Peter Buchen

The Maximum Entropy Distribution of an Asset Inferred from Option Prices

This paper describes the application of the Principle of Maximum Entropy to the estimation of the distribution of an underlying asset from a set of option prices. The resulting distribution is least committal with respect to unknown or missing information and is, hence, the least prejudiced. The maximum entropy distribution is the only information about the asset that can be inferred from the price data alone. An extension to the Principle of Minimum Cross-Entropy allows the inclusion of prior knowledge of the asset distribution. We show that the maximum entropy distribution is able to accurately fit a known density, given simulated option prices at different strikes.

The pricing of dual-expiry exotics

Abstract This paper develops a new technique for pricing a class of exotic options that are characterized by two expiry dates. Examples of such exotics include compound options, chooser options, extendable options, shout options, partial barrier options and others. The method, based on the partial differential equation approach to option pricing, however requires no formal solution of such equations. Instead, the method exploits the observation that dual expiry options have payoffs that can be perfectly replicated by a particular set of first and second order binary options. Hence, in order to avoid arbitrage, the exotic option prices are obtained by static replication with respect to this family of binaries. The representation of prices in terms of these binaries is also quite general and does not depend on any particular underlying asset price dynamics. Closed form expressions agreeing with published results are given for the case of log-normal asset price dynamics and standard Black–Scholes assumptions.

A New Approach to Pricing Double-Barrier Options with Arbitrary Payoffs and Exponential Boundaries

Abstract We consider in this article the arbitrage free pricing of double knock-out barrier options with payoffs that are arbitrary functions of the underlying asset, where we allow exponentially time-varying barrier levels in an otherwise standard Black–Scholes model. Our approach, reminiscent of the method of images of electromagnetics, considerably simplifies the derivation of analytical formulae for this class of exotics by reducing the pricing of any double-barrier problem to that of pricing a related European option. We illustrate the method by reproducing the well-known formulae of Kunitomo and Ikeda (1992) for the standard knock-out double-barrier call and put options. We give an explanation for the rapid rate of convergence of the doubly infinite sums for affine payoffs in the stock price, as encountered in the pricing of double-barrier call and put options first observed by Kunitomo and Ikeda (1992).

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.

Perturbation formulas for linear dispersive waves in weak spatially non-uniform media

Abstract First-order perturbation formulas for frequency, wave number, group velocity, amplitude and phase are obtained for linear dispersive waves propagating in weak spatially (but otherwise general) non-uniform media. The formulas are derived from Whithams theory under the assumptions of slow wave modulations and weak (i.e. small scale and smooth) spatial nonuniformity. The analysis presented here indicates that the perturbations at any location of the medium depend on both the local heterogeneity and its spatial average over the propagation path. We illustrate the theory with the well-known example of flexural waves along a non-uniform beam to demonstrate that our theory is identical to a joint WKBJ-stationary phase representation of the solution.

CLA’s, PLA’s and a new method for pricing general passport options

This paper is primarily concerned with pricing a general passport option (GPO) within the standard Black–Scholes framework. We show that in all possible cases of the allowed trading strategy, the price can be decomposed into simple portfolios of standard European calls and puts and a contract we call a ‘PLA’ or a put on the log-asset price. For completeness, we also introduce the call on the log-asset price (or CLA) and explore their properties and applications. The decomposition of the GPO into its constituent parts is achieved with the help of the Method of Images to convert certain barrier option payoffs into equivalent European payoffs. This technique considerably simplifies the calculation and adds significant transparency to what is otherwise regarded as very complex problem. Curiously, a spin-off of the method to price the GPO suggests an alternative and simpler way to price lookback options.

A VALUATION FORMULA FOR MULTI-ASSET, MULTI-PERIOD BINARIES IN A BLACK–SCHOLES ECONOMY

We present a new valuation formula for a generic, multi-period binary option in a multi-asset Black�Scholes economy. The payoff of this so-called M-binary is the most general possible, subject to the condition that a simple analytic expression exists for the present value. Portfolios of M-binaries can be used to statically replicate many European exotics for which there exist closed-form Black�Scholes prices.

Foundations of Option Pricing

In this chapter we lay down the foundations of option and derivative security pricing in the classical Black-Scholes (BS) paradigm. The two key assumptions in this approach are the absence of arbitrage and the modelling of asset prices by geometrical Brownian motion (gBm). No arbitrage is the driving mechanism for much of modern finance and still plays a dominant role in models extended beyond the BS framework. It is generally recognised that gBm, which implies Gaussian logreturns of asset prices, while good for analysis is rather a poor description of reality. Financial data very often show stylised features in their log-return distributions that are highly non-Gaussian. These include, heavy-tails (leptokurtosis), skewness, stochastic volatility and long-range dependence. Unfortunately, no model of asset prices capturing all, or even some of these features, is widely accepted by either theorists or practitioners.