2012

Two Models of Stochastic Loss Given Default

We propose two structural models for stochastic losses given default which allow to model the credit losses of a portfolio of defaultable financial instruments. The credit losses are integrated into a structural model of default events accounting for correlations between the default events and the associated losses. We show how the models can be calibrated and analyze the impact of correlations between the occurrences of defaults and recoveries by testing our models for a representative sample portfolio.

q-fin.RM↗
arXiv2012

Modeling and Pricing of Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities

In this paper, we model financial markets with semi-Markov volatilities and price covarinace and correlation swaps for this markets. Numerical evaluations of vari- nace, volatility, covarinace and correlations swaps with semi-Markov volatility are presented as well. The novelty of the paper lies in pricing of volatility swaps in closed form, and pricing of covariance and correlation swaps in a market with two risky assets.

q-fin.PR↗
arXiv2012

Collateralized CVA Valuation with Rating Triggers and Credit Migrations

In this paper we discuss the issue of computation of the bilateral credit valuation adjustment (CVA) under rating triggers, and in presence of ratings-linked margin agreements. Specifically, we consider collateralized OTC contracts, that are subject to rating triggers, between two parties -- an investor and a counterparty. Moreover, we model the margin process as a functional of the credit ratings of the counterparty and the investor. We employ a Markovian approach for modeling of the rating transitions of the two parties to the contract. In this framework, we derive the representation for bilateral CVA. We also introduce a new component in the decomposition of the counterparty risky price: namely the rating valuation adjustment (RVA) that accounts for the rating triggers. We give two examples of dynamic collateralization schemes where the margin thresholds are linked to the credit ratings of the parties. We account for the rehypothecation risk in the presence of independent amounts. Our results are illustrated via computation of various counterparty risk adjustments for a CDS contract and for an IRS contract.

q-fin.PR↗
arXiv2012

Binomial Tree Model for Convertible Bond Pricing within Equity to Credit Risk Framework

In the present paper we fill an essential gap in the Convertible Bonds pricing world by deriving a Binary Tree based model for valuation subject to credit risk. This model belongs to the framework known as Equity to Credit Risk. We show that this model converges in continuous time to the model developed by Ayache, Forsyth and Vetzal [2003]. To this end, both forms of credit risk modeling, the so-called reduced (constant intensity of default model for the underlying) and the so-called synthesis (variable intensity of default model for the underlying) are considered. We highlight and quantify certain issues that arise, as transition probability analysis and threshold values of model inputs (tree step, underlying stock price, etc.). This study may be considered as an alternative way to develop the price dynamics model of Ayache et al. [2003] for convertible bonds in credit risk environment.

q-fin.PR↗
arXiv2012

Preliminary remarks on option pricing and dynamic hedging

An elementary arbitrage principle and the existence of trends in financial time series, which is based on a theorem published in 1995 by P. Cartier and Y. Perrin, lead to a new understanding of option pricing and dynamic hedging. Intricate problems related to violent behaviors of the underlying, like the existence of jumps, become then quite straightforward by incorporating them into the trends. Several convincing computer experiments are reported.

q-fin.PR↗