Exact solutions via invariant approach for Black‐Scholes model with time‐dependent parameters

We analyze the Black‐Scholes model with time‐dependent parameters, and it is governed by a parabolic partial differential equation (PDE). First, we compute the Lie symmetries of the Black‐Scholes model with time‐dependent parameters. It admits 6 plus infinite many Lie symmetries, and thus, it can be reduced to the classical heat equation. We use the invariant criteria for a scalar linear (1+1) parabolic PDE and obtain 2 sets of equivalence transformations. With the aid of these equivalence transformations, the Black‐Scholes model with time‐dependent parameters transforms to the classical heat equation. Moreover, the functional forms of the time‐dependent parameters in the PDE are determined via this method. Then we use the equivalence transformations and known solutions of the heat equation to establish a number of exact solutions for the Black‐Scholes model with time‐dependent parameters.     

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.     

Stock options as barrier contingent claims

A comprehensive model is suggested that values securities as options and consequently ordinary stock options as compound options. Extending the basic Black–Scholes model, it can incorporate common contractual features and stylized facts. More specifically, a closed form solution is derived for the price of a call option on a down‐and‐out call. It is then shown how the result obtained can be generalized in order to price options on complex corporate securities, allowing among other things for corporate taxation, costly financial distress and deviations from the absolute priority rule. The characteristics of the model are illustrated with numerical examples.     

A comprehensive mathematical approach to exotic option pricing

This work illustrates how several new pricing expressions for exotic options can be derived within a Lévy framework by employing a unique pricing expression. To the purpose, a unifying formula is obtained by solving some nested Cauchy problem for pseudodifferential equations generalizing Black–Scholes PDE. The main result extends (Agliardi R. The quintessential option pricing formula under Lévy processes. Applied Mathematics Letters 2009; 22:1626‐1631) and is a powerful tool for generating new valuation expressions. Several examples of pricing formulas under the Lévy processes are provided to illustrate the flexibility of the method. Some of them are new in the financial literature. Finally, many existing pricing formulas of the traditional Gaussian model are easily obtained as a by‐product. Copyright © 2012 John Wiley & Sons, Ltd.     

企业经营者股票期权薪酬机制的模型探讨

采用 Black-Scholes期权定价理论 ,建立了激励机制下企业经营者股票期权薪酬机制的分析、操作模型     

Are we using the wrong letters? An analysis of executive stock option Greeks

Greek letters, in particular delta and vega based on the Black–Scholes model (BS), have been widely used to estimate the sensitivity of CEO wealth to changes in stock price (delta) and stock return volatility (vega) and to evaluate the executive stock options (ESOs) granted on the basis of performance and risk. However, the BS model does not take into account the main features of ESOs and therefore the delta and vega values it produces are not valid. The Cvitanic–Wiener–Zapatero model (CWZ) is an alternative model to Black–Scholes for valuing ESOs. It has a closed formula and considers the main features of ESOs. We carry out a sensitivity analysis to show that research on option-based compensation and its risk-taking effects is not robust in ESO pricing models. The sensitivity analysis consists of comparing the impact of the common parameters of the BS and CWZ models, as well as the effect of the specific parameters of the CWZ model, on the sensitivity of CEO wealth to stock price and stock volatility. Additionally, using panel data methodology, we develop an empirical analysis to illustrate the influence of stock return volatility and different corporate policies on both CEO wealth sensitivities.     

The Minimal Entropy Martingale Measure and Numerical Option Pricing for the Barndorff–Nielsen–Shephard Stochastic Volatility Model

Abstract

We develop and apply a numerical scheme for pricing options in the stochastic volatility model proposed by Barndorff–Nielsen and Shephard. This non-Gaussian Ornstein–Uhlenbeck type of volatility model gives rise to an incomplete market, and we consider the option prices under the minimal entropy martingale measure. To numerically price options with respect to this risk neutral measure, one needs to consider a Black and Scholes type of partial differential equation, with an integro-term arising from the volatility process. We suggest finite difference schemes to solve this parabolic integro-partial differential equation, and derive appropriate boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black and Scholes formula for call options, and the implications of the stochastic volatility on the shape of the volatility smile.     

Parabolic variational inequalities: The Lagrange multiplier approach

Parabolic variational inequalities are discussed and existence and uniqueness of strong as well as weak solutions are established. Our approach is based on a Lagrange multiplier treatment. Existence is obtained as the unique asymptotic limit of solutions to a family of appropriately regularized nonlinear parabolic equations. Two regularization techniques are presented resulting in feasible and unfeasible approximations respectively. Monotonicity results of the regularized solutions and convergence rate estimate are established. The results are applied to the Black–Scholes model for American options. The case of the bilateral constraints is also treated. Numerical results for the Black–Scholes model are presented and prove the practical efficiency of our results.     

Approximate indifference pricing in exponential Lévy models

Financial markets based on Lévy processes are typically incomplete and option prices depend on risk attitudes of individual agents. In this context, the notion of utility indifference price has gained popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to apply in practice, due to the high computational cost of solving the non-linear partial integro-differential equation associated to the indifference price. In this work, we develop closed-form approximations to exponential utility indifference prices in exponential Lévy models. To this end, we first establish a new non-asymptotic approximation of the indifference price which extends earlier results on small risk aversion asymptotics of this quantity. Next, we use this formula to derive a closed-form approximation of the indifference price by treating the Lévy model as a perturbation of the Black–Scholes model. This extends the methodology introduced in a recent paper for smooth linear functionals of Lévy processes (?erný et al. 2013) to non-linear and non-smooth functionals. Our formula represents the indifference price as the linear combination of the Black–Scholes price and correction terms which depend on the variance, skewness and kurtosis of the underlying Lévy process, and the derivatives of the Black–Scholes price. As a by-product, we obtain a simple approximation for the spread between the buyer’s and the seller’s indifference price. This formula allows to quantify, in a model-independent fashion, how sensitive a given product is to jump risk when jump size is small.     

Evaluation of insurance products with guarantee in incomplete markets

Life insurance products are usually equipped with minimum guarantee and bonus provision options. The pricing of such claims is of vital importance for the insurance industry. Risk management, strategic asset allocation, and product design depend on the correct evaluation of the written options. Also regulators are interested in such issues since they have to be aware of the possible scenarios that the overall industry will face. Pricing techniques based on the Black & Scholes paradigm are often used, however, the hypotheses underneath this model are rarely met.To overcome Black & Scholes limitations, we develop a stochastic programming model to determine the fair price of the minimum guarantee and bonus provision options. We show that such a model covers the most relevant sources of incompleteness accounted in the financial and insurance literature. We provide extensive empirical analyses to highlight the effect of incompleteness on the fair value of the option, and show how the whole framework can be used as a valuable normative tool for insurance companies and regulators.     

A superconvergent fitted finite volume method for Black–Scholes equations governing European and American option valuation

We develop a superconvergent fitted finite volume method for a degenerate nonlinear penalized Black–Scholes equation arising in the valuation of European and American options, based on the fitting idea in Wang [IMA J Numer Anal 24 (2004), 699–720]. Unlike conventional finite volume methods in which the dual mesh points are naively chosen to be the midpoints of the subintervals of the primal mesh, we construct the dual mesh judiciously using an error representation for the flux interpolation so that both the approximate flux and solution have the second‐order accuracy at the mesh points without any increase in computational costs. As the equation is degenerate, we also show that it is essential to refine the meshes locally near the degenerate point in order to maintain the second‐order accuracy. Numerical results for both European and American options with constant and nonconstant coefficients will be presented to demonstrate the superconvergence of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1190–1208, 2015     

Pricing a European Basket Option in the Presence of Proportional Transaction Costs

A crucial assumption in the Black–Scholes theory of options pricing is the no transaction costs assumption. However, following such a strategy in the presence of transaction costs would lead to immediate ruin. This paper presents a stochastic control approach to the pricing and hedging of a European basket option, dependent on primitive assets whose prices are modelled as lognormal diffusions, in the presence of costs proportional to the size of the transaction. Under certain assumptions on the individual preferences, it is able to reduce the dimensionality of the resulting control problem. This facilitates considerably the study of the value function and the characterisation of the optimal trading policy. For solution of the problem a perturbation analysis scheme is utilized to derive a non‐trivial, asymptotically optimal result. The findings reveal that this result can be expressed by means of a small correction to the corresponding solution of the frictionless Black–Scholes type problem, resembling a multi‐dimensional ‘bandwidth’ around the vanilla case, which, moreover, is readily tractable.     

Fractional model and solution for the Black‐Scholes equation

This work presents a new model of the fractional Black‐Scholes equation by using the right fractional derivatives to model the terminal value problem. Through nondimensionalization and variable replacements, we convert the terminal value problem into an initial value problem for a fractional convection diffusion equation. Then the problem is solved by using the Fourier‐Laplace transform. The fundamental solutions of the derived initial value problem are given and simulated and display a slow anomalous diffusion in the fractional case.     

A BLACK－SCHOLES FORMULA FOR OPTIONPRICINGWITHDIVIDENDS

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Hedging lookback and partial lookback options using Malliavin calculus

The paper considers a Black and Scholes economy with constant coefficients. A contingent claim is said to be simple if the payoff at maturity is a function of the value of the underlying security at maturity. To replicate a simple contingent claim one uses so called delta-hedging, and the well-known strategy is derived from Itô calculus and the theory of partial differentiable equations. However, hedging path-dependent options require other tools since the price processes, in general, no longer have smooth stochastic differentials. It is shown how Malliavin calculus can be used to derive the hedging strategy for any kind of path-dependent options, and in particular for lookback and partial lookback options.     

Existence of a fundamental solution of partial differential equations associated to Asian options

We prove the existence and uniqueness of the fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in the Black & Scholes setting for the pricing problem relevant to path dependent options. We improve previous results in that we provide a closed form expression for the solution of the Cauchy problem under weak regularity assumptions on the coefficients of the differential operator. Our method is based on a limiting procedure, whose convergence relies on some barrier arguments and uniform a priori estimates recently discovered.     

Bounds for the price of a European-style Asian option in a binary tree model

Inspired by the ideas of Rogers and Shi [J. Appl. Prob. 32 (1995) 1077], Chalasani et al. [J. Comput. Finance 1(4) (1998) 11] derived accurate lower and upper bounds for the price of a European-style Asian option with continuous averaging over the full lifetime of the option, using a discrete-time binary tree model. In this paper, we consider arithmetic Asian options with discrete sampling and we generalize their method to the case of forward starting Asian options. In this case with daily time steps, the method of Chalasani et al. is still very accurate but the computation can take a very long time on a PC when the number of steps in the binomial tree is high. We derive analytical lower and upper bounds based on the approach of Kaas et al. [Insurance: Math. Econ. 27 (2000) 151] for bounds for stop-loss premiums of sums of dependent random variables, and by conditioning on the value of underlying asset at the exercise date. The comonotonic upper bound corresponds to an optimal superhedging strategy. By putting in less information than Chalasani et al. the bounds lose some accuracy but are still very good and they are easily computable and moreover the computation on a PC is fast. We illustrate our results by different numerical experiments and compare with bounds for the Black and Scholes model [J. Pol. Econ. 7 (1973) 637] found in another paper [Bounds for the price of discretely sampled arithmetic Asian options, Working paper, Ghent University, 2002]. We notice that the intervals of Chalasani et al. do not always lie within the Black and Scholes intervals. We have proved that our bounds converge to the corresponding bounds in the Black and Scholes model. Our numerical illustrations also show that the hedging error is small if the Asian option is in the money. If the option is out of the money, the price of the superhedging strategy is not as adequate, but still lower than the straightforward hedge of buying one European option with the same exercise price.     

Options to expand and to contract in combination

This work presents a real options approach to the valuation of multiple investment projects, focusing on the case of option to expand and/or to contract. Proper valuation formulas are obtained by solving Black–Scholes PDE and the impact of strategic interaction among multiple options is studied.     

Hedging Large Portfolios of Options in Discrete Time*

The problem studied is that of hedging a portfolio of options in discrete time where underlying security prices are driven by a combination of idiosyncratic and systematic risk factors. It is shown that despite the market incompleteness introduced by the discrete time assumption, large portfolios of options have a unique price and can be hedged without risk. The nature of the hedge portfolio in the limit of large portfolio size is substantially different from its continuous time counterpart. Instead of linearly hedging the total risk of each option separately, the correct portfolio hedge in discrete time eliminates linear as well as second and higher order exposures to the systematic risk factors only. The idiosyncratic risks need not be hedged, but disappear through diversification. Hedging portfolios of options in discrete time thus entails a trade‐off between dynamic and cross‐sectional hedging errors. Some computations are provided on the outcome of this trade‐off in a discrete‐time Black–Scholes world.     

A possible way of estimating options with stable distributed underlying asset prices

Option pricing theory is considered when the underlying asset price satisfies a stochastic differential equation which is driven by random motions generated by stable distributions. The properties of the stable distributions are discussed and their connection with the theory of fractional Brownian motion is noted. This approach attempts to generalize the classical Black–Scholes formulation, to allow for the presence of fat tails in the distribution of log prices which leads to a diffusion equation involving fractional Brownian motion. The resulting option pricing via a hedging strategy approach is independently derived by constructing a backward Kolmogorov equation for a simple trinomial model where the probabilities are assumed to satisfy a particular fractional Taylor series due to Dzherbashyan and Nersesyan. To effect this development, some knowledge of fractional integration and differentiation is required so this is briefly reviewed. Consideration is also given to a different hedging strategy approach leading to a fractional Black–Scholes equation involving the market price of risk. Modification to the model is also considered such as the impact of transaction costs. A simple example of American options is also considered.