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An easy method for calculating parametric value at risk is given, using significant decay factors. Please note that it assumes constant weightings of securities, which is not true in general for portfolios but is true for 1 equity portfolios and certain benchmark constructions. (e.g, 30% MSCI world, 70% Barclays Gloagg)
Nervously this is my first short paper since I left university, please give me your comments. (It's already well known etc) Discuss this paper
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by Lan Zhang This paper is about how to estimate the integrated covariance hX, Y iT of two assets over a fixed time horizon [0, T], when the observations about X and Y are “contaminated” and when such noisy observations are at discrete, but not synchronized, times. We show that the usual previous-tick covariance estimator is biased, and the size of the bias is more pronounced for less liquid assets. This is an analytic characeterization of the Epps effect. We also provide optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms, including nonsynchronous trading, microstructure noise, and time discretization. Finally, a two-scales covariance estimator is provided which simulaneously cancels (to first order) the Epps effect and the effect of microstructure noise. The gain is demonstrated in data. KEY WORDS: Bias-variance tradeoff; Epps effect; High frequency data; Measurement error; Market Microstructure; Martingale; Nonsynchronous trading; Realized covariance; Realized variance; Two scales estimation.
In this article we discuss the Heston model with stochastic interest rates driven by Hull-White (HW) or Cox-Ingersoll-Ross (CIR) processes. We define a so-called volatility compensator which guarantees that the Heston hybrid model with a non-zero correlation between the equity and interest rate processes is properly defined. Moreover, we propose an approximation for the characteristic function, so that pricing of basic derivative products can be efficiently done using Fourier techniquesWe also discuss the effect of the approximations on the instantaneous correlations, and check the influence of the correlation between stock and interest rate on the implied volatilities.
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by Zaizhi Wang Abstract: This paper tackles the issue of approximated formula for stochastic model with time dependent model parameters, using an averaging principle. The idea lies in finding a similar model but with constant parameters that is the closest to our initial process, along the same lines as results proven by Gy Discuss this paper
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by David LeRay Financial mathematics is a branch of mathematics that assesses the risk and value of various ancial instruments. Banks, companies, and other institu-tions mitigate their risk through nancial instruments known as derivatives, that derive their value from some underlying asset. The equations that arise from pricing and modeling can be very complex, leading to the necessity of numerical methods. This project studied the use of certain numerical methods for the pricing of a particular type of option called an Asian option. Asian options can provide favorable risk pro les because the payout is determined based on the average value over a time period, rather than the nal value. The price of an Asian option is governed by a partial di erential equation in three variables: stock price, average price over the current time interval, and time. The solution method was rst to discretize the partial di erential equation into a system of ordinary di erential equations. Next, the ODE system was integrated using a sti -ODE solver available in MATLAB. Enhancements to this solution method include specifying the sparsity pattern, implementing an iterative linear solver (GMRES [2]) in place of MATLAB's built-in direct linear solver, and using preconditioning to improve the solution characteris-tics of that solver. Discuss this paper
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Aothors : Tian-Shyr Dai, Guan-Shieng Huang and Yuh-Dauh Lyuu
Asian options can be priced on the unrecombining binomial tree. Unfortunately, without approximation, the running time is exponential. This paper presents efficient and extremely accurate approximation algorithms for Asian options on the binomial tree. Discuss this paper
In an equity swap, two parties make a series of payments to each other with at least one set of payments determined by a stock or index return. The other set of payments can be a fixed or floating rate or the return on another stock or index. Equity swaps are used to substitute for a direct transaction in stock. This article explores equity swaps, describing variations, applications, and advantages and disadvantages. It also presents and illustrates formulas for pricing and valuation and provides empirical evidence comparing the performance of equity swaps against comparable strategies involving direct investment in equity. Discuss this paper
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by Craig Liu and Deng-Feng Wang In this work, we consider the issue of pricing exchange options and spread options with stochastic interest rates. We provide the closed form solution for the exchange option price when interest rate is stochastic. Our result holds when interest rate is modeled with a stochastic term structure of general form, which includes Vasicek model, CIR term structure, and other well-known term structure models as specialccases. In particular, we have discussed the possibility of using our closed form solution as a control variate in pricing spread options with stochastic interest rate Discuss this paper
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by Yacine Aijt-Sahalia and Robert Kimmel We develop and implement a technique for maximum likelihood estimation in closed-form of multivariate a Discuss this paper
This paper proposes a unified state-space formulation for parameter estimation of exponential-affine term structure models. This class of models, charaterized by Duffie and Kan (1993), contains models such as Vasicek (1977), Cox, Ingersoll and Ross (1985) and Chen and Scott (1992), among others. The proposed method uses an approximate linear Kalman filter which only requires specifying the conditional mean and variance of the system in an approximate sense. The method allows for measurement errors in the observed yields to maturitiy, and can simultaneously deal with many yields on bonds with different maturities. A Monte Carlo study indicates thet the proposed method is a reliable procedure for moderate sample sizes. An empirical analysis of three existing exponential-affine term structure models is carried out using monthly U.S. Treasury yield data with four different maturities. Our test results indicate a strong rejection of all three models. Discuss this paper
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Joe Ganley Giorgio Trebeschi World markets in the 1990s appear to have been subject to greater turbulence and to more shocks than hitherto. At the same time we observe a wide variety of market structures and trading platforms. This raises the question of whether, for a common shock, markets will respond differently. In particular, is it possible to rank the performance of different market structures during turbulent trading conditions? We examine the performance of four equity markets Discuss this paper
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by Carol Alexandar and Anea Dimitriu Abstract: This paper examines, from a market efficiency perspective, the performance of a simple dynamic equity indexing strategy based on cointegration. A consistent 'abnormal' return in excess of the benchmark is demonstrated over different time horizons and in different real world and simulated stock markets. A measure of stock price dispersion is shown to be a leading indicator for the abnormal return and their relationship is modelled as a Markov switching process of two market regimes. We find that the entire abnormal return is associated with the high volatility regime as the indexing model implicitly adopts a strategic position that pays off during market crashes, whilst effectively tracking the benchmark in normal market circumstances. Therefore we find no evidence of market inefficiency. Nevertheless our results have implications for equity fund managers: we show how, without any stock selection, solely through a smart optimization that has an implicit element of market timing, the benchmark performance can be significantly enhanced.
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Ramaprasad Bhar, Carl Chiarella This paper considers the Heath-Jarrow-Morton (HJM) model of the term structure of interest rates for a fairly general specification of forward rate volatility, including stochastic variables. Estimation of this volatility function is at the heart of the identification of the HJM model. Reduction of the model to state space form is discussed and use of the Kalman filter as an estimation technique is proposed. Since typical data sets are small, a bootstrap procedure is used to determine the statistical significance of the estimates. A Monte-Carlo experiment is used to compare the bootstrap and “true” smallsample distributions of the estimates of the parameters of the volatility function.
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We present two robust extensions of the CreditGrades model: the first one assumes that the variance of returns on the firm Discuss this paper
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Samuli Ikonen Jari Toivanen Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two-dimensional parabolic partial differential inequality. A nite difference discretization on nonuniform grids leading to linear complementarity problems with M-matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear quations in the operator splitting method and in the penalty method are solved using a multigrid method. The projected multigrid method and the componentwise splitting method lead to a sequence of linear complementarity problems with one-dimensional differential operators which are solved using the Brennan and Schwartz algorithm. The numerical experiments compare the accuracy and speed of the considered thods. The accuracies of all methods appear to be similar. Thus,the additional approximations made in the operator splitting method, in the penalty method, and in the componentwise splitting method do not increase the error essentially. The componentwise splitting method is the fastest one. All multigrid based methods have similar rapid grid independent convergence rates. They are from two to four times slower that the componentwise splitting method. On the coarsest grid the speed of the projected SOR is comparable with the multigrid methods while on ner grids it is several times slower. Discuss this paper
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Christian Wallner Uwe Wystup Abstract: No front-office software can survive without providing derivatives of option prices with respect to underlying market or model parameters, the so called Greeks. If a closed form solution for an option exists, Greeks can be computed analytically and they are numerically stable. However, for American style options, there is no closed-form solution. The price is computed by binomial trees, finite difference methods or an analytic approximation. Taking derivatives of these prices leads to instable numerics or misleading results, specially for Greeks of higher order. We compare the computation of the Greeks in various pricing methods and conclude with the recommendation to use Leisen-Reimer trees. Discuss this paper
LEIF B.G. ANDERSEN Banc of America Securities -------------------------------------------------------------------------------- January 23, 2007
Abstract: Stochastic volatility models are increasingly important in practical derivatives pricing applications, yet relatively little work has been undertaken in the development of practical Monte Carlo simulation methods for this class of models. This paper considers several new algorithms for time-discretization and Monte Carlo simulation of Heston-type stochastic volatility models. The algorithms are based on a careful analysis of the properties of affine stochastic volatility diffusions, and are straightforward and quick to implement and execute. Tests on realistic model parameterizations reveal that the computational efficiency and robustness of the simulation schemes proposed in the paper compare very favorably to existing methods.
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Robert Engle and Robert Ferstenberg NBER Working Paper No. 12165 ABSTRACT Transaction costs in trading involve both risk and return. The return is associated with the cost of immediate execution and the risk is a result of price movements during a more gradual trading. The paper shows that the trade-off between risk and return in optimal execution should reflect the same risk preferences as in ordinary investment. The paper develops models of the joint optimization of positions and trades, and shows conditions under which optimal execution does not depend upon the other holdings in the portfolio. Optimal execution however may involve trades in assets other than those listed in the order; these can hedge the trading risks. The implications of the model for trading with reversals and continuations are developed. The model implies a natural measure of liquidity risk.
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Wim Schoutens Abstract In this paper we overview the pricing of several so-called exotic options in the nowdays quite popular exponential Levy models. Discuss this paper
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Cristina Sommacampagna Abstract In this paper we develop a new approach to Value-at-Risk estimating the betas of the assets in the portfolio with the Kalman filter. This technique is applied to a portfolio of assets of an insurance company and is compared with the performances of two traditional methodologies: the approach based on the variance-covariance matrix of returns and the approach based on OLS Sharpe betas. The back testing analysis shows that the proposed technique is able to capture the dynamics of financial markets and is flexible enough to match the hedging purposes of a financial institution. Keywords: Value-at-Risk; Kalman filter; Sharpe beta.
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