Black-Scholes Framework Research Articles

The Use of Statistical Tests to Calibrate the Black-Scholes Asset Dynamics Model Applied to Pricing Options with Uncertain Volatility

A new method for calibrating the Black-Scholes asset price dynamics model is proposed. The data used to test the calibration problem included observations of asset prices over a finite set of (known) equispaced discrete time values. Statistical tests were used to estimate the statistical significance of the two parameters of the Black-Scholes model: the volatility and the drift. The effects of these estimates on the option pricing problem were investigated. In particular, the pricing of an option with uncertain volatility in the Black-Scholes framework was revisited, and a statistical significance was associated with the price intervals determined using the Black-Scholes-Barenblatt equations. Numerical experiments involving synthetic and real data were presented. The real data considered were the daily closing values of the S&P500 index and the associated European call and put option prices in the year 2005. The method proposed here for calibrating the Black-Scholes dynamics model could be extended to other science and engineering models that may be expressed in terms of stochastic dynamical systems.

A note on first-passage times of continuously time-changed Brownian motion

The probability of a Brownian motion with drift to remain between two constant barriers (for some period of time) is known explicitly. In mathematical finance, this and related results are required, for example, for the pricing of single-barrier and double-barrier options in a Black–Scholes framework. One popular possibility to generalize the Black–Scholes model is to introduce a stochastic time scale. This equips the modelled returns with desirable stylized facts such as volatility clusters and jumps. For continuous time transformations, independent of the Brownian motion, we show that analytical results for the double-barrier problem can be obtained via the Laplace transform of the time change. The result is a very efficient power series representation for the resulting exit probabilities. We discuss possible specifications of the time change based on integrated intensities of shot-noise type and of basic affine process type.

Matching Sensitivities to Discrete Dividends: A New Approach for Pricing Vanillas

The incorporation of a dividend yield in the classical Black-Scholes framework results in a minor modification of the well-known formula, since the lognormality of the underlying asset is preserved. However, market-makers prefer to work with cash dividends with fixed value instead of a dividend yield. Since there is no closed-form solution for the price of a European call in this case, many approximations have been proposed in the literature. Here, we present a new approach based on an exact analytical formula for the sensitivity to dividends of a European option. We use this result to elaborate a price proxy having the same Taylor expansion as the exact price with respect to the dividends. The resulting proxy has no computational complexity (the same as for the usual Black-Scholes formula), and numerical tests show the excellent accuracy of the method for all practical cases. Copyright © 2011 Wilmott Magazine Ltd.

AN EFFICIENT BINOMIAL TREE METHOD FOR CLIQUET OPTIONS

This work proposes a binomial method for pricing the cliquet options, which provide a guaranteed minimum annual return. The proposed binomial tree algorithm simplifies the standard binomial approach, which is problematic for cliquet options in the computational point of view, or other recent methods, which may be of intricate algorithm or require pre- or post-processing computations. Our method is simple, efficient and reliable in a Black-Scholes framework with constant interest rates and volatilities.

Valuation of Ratchet Equity-Indexed Annuities

Ratchet EIAs are the most popular equity-indexed annuities (EIAs) because returns are credited periodically with a guaranteed minimum and the account value never decreases once the return is credited. Pricing ratchet EIAs, however, is challenging. This paper derives pricing formulas that cover more contract features of ratchet EIAs than any in the literature. We obtain closed-form solutions in the Black-Scholes framework for both compound and simple versions of annual-reset ratchet products that may have a return cap and employ two types of geometric return averaging. Our numerical results demonstrate the impacts of individual contract features on contract value and the interactions among these features.

A Finite-Dimensional Approximation for Pricing Moving Average Options

We propose a method for pricing American options whose payoff depends on the moving average of the underlying asset price. The method uses a finite-dimensional approximation of the infinite-dimensional dynamics of the moving average process based on a truncated Laguerre series expansion. The resulting problem is a finite-dimensional optimal stopping problem, which we propose solving with a least squares Monte Carlo approach. We analyze the theoretical convergence rate of our method and present numerical results in the Black–Scholes framework.

The Valuation of Guaranteed Lifelong Withdrawal Benefit Options in Variable Annuity Contracts and the Impact of Mortality Risk

In light of the growing importance of the variable annuities market, in this paper we introduce a theoretical model for the pricing and valuation of guaranteed lifelong withdrawal benefit (GLWB) options embedded in variable annuity products. As the name suggests, this option offers a lifelong withdrawal guarantee; therefore, there is no limit on the total amount that is withdrawn over the term of the policy because if the account value becomes zero while the insured is still alive, he or she continues to receive the guaranteed amount annually until death. Any remaining account value at the time of death is paid to the beneficiary as a death benefit. We offer a specific framework to value the GLWB option in a market-consistent manner under the hypothesis of a static withdrawal strategy, according to which the withdrawal amount is always equal to the guaranteed amount. The valuation approach is based on the decomposition of the product into living and death benefits. The model makes use of the standard no-arbitrage models of mathematical finance, which extend the Black-Scholes framework to insurance contracts, assuming the fund follows a geometric Brownian motion and the insurance fee is paid, on an ongoing basis, as a proportion of the assets. We develop a sensitivity analysis, which shows how the value of the product varies with the key parameters, including the age of the policyholder at the inception of the contract, the guaranteed rate, the risk-free rate, and the fund volatility. We calculate the fair fee, using Monte Carlo simulations under different scenarios. We give special attention to the impact of mortality risk on the value of the option, using a flexible model of mortality dynamics, which allows for the possible perturbations by mortality shock of the standard mortality tables used by practitioners. Moreover, we evaluate the introduction of roll-up and step-up options and the effect of the decision to delay withdrawing. Empirical analyses are performed, and numerical results are provided.

On Pricing Equity-Linked Investment Products with a Threshold Expense Structure

This paper considers a certain expense structure where a vendor of equity-linked investment product will collect its expenses continuously from the investor's account whenever the investment performance exceeds a certain threshold level. Under the Black-Scholes framework, we derive compact convolution formulas for evaluating the total expenses to be collected during the investment period by using the joint Laplace transform of the Brownian motion and its excursion time. We provide numerical examples for illustration.

Pricing Multiple Triggers Contingent Claims

Plain vanilla options have a single underlying asset and a single condition on the payoff at the expiration date. For this class of options, a well-known result of Duffie, Pan and Singleton (2000) shows how to invert the characteristic function to obtain a closed-form formula for their prices. However, multiple-asset and multiple-condition derivatives such as rainbow options cannot be priced within this framework. Utilizing inversion of the Fourier transform – and resorting to neither the Black-Scholes framework nor the affine models settings – the authors provide an analytical solution for options whose payoffs depend on two or more conditions. Numerical experiments based on the multiple-asset and multiple-condition derivatives are provided to illustrate the usefulness of the proposed approach.

Pricing bounds for discrete arithmetic Asian options under Lévy models

Analytical bounds for Asian options are almost exclusively available in the Black–Scholes framework. In this paper we derive bounds for the price of a discretely monitored arithmetic Asian option when the underlying asset follows an arbitrary Lévy process. Explicit formulas are given for Kou’s model, Merton’s model, the normal inverse Gaussian model, the CGMY model and the variance gamma model. The results are compared with the comonotonic upper bound, existing numerical results, Monte carlo simulations and in the case of the variance gamma model with an existing lower bound. The method outlined here provides lower and upper bounds that are quick to evaluate, and more accurate than existing bounds.

Pricing maturity guarantee with dynamic withdrawal benefit

Motivated by the importance of withdrawal benefits for enhancing sales of variable annuities, we propose a new equity-linked product which provides a dynamic withdrawal benefit (DWB) during the contract period and a minimum guarantee at contract maturity. The term DWB is coined to reflect the duality between it and dynamic fund protection. Under the Black–Scholes framework and using results pertaining to reflected Brownian motion, we obtain explicit pricing formulas for the DWB payment stream and the maturity guarantee. These pricing formulas are also derived by means of Esscher transforms, which is another seminal contribution by Gerber to finance. In particular, we show that there are closed-form formulas for pricing European put and call options on a traded asset whose price can be modeled as the exponential of a reflected Brownian motion.

THE WIENER-HOPF TECHNIQUE AND DISCRETELY MONITORED PATH-DEPENDENT OPTION PRICING

Fusai, Abrahams, and Sgarra (2006) employed the Wiener-Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black-Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path-dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first-touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener-Hopf technique yields a novel formula for double-barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Pade approximation. A Gaussian Black-Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Levy processes generally.

A Model-Free Approach to Delta Hedging

We utilize the existence of trends for financial time series (Fliess M., Join C.: A Mathematical Proof of the Existence of Trends in Financial Time Series, http://ssrn.com/abstract=1459662) in order to propose a model-free setting for delta hedging. It avoids most of the shortcomings encountered with the now classic Black-Scholes framework. Several convincing computer simulations are presented. Some of them are dealing with abrupt changes, i.e., jumps.

Pricing cliquet options by tree methods

This paper focuses on the problem of pricing the cliquet options which provide a guaranteed minimum annual return. The tree method which we propose simplifies the standard binomial Cox–Ross–Rubinstein approach which, in this context, is problematic from a computational point of view. Our technique provides very efficient and reliable evaluations in a Black-Scholes framework with piecewise constant interest rates and volatilities.

PDE methods for maximum drawdown

Maximum drawdown is a risk measure that plays an important role in portfolio management. In this paper, we address the question of computing the expected value of the maximum drawdown using a partial differential equation approach. First, we derive a two-dimensional convection diffusion pricing equation for the maximum drawdown in the Black–Scholes framework. Due to the properties of the maximum drawdown, this equation has a non-standard boundary condition. We apply an alternating direction implicit method to solve the equation numerically. We also discuss stability and convergence of the numerical method.

Two Exotic Lookback Options

This paper formally analyses two exotic options with lookback features, referred to as extreme spread lookback options and look‐barrier options, first introduced by Bermin. The holder of such options receives partial protection from large price movements in the underlying, but at roughly the cost of a plain vanilla contract. This is achieved by increasing the leverage through either floating the strike price (for the case of extreme spread options) or introducing a partial barrier window (for the case of look‐barrier options). We show how to statically replicate the prices of these hybrid exotic derivatives with more elementary European binary options and their images, using new methods first introduced by Buchen and Konstandatos. These methods allow considerable simplification in the analysis, leading to closed‐form representations in the Black–Scholes framework.

Pricing American Call Options Under the Assumption of Stochastic Dividends - An Application of the Korn-Rogers-Model

In nancial mathematics stock prices are usually modelled directly as a result of supply and demand and under the assumption that dividends are paid continuously. In contrast economic theory gives us the dividend discount model assuming that the stock price equals the present value of its future dividends. These two models need not to contradict each other - in their paper Korn and Rogers (2005) introduce a general dividend model preserving the stock price to follow a stochastic process and to be equal to the sum of all its discounted dividends. In this paper we specify the model of Korn and Rogers in a Black-Scholes framework in order to derive a closed-form solution for the pricing of American Call options under the assumption of a known next dividend followed by several stochastic dividend payments during the option's time to maturity.

Commodity volatility modelling and option pricing with a potential function approach

We consider a novel approach to modelling of commodity prices and apply it to commodity option pricing and volatility estimation. This approach is particularly suited for prices with multiple attraction regions, such as crude oil and other energy and agricultural commodities. The price is modelled as a diffusion process governed by a potential function with minima at the attraction points. When applied to crude oil prices, the method captures characteristic behaviour of the prices remarkably well. Pricing of European options on spot and futures commodity contracts is developed within the potential model, and compared to the Black–Scholes framework. The approach provides a new way of estimating the volatility, which is particularly useful when option prices (and hence implied volatilities) are not readily available; this is often the case for commodity markets. European options on physical commodities and commodity futures are priced using the volatility forecasts obtained from the model. The performance of the model is evaluated on the basis of the hedging costs of an option. For options on crude oil, the method outperforms – in terms of hedging costs—the Black–Scholes approach with historical volatility.

Four Things You Might Not Know About the Black-Scholes Formula

1. Rolf Poulsen 1. An associate professor in the Department of Mathematical Sciences at the University of Copenhagen in Copenhagen, Denmark. (rolf{at}math.ku.dk) More than 40 years of playing around with the Black-Scholes equation has produced a large number of insights about its properties. Some are just cute little mathematical tidbits; others may have considerable value for practitioners, even if they amount to no more than useful rules-of-thumb or computational tricks; and some actually reflect deep mathematical properties of the Black-Scholes framework. In this article, Poulsen discusses several such neat little results about the option pricing model. TOPICS: [Options][1], [statistical methods][2], [fundamental equity analysis][3] [1]: https://www.pm-research.com/topic/options [2]: https://www.pm-research.com/topic/statistical-methods [3]: https://www.pm-research.com/topic/fundamental-equity-analysis

The obstacle problem for a class of hypoelliptic ultraparabolic equations

We study the obstacle problem for a class of degenerate parabolic operators with continuous coefficients. This problem arises in the Black–Scholes framework when considering path-dependent American options. We prove the existence of a unique strong solutionuto the Cauchy and Cauchy–Dirichlet problems, under rather general assumptions on the obstacle function. We also show thatuis a solution in the viscosity sense.