[Paper Review] The Pricing of Multiple-Expiry Exotics

O Hyong-Chol, Mun-Chol Kim|arXiv (Cornell University)|Feb 14, 2013

Stochastic processes and financial applications6 references3 citations

TL;DR

This paper introduces higher-order binary options as a unified framework for pricing multiple-expiry exotic derivatives, extending Buchen's dual-expiry method to options with more than three expiry dates. By expressing complex payoffs as linear combinations of n-th order binary options and using static replication, the authors derive closed-form pricing formulas via PDE solutions, enabling efficient valuation of Bermudan, extendable, and multiple-shout options under constant volatility and interest rates.

ABSTRACT

In this paper we extend Buchen's method to develop a new technique for pricing of some exotic options with several expiry dates(more than 3 expiry dates) using a concept of higher order binary option. At first we introduce the concept of higher order binary option and then provide the pricing formulae of $n$-th order binaries using PDE method. After that, we apply them to pricing of some multiple-expiry exotic options such as Bermudan option, multi time extendable option, multi shout option and etc. Here, when calculating the price of concrete multiple-expiry exotic options, we do not try to get the formal solution to corresponding initial-boundary problem of the Black-Scholes equation, but explain how to express the expiry payoffs of the exotic options as a combination of the payoffs of some class of higher order binary options. Once the expiry payoffs are expressed as a linear combination of the payoffs of some class of higher order binary options, in order to avoid arbitrage, the exotic option prices are obtained by static replication with respect to this family of higher order binaries.

Motivation & Objective

• To develop a unified pricing framework for multiple-expiry exotic options with more than three expiry dates.
• To extend Buchen’s dual-expiry binary option method to higher-order binaries using PDE techniques.
• To enable static replication of complex exotic payoffs by expressing them as linear combinations of higher-order binary payoffs.
• To provide closed-form pricing formulas for n-th order binary options using the Black-Scholes PDE.
• To demonstrate the method on concrete instruments like Bermudan, multi-time extendable, and multiple-shout options.

Proposed method

• Introduce n-th order binary options inductively, where the payoff of an n-th order binary is a (n-1)-th order binary.
• Derive pricing formulas for n-th order binaries using the Black-Scholes PDE and the Feynman-Kac representation.
• Express the terminal payoff of multiple-expiry exotics as a piecewise linear combination of higher-order binary payoffs.
• Apply static replication: since the payoff is replicated by a portfolio of higher-order binaries, the option price equals the present value of the replicating portfolio.
• Use known pricing formulas for first- and second-order binaries (e.g., $ A_K^+ $, $ Q_K^{+} $, $ Q_{KK}^{-+} $) to build higher-order expressions recursively.
• Solve the initial-boundary value problem for the Black-Scholes PDE by transforming it into a constant-coefficient PDE via $ y = \ln x $, then apply the expectation formula.

Experimental results

Research questions

• RQ1 How can Buchen’s dual-expiry binary method be generalized to options with more than three expiry dates?
• RQ2 What is the mathematical structure of n-th order binary options, and how can they be priced using PDEs?
• RQ3 Can complex multiple-expiry exotic payoffs be decomposed into linear combinations of higher-order binary payoffs?
• RQ4 What is the static replication strategy for multiple-expiry exotics using higher-order binaries?
• RQ5 How do pricing formulas for multiple-expiry exotics emerge from the combination of higher-order binary prices?

Key findings

• The price of a fixed-time twice shout call option at time $ t < T_0 $ is given by a combination of $ Q_{KKK}^{-{-}+} $, $ Q_{KK}^{-+} $, $ A_{KK}^{-+} $, $ A_K^+ $, and $ B_K^+ $ terms with time-dependent discount factors.
• For $ x > K $, the price of the twice-shout option at $ T_0 $ is $ G(T_0,T_1,T_2) \cdot x - K \cdot e^{-r(T_2-T_0)} $, where $ G $ is a composite function of cumulative normal and bivariate normal distributions.
• The coefficient $ G(T_0,T_1,T_2) $ combines $ e^{-r(T_2-T_1)} + g(T_1,T_2) $, $ e^{-q(T_1-T_0)}N(d^+) $, $ e^{-r(T_2-T_0)}N(-d^-) $, and a bivariate normal term $ g_1 $.
• The method avoids solving the full initial-boundary value problem by directly expressing the payoff as a combination of higher-order binary payoffs, enabling static replication.
• The framework generalizes Buchen’s approach to multiple expiries and provides a systematic way to price complex exotics using recursive binary decomposition.
• The model assumes constant interest rate, dividend yield, and volatility, but the extension to time-dependent coefficients is straightforward.