On Higher Derivatives of Expectations

It is understood that derivatives of an expectation $E [\phi(S(T)) | S(0) = x]$ with respect to $x$ can be expressed as $E [\phi(S(T)) \pi | S(0) = x]$, where $S(T)$ is a stochastic variable at time $T$ and $\pi$ is a stochastic weighting function (weight) independent of the form of $\phi$. Derivatives of expectations of this form are encountered in various fields of knowledge. We establish two results for weights of higher order derivatives under the dynamics given by (\ref{dynamics}). Specifically, we derive and solve a recursive relationship for generating weights. This results in a tractable formula for weights of any order.price sensitivities, greeks, malliavin calculus

Efficient computation of option price sensitivities for options of American style

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. --American options,Greeks,Leisen-Reimer trees

Closed formula for options with discrete dividends and its derivatives

We present a closed pricing formula for European options under the BlackScholes model and formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and by expressing the spatial derivatives as expectations under special measures, as in Carr, together with an unusual change of measure technique that relies on the replacement of the initial condition. The closed formulas are attained for the case where no dividend payment policy is considered. Despite its small practical relevance, a digital dividend policy case is also considered which yields approximation formulas. The results are readily extensible to time dependent volatility models but no so for local-vol type models. For completeness, we reproduce the numerical results in Vellekoop and Nieuwenhuis using the formulas here obtained. The closed formulas presented here allow a fast calculation of prices or implied volatilities when compared with other valuation procedures that rely on numerical methods. --equity option,discrete dividend,hedging,analytic formula

Scaling invariance and contingent claim pricing

[MAS R-9914] Prices of tradables can only be expressed relative to eachother at any instant of time. This fundamental fact should thereforealso hold for contingent claims, i.e. tradable instruments, whoseprices depend on the prices of other tradables. We show that thisproperty induces local scale-invariance in the problem of pricingcontingent claims. Due to this symmetry we do {it not/} require anymartingale techniques to arrive at the price of a claim. If thetradables are driven by Brownian motion, we find, in a natural way,that this price satisfies a PDE. Both possess a manifestgauge-invariance. A unique solution can only be given when we imposerestrictions on the drifts and volatilities of the tradables, i.e.the underlying market structure. We give some examples of theapplication of this PDE to the pricing of claims. In the Black-Scholesworld we show the equivalence of our formulation with the standardapproach. It is stressed that the formulation in terms of tradablesleads to a significant conceptual simplification of thepricing-problem.#[MAS R-9919] This article is the second one in a series on the use of scaling invariance in finance. In the first paper, we introduced a new formalism for the pricing of derivative securities, which focusses on tradable objects only, and which completely avoids the use of martingale techniques. In this article we show the use of the formalism in the context of path-dependent options. We derive compact and intuitive formulae for the prices of a whole range of well known options such as arithmetic and geometric average options, barriers, rebates and lookback options. Some of these have not appeared in the literature before. For example, we find rather elegant formulae for double barrier options with moving barriers, continuous dividends and all possible configurations of the barriers. The strength of the formalism reveals itself in the ease with which these prices can be derived. This allowed us to pinpoint some mistakes regarding geometric mean options, which frequently appear in the literature. Furthermore, symmetries such as put-call transformations appear in a natural way within the framework

Speed and Accuracy Comparison of Noncentral Chi-Square Distribution Methods for Option Pricing and Hedging under the CEV Model

Pricing options and evaluating greeks under the constant elasticity of variance (CEV) model require the computation of the noncentral chi-square distribution function. In this article, we compare the performance in terms of accuracy and computational time of alternative methods for computing such probability distributions against an xternally tested benchmark. In addition, we present closed-form solutions for computing greek measures under the CEV option pricing model for both beta 2, thus being able to accommodate direct leverage effects as well as inverse leverage effects that are frequently observed in the options markets

Pricing and hedging bond options and sinking-fund bonds under the CIR model

This article derives simple closed-form solutions for computing Greeks of zero-coupon and coupon-bearing bond options under the CIR interest rate model, which are shown to be accurate, easy to implement, and computationally highly e cient. These novel analytical solutions allow us to extend the literature in two other directions. First, the static hedging portfolio approach is used for pricing and hedging American-style plain-vanilla zero-coupon bond options under the CIR model. Second, we derive analytically the comparative static properties of sinking-fund bonds under the same interest rate modeling setup.Manuela Larguinho and Carlos A. Braumann belong to the research center CIMA (Centro de Investigação em Matemática e Aplicações, Instituto de Investigação e Formação Avançada, Universidade de Évora), supported by FCT (Fundação para a Ciência e a Tecnologia, Portugal), project UID/04674/2020. José Carlos Dias belongs to the Business Research Unit (BRU-IUL) and acknowledges the support provided by FCT [grant number UIDB/00315/2020]

Essays on option pricing under alternative one-dimensional diffusions

Essays on Option Pricing under Alternative One-Dimensional Diffusions Given its analytical attractiveness, the process most commonly used in the financial and real options literature is the geometric Brownian motion. However, this assumption embodies some unrealistic implications for the dynamical behavior of the underlying asset price. To overcome this issue, alternative stochastic processes have been considered in the valuation of financial and real options. This thesis examines financial and real options using alternative one-dimensional diffusions, namely the constant elasticity of variance (CEV) and mean-reverting CEV diffusions. This thesis has two main purposes. First, it derives closed-form solutions for computing Greeks of European-style options under both the CEV and CIR (Cox, Ingersoll and Ross) models. Second, it analyzes the optimal entry and exit policy of a firm in the presence of output price uncertainty and costly reversibility of investment under a generalized class of one-dimensional diffusions and shows how the hysteretic band is affected; **** Resumo: Ensaios sobre a Avaliação de Opções sob Difusões Unidimensionais Alternativas Na avaliação de opções financeiras e reais, o processo mais utilizado na literatura ´e o movimento Browniano geométrico. Contudo, esta suposição ao incorpora algumas implicações irrealistas para o comportamento dinâmico do preço do activo subjacente. Para ultrapassar estas limitações, têm sido considerados processos estocásticos alternativos para a avaliação de opções financeiras e reais. Esta tese analisa opções, financeiras e reais, utilizando difusões unidimensionais alternativas, nomeadamente as difusões elasticidade constante da vari ˆancia (CEV - constant elasticity of variance) e CEV com reversão `a média. Esta tese tem dois objectivos principais. Primeiro, derivar soluções analíticas para calcular as letras gregas de opções de tipo Europeu para os modelos CEV e CIR (Cox, Ingersoll e Ross). Segundo, analisar a política óptima de entrada e de saída de uma empresa na presença de incerteza no prec¸o do output e de reversibilidade dos custos de investimento, para uma classe generalizada de difusões unidimensionais, e mostrar a influência sobre a banda de histerese económica

Valuation of derivative securities using stochastic analytic and numerical methods

This thesis details methods and procedures to compute prices and hedging strategies for derivative securities in financial mathematics using stochastic analytic, numerical and variance reduction techniques. Results are obtained on explicit hedge ratio representations for non-smooth payoff functionals and mult idimensional diffusion processes with stopping boundaries. These methods are used to determine hedge ratios for the maximum of several assets and lookback options. A number of powerful variance reduction techniques are described. These include the use of measure transformations, discrete versions of importance sampling estimators, control variates based on Ito integral representations, stratified sampling and quasi Monte Carlo. For many of these techniques explicit formulas for the variance of the resulting estimators are obtained. Pricing and hedging procedures are developed for a class of foreign exchange barrier options under stochastic volatility. These procedures are applied to the calculation of down-and-out call options for the Heston model. A general methodology for pricing discount bonds and options on discount bonds for multifactor term structure models is established. This approach is used for both European and American style securities for a version of the two-factor Fong and Vasicek model, extended to include time dependent parameters. For American pricing an exact representation of the early exercise premium is derived for a class of one-factor models. This enables both American option prices and the corresponding two-dimensional critical exercise boundary to be computed for the extended Fong and Vasicek model