The early exercise premium in American put option prices

This study estimates the value of the early exercise premium in American put option prices using Swedish equity options data. The value of the premium is found as the deviation of the American put price from European put-call parity, and in addition a theoretical estimate of the premium is computed. The empirically found premium is also used in a modified version of the control variate approach to value American puts. The results indicate a substantial value of the early exercise premium, where the premium derived from put-call parity is higher than the theoretical premium. The premium also increases with moneyness and time left to expiration, while the effect of interest rate and volatility depends on the moneyness of the option. The modified control variate technique works reasonably well relative to the theoretical models. In particular, for deep in-the-money options, this technique is superior.     

Evaluation of the MEMM, Parameter Estimation and Option Pricing for Geometric Lévy Processes

In this paper, we shall propose a useful approach to evaluate concretely the MEMM (minimal entropy martingale measure) for the typical geometric Lévy processes such as compound Poisson, stable, VG (Variance Gamma), CGMY (Carr-Geman-Madan-Yor), NIG (Normal Inverse Gaussian), etc. In addition, we shall estimate the parameters of geometric Lévy processes and value the European call option and Asian call option using the Nikkei financial data.     

Fourier transformation and the pricing of average-rate derivatives

In this article we propose a method to compute the density of the arithmetic average of a Markov process. This approach is then applied to the pricing of average rate options (Asian options). It is demonstrated that as long as a closed form formula is available for the discount bond price when the underlying process is treated as the riskless interest rate, analytical formulas for the density function of the arithmetic average and the Asian option price can be derived. This includes the affine class of term structure models. The Cox et al. (1985) square root interest rate process is used as an example. When the underlying process follows a geometric Brownian motion, a very efficient numerical method is proposed for computing the density function of the average. Extensions of the techniques to the cases of multiple state variables are also discussed.      

Evaluation of the Asian Option by the Dual Martingale Measure

In this short paper, we shall consider the arbitrage free Asian call option pricing under the standard Black-Scholes setting. Yor [11] studied this problem by using the bond as numéraire, whereas we use the stock as numéraire which enables us to construct a single variable Markov process for Asian option pricing. Then we show the results obtained by Yor easily through the backward equation treatment for this one dimensional Markov process. Furthermore we shall show the related results for Asian option pricing derived by German-Yor [4] and Eydeland-German [3] through our approach.     

Partial differential equations for Asian option prices

In this article, we consider fixed and floating strike European style Asian call and put options. For such options, there is no convenient closed-form formula for the prices. Previously, Rogers and Shi, Vecer, and Dubois and Lelièvre have derived partial differential equations with one state variable, with the stock price as numeraire, for the option prices. In this paper, we derive a whole family of partial differential equations, each with one state variable with the stock price as numeraire, from which Asian options can be priced. Any one of these partial differential equations can be transformed into any other. This family includes four partial differential equations which have a particularly simple form including the three found by Rogers and Shi, Vecer, and Dubois and Lelièvre. Our analysis includes the case of a dividend yield; we also include the case of in progress Asian options with floating strike, whereby we discuss the new equation proposed by Vecer, which uses the average asset as numeraire. We perform an error analysis on the four special partial differential equations and Vecer’s new equation and find that their truncation errors are all of the same order. We also perform numerical comparisons of the five partial differential equations and conclude, as expected, that Vecer’s equations and that of Dubois and Lelièvre do better when the volatility is low but that with higher volatilities the performance of all five equations is similar. Vecer’s equations are unique in possessing a certain martingale property and as they perform numerically well or better than the others, must be considered the preferred choice.     

The square-root process and Asian options

Although the square-root process has long been used as an alternative to the Black–Scholes geometric Brownian motion model for option valuation, the pricing of Asian options on this diffusion model has never been studied analytically. However, the additivity property of the square-root process makes it a very suitable model for the analysis of Asian options. In this paper, we develop explicit prices for digital and regular Asian options. We also obtain distributional results concerning the square-root process and its average over time, including analytic formulae for their joint density and moments. We also show that the distribution is actually determined by those moments.     

“Modeling the Impact of Genetics on Insurance”, Angus S. Macdonald,January, 1999

Abstract

Current formulas in credibility theory often estimate expected claims as a function of the sample mean of the experience claims of a policyholder. An actuary may wish to estimate future claims as a function of some statistic other than the sample arithmetic mean of claims, such as the sample geometric mean. This can be suggested to the actuary through the exercise of regressing claims on the geometric mean of prior claims. It can also be suggested through a particular probabilistic model of claims, such as a model that assumes a lognormal conditional distribution. In the first case, the actuary may lean towards using a linear function of the geometric mean, depending on the results of the data analysis. On the other hand, through a probabilistic model, the actuary may want to use the most accurate estimator of future claims, as measured by squared-error loss. However, this estimator might not be linear.

In this paper, I provide a method for balancing the conflicting goals of linearity and accuracy. The credibility estimator proposed minimizes the expectation of a linear combination of a squared-error term and a second-derivative term. The squared-error term measures the accuracy of the estimator, while the second-derivative term constrains the estimator to be close to linear. I consider only those families of distributions with a one-dimensional sufficient statistic and estimators that are functions of that sufficient statistic or of the sample mean. Claim estimators are evaluated by comparing their conditional mean squared errors. In general, functions of the sufficient statistics prove to be better credibility estimators than functions of the sample mean.     

Efficient estimation of lower and upper bounds for pricing higher-dimensional American arithmetic average options by approximating their payoff functions

In this paper, we develop an efficient payoff function approximation approach to estimating lower and upper bounds for pricing American arithmetic average options with a large number of underlying assets. The crucial step in the approach is to find a geometric mean which is more tractable than and highly correlated with a given arithmetic mean. Then the optimal exercise strategy for the resultant American geometric average option is used to obtain a low-biased estimator for the corresponding American arithmetic average option. This method is particularly efficient for asset prices modeled by jump-diffusion processes with deterministic volatilities because the geometric mean is always a one-dimensional Markov process regardless of the number of underlying assets and thus is free from the curse of dimensionality. Another appealing feature of our method is that it provides an extremely efficient way to obtain tight upper bounds with no nested simulation involved as opposed to some existing duality approaches. Various numerical examples with up to 50 underlying stocks suggest that our algorithm is able to produce computationally efficient results.     

Option values under stochastic volatility: Theory and empirical estimates

This paper numerically solves the call option valuation problem given a fairly general continuous stochastic process for return volatility. Statistical estimators for volatility process parameters are derived, and parameter estimates are calculated for several individual stocks and indices. The resulting estimated option values do not differ dramatically from Black-Scholes values in most cases, although there is some evidence that for longer-maturity index options, Black-Scholes overvalues out-of-the-money calls in relation to in-the-money calls.     

The geometric chain-ladder

Abstract

The log normal reserving model is considered. The contribution of the paper is to derive explicit expressions for the maximum likelihood estimators. These are expressed in terms of development factors which are geometric averages. The distribution of the estimators is derived. It is shown that the analysis is invariant to traditional measures for exposure.     

Estimating cost of capital in firm valuations with arithmetic or geometric mean – or better use the Cooper estimator?

We test the extent and determinants of bias effects of the arithmetic as well as the geometric mean estimator and the estimator of Cooper [1996. Arithmetic versus geometric mean estimators: Setting discount rates for capital budgeting. European Financial Management 2 (July): 157–67] regarding discount rate estimation for firm valuation by way of a bootstrap approach for 13 different countries. The Cooper estimator is superior to both the geometric and the (conventional) arithmetic mean estimator. However, a ‘truncated’ version of the arithmetic mean estimator leads generally to better estimation outcomes than the Cooper estimator. This means that, in order to reduce problems of upward-biased firm value estimates, expected cash flows beyond a certain time horizon are completely neglected in terminal value estimation. Such an approach seems particularly reasonable for the valuation of young growth companies as well as for companies from quickly developing countries such as Brazil, China, or Thailand, because the bias in terminal value estimation is increasing in the growth rate of future expected cash flows.     

Optimal importance sampling with explicit formulas in continuous time

In the Black–Scholes model, consider the problem of selecting a change of drift which minimizes the variance of Monte Carlo estimators for prices of path-dependent options. Employing large deviations techniques, the asymptotically optimal change of drift is identified as the solution to a one-dimensional variational problem, which may be reduced to the associated Euler–Lagrange differential equation. Closed-form solutions for geometric and arithmetic average Asian options are provided. The authors acknowledge the support of the National Science Foundation under grants DMS-0532390 (Guasoni) and DGE-0221680 (Robertson) at Boston University.     

Efficient exposure computation by risk factor decomposition

The focus of this paper is the efficient computation of counterparty credit risk exposure on portfolio level. Here, the large number of risk factors rules out traditional PDE-based techniques and allows only a relatively small number of paths for nested Monte Carlo simulations, resulting in large variances of estimators in practice. We propose a novel approach based on Kolmogorov forward and backward PDEs, where we counter the high dimensionality by a generalization of anchored-ANOVA decompositions. By computing only the most significant terms in the decomposition, the dimensionality is reduced effectively, such that a significant computational speed-up arises from the high accuracy of PDE schemes in low dimensions compared to Monte Carlo estimation. Moreover, we show how this truncated decomposition can be used as control variate for the full high-dimensional model, such that any approximation errors can be corrected while a substantial variance reduction is achieved compared to the standard simulation approach. We investigate the accuracy for a realistic portfolio of exchange options, interest rate and cross-currency swaps under a fully calibrated 10-factor model.     

On the asymptotic behavior of the prices of Asian options

In this paper, we study the price of a long term Asian option the pay-off of which is determined by the average price of the underlying asset during the last fixed number of days of its life. As one can imagine, it converges to the price of a plain vanilla option as the time to maturity increases. We explicitly obtained the asymptotic difference which will be useful for computing the price of Asian option in practice.     

A P.D.E. approach to Asian options: analytical and numerical evidence

We first derive a one-state-variable partial differential equation, easy to implement, which characterizes the price of a European type Asian option. This result is explained and related to previous literature. We then derive new results on the hedging of an Asian option and propose analytical and numerical analysis on the comparison between Asian and European options. Our methodology which applies to “fixed-strike” Asian options as well as to “floating-strike” Asian options completes and clarifies various results in the literature. In this paper we focus on “backward-starting” Asian options. Our approach is quite general however, and we explain how to adapt our main results to the case of “forward-starting” Asian options.     

Asian options and meromorphic Lévy processes

One method to compute the price of an arithmetic Asian option in a Lévy driven model is based on an exponential functional of the underlying Lévy process: If we know the distribution of the exponential functional, we can calculate the price of the Asian option via the inverse Laplace transform. In this paper, we consider pricing Asian options in a model driven by a general meromorphic Lévy process. We prove that the exponential functional is equal in distribution to an infinite product of independent beta random variables, and its Mellin transform can be expressed as an infinite product of gamma functions. We show that these results lead to an efficient algorithm for computing the price of the Asian option via the inverse Mellin–Laplace transform, and we compare this method with some other techniques.     

Combining fair pricing and capital requirements for non-life insurance companies

The aim of this article is to identify fair equity-premium combinations for non-life insurers that satisfy solvency capital requirements imposed by regulatory authorities. In particular, we compare target capital derived using the value at risk concept as planned for Solvency II in the European Union with the tail value at risk concept as required by the Swiss Solvency Test. The model framework uses Merton’s jump-diffusion process for the market value of liabilities and a geometric Brownian motion for the asset process; fair valuation is conducted using option pricing theory. We show that even if regulatory requirements are satisfied under different risk measures and parameterizations, the associated costs of insolvency – measured with the insurer’s default put option value – can differ substantially.     

Skew Brownian Motion and Pricing European Options

Abstract

The volatility smile and systematic mispricing of the Black–Scholes option pricing model are the typical motivation for examining stochastic processes other than geometric Brownian motion to describe the underlying stock price. In this paper a new stochastic process is presented, which is a special case of the skew-Brownian motion of Itô and McKean. The process in question is the sum of a standard Brownian motion and an independent reflecting Brownian motion that is similar in construction to the stochastic representation of a skew-normal random variable. This stochastic process is taken in its exponential form to price European options. The derived option price nests the Black–Scholes equation as a special case and is flexible enough to accommodate stochastic volatility as well as stochastic skewness.     

The Model-Free Implied Volatility and Its Information Content

Britten-Jones and Neuberger (2000) derived a model-free impliedvolatility under the diffusion assumption. In this article,we extend their model-free implied volatility to asset priceprocesses with jumps and develop a simple method for implementingit using observed option prices. In addition, we perform a directtest of the informational efficiency of the option market usingthe model-free implied volatility. Our results from the Standard& Poor’s 500 index (SPX) options suggest that themodel-free implied volatility subsumes all information containedin the Black–Scholes (B–S) implied volatility andpast realized volatility and is a more efficient forecast forfuture realized volatility.