## Pricing options on multiple underlyings

In this section we show how to extend the Monte Carlo simulation technique to higher dimensions. The problem is not only that one has to deal with higher dimensional integrals, but also that one has to incorporate the underlying correlation structure between the considered securities. In our framework we need the covariance matrix of the log returns on an annual basis. In general, a basket option is an option on several underlyings (a basket of underlyings). Basket options can be European-,...

## Bibliography

Generalized autoregressive conditional heteroscedas-ticity, Journal of Econometrics 31 307-327. B hlmann, P. and McNeil, A.J. (1999). Nonparametric GARCH-models, Manuscript, ETH Zurich, http www.math.ethz.ch mcneil. Carroll, R.J. and Hall, P. (1988). Optimal rates of convergence for deconvolut-ing a density, J. Amer. Statist. Assoc. 83 1184-1186. Fan, J. and Truong, Y.K. (1993). Nonparametric regression with errors-in-variables, Ann. Statist. 19 1257-1272. Franke, J., H...

## Gbpusd Simulation

Simulated variance and covariance processes, both bivari-ate (blue) and univariate case (green), 105St. Figure 10.3. Simulated variance and covariance processes, both bivari-ate (blue) and univariate case (green), 105St. For this at each point in time an observation et is drawn from a multivariate normal distribution with variance t. Given these observations, t is updated according to (10.7). Then, a new residual is drawn with covariance Xt+1. We apply this procedure for T 3000....

## Generating Scenarios Monte Carlo Valueat Risk

Assume now that the copula C has been selected. For risk management purposes, we are interested in the Value-at-Risk of a position. While analytical methods for the computation of the Value-at-Risk exist for the multivariate normal distribution (i.e. for the Gaussian copula), we will in general have to use numerical simulations for the computation of the VaR. To that end, we need to generate pairs of random variables (Xi,X2) F(C), which form scenarios of possible changes of the risk factor. The...

## Inversion of the cdf minus the Gaussian Approximation

Assume that F is a cdf with mean m and standard deviation a, then F(x) - (x M,a) e-ixt- ( (t) - e i-ff2i2 2) dt (1.38) holds, where (. m, a) is the normal cdf with mean m and standard deviation a and eiMi-CT 4 2 its characteristic function. (Integrating the inversion formula (1.16) w.r.t. x and applying Fubini's theorem leads to (1.38).) Applying the Fourier inversion to F(x) (x m, a) instead of F(x) solves the (numerical) problem that t (t) has a pole at 0. Alternative distributions with known...

## Simulation and a Comparison of the SPDs

The example used here to show the procedure of generating the IBT, is taken from Derman and Kani (1994). Assume that the current value of the stock is S 100, the annually compounded riskless interest rate is r 3 per year for all time expirations, the stock has zero dividend. The annual BS implied volatility of an at-the-money call is assumed to be a 10 , and the BS implied volatility increases (decreases) linearly by 0.5 percentage points with every 10 point drop (rise) in the strike. From the...

## Data Analysis with XploRe

We store the time series of the different yield curves in individual files. The file names, the corresponding industries and ratings and the names of the matrices used in the XploRe code are listed in Table 3.2. Each file contains data for the maturities 3M to 10Y in columns 4 to 12. XploRe creates matrices from the data listed in column 4 of Table 3.2 and produces summary statistics for the different yield curves. As example files the data sets for US treasury and industry bonds with rating...

## Pros and Cons of Delta Gamma Approximations

Both assumptions of the Delta-Gamma-Normal approach - Gaussian innovations and a reasonably good quadratic approximation of the value function V - have been questioned. Simple examples of portfolios with options can be constructed to show that quadratic approximations to the value function can lead to very large errors in the computation of VaR Britton-Jones and Schae-fer, 1999 . The Taylor-approximation 1.1 holds only locally and is questionable from the outset for the purpose of modeling...

## VaR Estimation and Backtesting with XploRe

In this section we explain, how a VaR can be calculated and a backtesting can be implemented with the help of XploRe routines. We present numerical results for the different yield curves. The VaR estimation is carried out with the help of the VaRest command. The VaRest command calculates a VaR for historical simulation, if one specifies the method parameter as EDF empirical distribution function . However, one has to be careful when specifying the sequence of asset returns which are used as...

## Example Analysis of DAX data

We now use the IBT to forecast the future price distribution of the real stock market data. We use DAX index option prices data at January 4, 1999, which are included in MD BASE, a database located at CASE Center for Applied Statistics and Economics at Humboldt-Universitat zu Berlin, and provide some dataset for demonstration purposes. In the following program, we estimate the BS implied volatility surface first, while the quantlet volsurf, Fengler, Hardle and Villa 2001 , is used to obtain...

## Frequently Used Notation

X f x is defined as R real numbers R d f R U to, to AT transpose of matrix A X D the random variable X has distribution D E X expected value of random variable X Var X variance of random variable X Std X standard deviation of random variable X Cov X, Y covariance of two random variables X and Y N yU, S normal distribution with expectation and covariance matrix S, a similar notation is used if S is the correlation matrix cdf denotes the cumulative distribution function pdf denotes the...

## How Precise Are Price Distributions Predicted by Implied Binomial Trees

In recent years, especially after the 1987 market crash, it became clear that the prices of the underlying asset do not exactly follow the Geometric Brow-nian Motion GBM model of Black and Scholes. The GBM model with constant volatility leads to a log-normal price distribution at any expiration date All options on the underlying must have the same Black-Scholes BS implied volatility, and the Cox-Ross-Rubinstein CRR binomial tree makes use of this fact via the construction of constant transition...

## General Properties of Delta GammaNormal Models

The change in the portfolio value, AV, can be expressed as a sum of independent random variables that are quadratic functions of standard normal random variables Yt by means of the solution of the generalized eigenvalue problem with X CY, S CTA and A diag A1, , Am . Packages like LAPACK Anderson, Bai, Bischof, Blackford, Demmel, Dongarra, Croz, Greenbaum, Hammarling, McKenney and Sorensen, 1999 contain routines directly for the generalized eigenvalue problem. Otherwise C and A can be computed...

## Implied Binomial Trees

A well known model for financial option pricing is a GBM with constant volatility, it has a log-normal price distribution with density, at any option expiration T, where St is the stock price at time t, r is the riskless interest rate, t T t is time to maturity, and a the volatility. The model also has the characteristic that all options on the underlying must have the same BS implied volatility. However, the market implied volatilities of stock index options often show the volatility smile,...

## An Example Application to DAX data

This section describes how to estimate the Black-Scholes and local polynomial SPD using options data on the German DAX index. The dataset was taken from the financial database MD BASE located at CASE Center for Applied Statistics and Economics at Humboldt-Universitat zu Berlin. Since MD BASE is a proprietary database, only a limited dataset is provided for demonstration purposes. This database is filled with options and futures data provided by Eurex. Daily series of 1, 3, 6 and 12 months...

## Volatility Smile

Implied volatility smile as observed on January 4th, 1999 Figure 6.1. Implied volatility smile as observed on January 4th, 1999 In Figure 6.1 we display the output for the strike dimension. The deviation from the BS model is clearly visible implied volatilities form a convex smile in strikes. One finds a curved shape also across different maturities. In combination with the strike dimension this yields a surface with pronounced curvature Figure 6.2 . The discontinuity of the ATM...

## Bootstrapping Markov Chains

The one-period transition matrix P is unknown and must be estimated. The estimator P is associated with estimation errors which consequently influence the estimated multi-period transition matrices. The traditional approach to quantify this influence turns out to be tedious since it is difficult to obtain the distribution of P P , which could characterize the estimation errors. Furthermore, the distribution of P P m , with has to be discussed in order to address the sensitivity of the estimated...

## Statistical Modeling for VaR

VaR methodologies can be classified in terms of statistical modeling decisions and approximation decisions. Once the statistical model and the estimation procedure is specified, it is a purely numerical problem to compute or approximate the Value at Risk. The modeling decisions are 1. Which risk factors to include. This mainly depends on a banks' business portfolio . But it may also depend on the availability of historical data. If data for a certain contract is not available or the quality is...

## Hrir2 Firi F2r2 for all rir2 G R

Thus, we obtain the independence copula C n by which becomes obvious from the following theorem THEOREM 2.2 Let Ri and R2 be random variables with continuous distribution functions Fi and F2 and joint distribution function H. Then Ri and R2 are independent if and only if Cr1rr2 n. From Sklar's Theorem we know that there exists a unique copula C with P Ri lt ri,R2 lt r2 H ri,r2 C Fi ri ,F2fa . 2.6 Independence can be seen using Equation 2.4 for the joint distribution function H and the...

## Monte Carlo Sampling Method

The partial Monte-Carlo method is a Monte-Carlo simulation that is performed by generating underlying prices given the statistical model and then valuing them using the simple delta-gamma approximation. We denote X as a vector of risk factors, AV as the change in portfolio value resulting from X, L as AV, a as a confidence level and l as a loss threshold. A first order derivative with regard to risk factors r second order derivative with regard to risk factors covariance matrix of risk factors...

## Contents

1 Approximating Value at Risk in Conditional Gaussian Models 3 1.1.1 The Practical 1.1.2 Statistical Modeling for VaR 4 1.1.3 VaR 1.1.4 Pros and Cons of Delta-Gamma Approximations 7 1.2 General Properties of Delta-Gamma-Normal Models 8 1.3 Cornish-Fisher 1.4 Fourier Inversion 1.4.1 Error Analysis 1.4.2 Tail 1.4.3 Inversion of the cdf minus the Gaussian Approximation 21 1.5 Variance Reduction Techniques in Monte-Carlo Simulation 24 1.5.1 Monte-Carlo Sampling 1.5.2 Partial Monte-Carlo with...

## Cornish Fisher Approximations 131 Derivation

The Cornish-Fisher expansion can be derived in two steps. Let denote some base distribution and its density function. The generalized Cornish-Fisher expansion Hill and Davis, 1968 aims to approximate an a-quantile of F in terms of the a-quantile of , i.e., the concatenated function F-1 o . The key to a series expansion of F-1 o in terms of derivatives of F and is Lagrange's inversion theorem. It states that if a function s t is implicitly defined by and h is analytic in c, then an analytic...

## VaR Approximations

In this paper we consider certain approximations of VaR in the conditional Gaussian class of models. We assume that the conditional expectation of Xt, yU,t, is zero and its conditional covariance matrix t is estimated and given at time t 1. The change in the portfolio value over the time interval t 1, t is then where the w are the portfolio weights and AS is the function that maps the risk factor vector Xt to a change in the value of the i-th security value over the time interval t 1,t , given...