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arXiv2010

Bilateral counterparty risk valuation for interest-rate products: impact of volatilities and correlations

The purpose of this paper is introducing rigorous methods and formulas for bilateral counterparty risk credit valuation adjustments (CVA's) on interest-rate portfolios. In doing so, we summarize the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, as developed more in detail in Brigo and Capponi (2008), including the default of the investor. We illustrate the symmetry in the valuation and show that the adjustment involves a long position in a put option plus a short position in a call option, both with zero strike and written on the residual net present value of the contract at the relevant default times. We allow for correlation between the default times of the investor and counterparty, and for correlation of each with the underlying risk factor, namely interest rates. We also analyze the often neglected impact of credit spread volatility. We include Netting in our examples, although other agreements such as Margining and Collateral are left for future work.

q-fin.PR↗
arXiv2009

Utility maximization in models with conditionally independent increments

We consider the problem of maximizing expected utility from terminal wealth in models with stochastic factors. Using martingale methods and a conditioning argument, we determine the optimal strategy for power utility under the assumption that the increments of the asset price are independent conditionally on the factor process.

q-fin.PM↗
arXiv2011

On the Performance of Delta Hedging Strategies in Exponential Lévy Models

We consider the performance of non-optimal hedging strategies in exponential Lévy models. Given that both the payoff of the contingent claim and the hedging strategy admit suitable integral representations, we use the Laplace transform approach of Hubalek et al. (2006) to derive semi-explicit formulas for the resulting mean squared hedging error in terms of the cumulant generating function of the underlying Lévy process. In two numerical examples, we apply these results to compare the efficiency of the Black-Scholes hedge and the model delta to the mean-variance optimal hedge in a normal inverse Gaussian and a diffusion-extended CGMY Lévy model.

q-fin.CP↗
arXiv2009

Resilience of Volatility

The problem of non-stationarity in financial markets is discussed and related to the dynamic nature of price volatility. A new measure is proposed for estimation of the current asset volatility. A simple and illustrative explanation is suggested of the emergence of significant serial autocorrelations in volatility and squared returns. It is shown that when non-stationarity is eliminated, the autocorrelations substantially reduce and become statistically insignificant. The causes of non-Gaussian nature of the probability of returns distribution are considered. For both stock and currency markets data samples, it is shown that removing the non-stationary component substantially reduces the kurtosis of distribution, bringing it closer to the Gaussian one. A statistical criterion is proposed for controlling the degree of smoothing of the empirical values of volatility. The hypothesis of smooth, non-stochastic nature of volatility is put forward, and possible causes of volatility shifts are discussed.

q-fin.ST↗
arXiv2010

Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends

We analyze the regularity of the optimal exercise boundary for the American Put option when the underlying asset pays a discrete dividend at a known time $t_d$ during the lifetime of the option. The ex-dividend asset price process is assumed to follow Black-Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. The solution to the associated optimal stopping problem can be characterised in terms of an optimal exercise boundary which, in contrast to the case when there are no dividends, may no longer be monotone. In this paper we prove that when the dividend function is positive and concave, then the boundary is non-increasing in a left-hand neighbourhood of $t_d$, and tends to $0$ as time tends to $t_d^-$ with a speed that we can characterize. When the dividend function is linear in a neighbourhood of zero, then we show continuity of the exercise boundary and a high contact principle in the left-hand neighbourhood of $t_d$. When it is globally linear, then right-continuity of the boundary and the high contact principle are proved to hold globally. Finally, we show how all the previous results can be extended to multiple dividend payment dates in that case.

q-fin.CP↗
arXiv2009

Asymptotic behavior of prices of path dependent options

In this paper, we give a numerical method for pricing long maturity, path dependent options by using the Markov property for each underlying asset. This enables us to approximate a path dependent option by using some kinds of plain vanillas. We give some examples whose underlying assets behave as some popular Levy processes. Moreover, we give some payoffs and functions used to approximate them.

q-fin.PR↗
arXiv2011

Appraisal of a contour integral method for the Black-Scholes and Heston equations

A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., to appear] for integrating semi-discrete advection-diffusion PDEs, is extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parameters are derived theoretically and tested numerically. Test examples presented are the Black-Scholes PDE in one space dimension and the Heston PDE in two dimensions. In the latter case efficiency is compared to ADI splitting schemes for solving this problem. In the examples it is found that the contour integral method is superior for the range of medium to high accuracy requirements. Further improvements to the current implementation of the contour integral method are suggested.

q-fin.CP↗
arXiv2010

What Causes Business Cycles? Analysis of the Japanese Industrial Production Data

We explore what causes business cycles by analyzing the Japanese industrial production data. The methods are spectral analysis and factor analysis. Using the random matrix theory, we show that two largest eigenvalues are significant. Taking advantage of the information revealed by disaggregated data, we identify the first dominant factor as the aggregate demand, and the second factor as inventory adjustment. They cannot be reasonably interpreted as technological shocks. We also demonstrate that in terms of two dominant factors, shipments lead production by four months. Furthermore, out-of-sample test demonstrates that the model holds up even under the 2008-09 recession. Because a fall of output during 2008-09 was caused by an exogenous drop in exports, it provides another justification for identifying the first dominant factor as the aggregate demand. All the findings suggest that the major cause of business cycles is real demand shocks.

q-fin.GN↗
arXiv2009

Early exercise boundary for American type of floating strike Asian option and its numerical approximation

In this paper we generalize and analyze the model for pricing American-style Asian options due to (Hansen and Jorgensen 2000) by including a continuous dividend rate $q$ and a general method of averaging of the floating strike. We focus on the qualitative and quantitative analysis of the early exercise boundary. The first order Taylor series expansion of the early exercise boundary close to expiry is constructed. We furthermore propose an efficient numerical algorithm for determining the early exercise boundary position based on the front fixing method. Construction of the algorithm is based on a solution to a nonlocal parabolic partial differential equation for the transformed variable representing the synthesized portfolio. Various numerical results and comparisons of our numerical method and the method developed by (Dai and Kwok 2006) are presented.

q-fin.PR↗
arXiv2009

Homogeneous Volatility Bridge Estimators

We present a theory of homogeneous volatility bridge estimators for log-price stochastic processes. The main tool of our theory is the parsimonious encoding of the information contained in the open, high and low prices of incomplete bridge, corresponding to given log-price stochastic process, and in its close value, for a given time interval. The efficiency of the new proposed estimators is favorably compared with that of the Garman-Klass and Parkinson estimators.

q-fin.ST↗
arXiv2009

Credit Default Swap Calibration and Equity Swap Valuation under Counterparty Risk with a Tractable Structural Model

In this paper we develop a tractable structural model with analytical default probabilities depending on some dynamics parameters, and we show how to calibrate the model using a chosen number of Credit Default Swap (CDS) market quotes. We essentially show how to use structural models with a calibration capability that is typical of the much more tractable credit-spread based intensity models. We apply the structural model to a concrete calibration case and observe what happens to the calibrated dynamics when the CDS-implied credit quality deteriorates as the firm approaches default. Finally we provide a typical example of a case where the calibrated structural model can be used for credit pricing in a much more convenient way than a calibrated reduced form model: The pricing of counterparty risk in an equity swap.

q-fin.PR↗
arXiv2009

Credit Default Swap Calibration and Counterparty Risk Valuation with a Scenario based First Passage Model

In this work we develop a tractable structural model with analytical default probabilities depending on a random default barrier and possibly random volatility ideally associated with a scenario based underlying firm debt. We show how to calibrate this model using a chosen number of reference Credit Default Swap (CDS) market quotes. In general this model can be seen as a possible extension of the time-varying AT1P model in Brigo and Tarenghi (2004). The calibration capability of the Scenario Volatility/Barrier model (SVBAT1P), when keeping time-constant volatility, appears inferior to the one of AT1P with time-varying deterministic volatility. The SVBAT1P model, however, maintains the benefits of time-homogeneity and can lead to satisfactory calibration results, as we show in a case study where we compare different choices on scenarios and parameters. Similarly to AT1P, SVBAT1P is suited to pricing hybrid equity/credit derivatives and to evaluate counterparty risk in equity payoffs, and more generally to evaluate hybrid credit/equity payoffs. We consider the equity return swap in Brigo and Tarenghi (2004) and show its valuation under SVBAT1P with the same CDS and equity calibration input used earlier for AT1P, and further we hint at equity default swap valuation in the conclusions.

q-fin.PR↗
arXiv2010

Adaptive Wave Models for Option Pricing Evolution: Nonlinear and Quantum Schrödinger Approaches

Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black--Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equation. The nonlinear approach comes in two flavors: (i) for the case of constant volatility, it is defined by a single adaptive nonlinear Schrödinger (NLS) equation, while for the case of stochastic volatility, it is defined by an adaptive Manakov system of two coupled NLS equations. The linear quantum approach is defined in terms of de Broglie's plane waves and free-particle Schrödinger equation. In this approach, financial variables have quantum-mechanical interpretation and satisfy the Heisenberg-type uncertainty relations. Both models are capable of successful fitting of the Black--Scholes data, as well as defining Greeks. Keywords: Black--Scholes option pricing, adaptive nonlinear Schrödinger equation, adaptive Manakov system, quantum-mechanical option pricing, market-heat potential PACS: 89.65.Gh, 05.45.Yv, 03.65.Ge

q-fin.PR↗
arXiv2010

Simulation de trajectoires de processus continus

Continuous time stochastic processes are useful models especially for financial and insurance purposes. The numerical simulation of such models is dependant of the time discrete discretization, of the parametric estimation and of the choice of a random number generator. The aim of this paper is to provide the tools for the practical implementation of diffusion processes simulation, particularly for insurance contexts.

q-fin.CP↗
arXiv2013

Reflected Backward Stochastic Difference Equations with Finite State and their applications

In this paper, we first establish the reflected backward stochastic difference equations with finite state (FS-RBSDEs for short). Then we explore the Existence and Uniqueness Theorem as well as the Comparison Theorem by "one step" method. The connections between FS-RBSDEs and optimal stopping time problems are investigated and we also show that the optimal stopping problems with multiple priors under Knightian uncertainty is a special case of our FS-RBSDEs. As a byproduct we develop the general theory of g-martingales in discrete time with finite state including Doob-Mayer Decomposition Theorem and Optional Sampling Theorem. Finally, we consider the pricing models of American Option in both complete and incomplete markets.

math.PR↗
arXiv2012

Using Premia and Nsp for Constructing a Risk Management Benchmark for Testing Parallel Architecture

Financial institutions have massive computations to carry out overnight which are very demanding in terms of the consumed CPU. The challenge is to price many different products on a cluster-like architecture. We have used the Premia software to valuate the financial derivatives. In this work, we explain how Premia can be embedded into Nsp, a scientific software like Matlab, to provide a powerful tool to valuate a whole portfolio. Finally, we have integrated an MPI toolbox into Nsp to enable to use Premia to solve a bunch of pricing problems on a cluster. This unified framework can then be used to test different parallel architectures.

cs.CE↗