H. Windcliff

Analysis of the stability of the linear boundary condition for the Black–Scholes equation

The linear asymptotic boundary condition, i.e. assuming that the second derivative of the value of the derivative security vanishes as the asset price becomes large, is commonly used in practice. To our knowledge, there have been no rigorous studies of the stability of these methods, despite the fact that the discrete matrix equations obtained using this boundary condition loses diagonal dominance for large timesteps. In this paper, we demonstrate that the discrete equations obtained using this boundary condition satisfy necessary conditions for stability for a finite difference discretization. Computational experiments also show that this boundary condition satisfies sufficient conditions for stability as well.

The 1/n Pension Investment Puzzle

Abstract This paper examines the so-called 1/n investment puzzle that has been observed in defined contribution plans whereby some participants divide their contributions equally among the available asset classes. It has been argued that this is a very naive strategy since it contradicts the fundamental tenets of modern portfolio theory. We use simple arguments to show that this behavior is perhaps less naive than it at first appears. It is well known that the optimal portfolio weights in a mean-variance setting are extremely sensitive to estimation errors, especially those in the expected returns. We show that when we account for estimation error, the 1/n rule has some advantages in terms of robustness; we demonstrate this with numerical experiments. This rule can provide a risk-averse investor with protection against very bad outcomes.

Shout options: a framework for pricing contracts which can be modified by the investor

A shout option may be broadly defined as a financial contract which can be modified by the holder according to specified rules. In a simple example, the holder could have the right to set the strike of an option equal to the current value of the underlying asset. In such a case, the holder effectively has the right to select when to take ownership of an at-the-money option. More generally, the holder could have multiple rights along these lines, in some cases with a limit placed on the number of rights which may be exercised within a given time period (e.g., four times per year). The value of these types of contracts can be estimated by solving a system of interdependent linear complementarity problems. This paper describes a general framework for the valuation of complex types of shout options. Numerical issues related to interpolation and choice of timestepping method are considered in detail. Some illustrative examples are provided.

Valuation of segregated funds: shout options with maturity extensions

Abstract One of the most popular investments available in the Canadian market today is a mutual fund with the added feature of a long term maturity guarantee. These types of investments are known as segregated funds. They often have very complex option features. For example, these contracts typically contain multiple embedded shout options which permit the holder to reset the guarantee level and the maturity date for which it applies many times during the life of the contract. These funds also provide mortality benefits if the investor dies prior to the maturity date. This paper explores the valuation of segregated funds using an approach based on the numerical solution of a set of linear complementarity problems. Our results indicate that the option components of some of these contracts seem to be underpriced, especially for riskier funds with relatively high volatilities. This assumes that investors exercise their options optimally. Non-optimal behaviour by investors of course reduces the values of the embedded options, and we provide some illustrative results along these lines. We also show that alternative contract specifications which generate similar present values may require substantially different proportionate fees, since the expected durations of the contracts can be quite different.

Numerical Methods and Volatility Models for Valuing Cliquet Options

Several numerical issues for valuing cliquet options using PDE methods are investigated. The use of a running sum of returns formulation is compared to an average return formulation. Methods for grid construction, interpolation of jump conditions, and application of boundary conditions are compared. The effect of various volatility modelling assumptions on the value of cliquet options is also studied. Numerical results are reported for jump diffusion models, calibrated volatility surface models, and uncertain volatility models.

An object-oriented framework for valuing shout options on high-performance computer architectures

Abstract A shout option is a financial contract which allows the holder to change the payoff during the lifetime of the contract. Complex versions of these options are embedded in financial products which offer various types of maturity guarantees such as segregated funds marketed by Canadian insurance companies. The value of these options can be determined by solving a collection of coupled partial differential equations. In this work we develop an object-oriented framework for valuing these contracts which is capable of exploiting modern high-performance computer architectures. We use this framework to study practical aspects of valuing and hedging these contracts.

Understanding the Behavior and Hedging of Segregated Funds Offering the Reset Feature

Abstract Segregated funds have become an extremely popular Canadian investment vehicle. These instruments provide long-term maturity guarantees and often include complex option features. One controversial aspect is the reset feature, which provides the ability to lock in market gains. Recently, regulators have announced that firms offering these products will be subject to new capital requirements. This paper discusses the effects of volatility, interest rates, investor optimality, and product design on the cost of providing a segregated fund guarantee. For each scenario, the authors provide the appropriate management expense ratio (MER) that should be charged and demonstrate the current liability using a given fixed MER. The paper also investigates intuitive reasons that cause the reset feature to require such a dramatic increase in the hedging costs. Finally, an approximate method for handling the reset feature is presented that can be computed very efficiently, provided the correct proportional fee is charged.

Simulations for Hedging Financial Contracts with Optimal Decisions

Simulations are powerful techniques for quantifying risk exposures. This paper presents a methodology for simulating the performance of hedging strategies for financial contracts with embedded optimization features. As a case study, we provide simulations of mutual fund guarantees offering a reset provision. In Canada, these types of contracts are known as segregated funds. The optimization component of these contracts is that the holder can choose when to lock in market gains, typically up to two or four times per calendar year. Recently, Canadian regulators have imposed new capital requirements for firms selling these contracts. However, these requirements can be reduced if hedging strategies are put in place. The techniques presented here would allow companies to evaluate their proposed hedging strategies and to quantify their remaining risk exposures. We study the effect of non-optimal investor behaviour on the hedging of these contracts. In particular, we present results for the heuristic use of the reset feature; for example, locking in whenever the underlying asset value has risen by 15% as recently suggested by a Canadian Institute of Actuaries task force on segregated funds.