BY ALEC HOLLIS
In the financial arena, volatility is virtually unavoidable. But there are ways to circumvent the risk and uncertainty. Interest rate volatility modeling is one of these workarounds. Read on to learn why such modeling techniques are vital to sound interest rate risk management at your credit union.
Conventional financial wisdom equates risk to volatility.Understanding volatility is to understand that we invest in markets characterized by uncertainty.Volatility in interest rates has a major impact on the pricing of financial instruments with embedded options, such as mortgages, callable bonds and interest rate derivatives. Therefore, modeling interest rate volatility is vital to sound interest rate risk management.
Modeling interest rate volatility is essential because options are widely present in a variety of financial instruments on a typical balance sheet.The presence of options means cash flows are contingent on the path of interest rates.Simple discounting methodologies cannot capture the effect of path dependencies, thus making it necessary to use interest rate volatility modeling techniques.Lattice-based models and the Monte Carlo stochastic approach are the most commonly used methods to model interest rate volatility.The nature of the instrument being valued determines which method is more appropriate.
In general, lattice-based valuation, most commonly binomial option pricing, is sufficient for weakly path-dependent instruments, but the Monte Carlo approach is necessary for strongly path-dependent instruments.“Weakly path-dependent instruments have cash flows that depend only upon prevailing interest rates,” whereas with “strongly path-dependent instruments… cash flows are reliant upon the path rates have traveled in the past as well as present interest rate levels.”Weakly path-dependent bonds include callable bonds, interest rate caps and floors, range floaters and swaptions, whereas strongly path-dependent instruments include mortgage loans, mortgage-backed securities, collateralized mortgage obligations and index-amortizing notes.
While a Monte Carlo simulation theoretically could be used to value weakly path-dependent instruments, lattice-based valuations provide satisfactory valuation results with fewer computational requirements. Lattice-based techniques use interest rate trees, which make assumptions about the future path of interest rates.Nodes are constructed, with each node representing a discrete point in time in which interest rates can follow one of two movements: up or down.The magnitude of each movement is determined by the interest rate volatility assumption in the model.The time interval of the nodes is called the “tree step.”Once the tree is created, the option payoffs are calculated at each node and the present value of these expected cash flows is calculated to derive the value of the option.The value of the option becomes larger as the volatility assumption is increased due to the larger payoffs created from the more extreme movements in interest rates.Remember, option values can never be negative, so greater volatility increases the upside payoffs but the downside is floored at zero.
Exhibit 1 shows an example of a two-period interest rate tree with a yearly step, producing a total of six nodes.In practice, interest rate trees are quite a bit larger because steps are typically in days – think tens of thousands of nodes!Step sizes typically range from daily to monthly, but they should not be greater than monthly.
Exhibit 1: Two-Period Interest Rate Tree with Yearly Steps
Now that we have explored lattice-based option pricing, let’s review the techniques of the Monte Carlo approach.The Monte Carlo approach is an application of stochastic modeling.Stochastic, by definition, means random and represents the converse of a deterministic process.A deterministic process includes no parameters for variability (volatility) or randomness, whereas a stochastic process allows for such randomness according to the parameters of the model.The major implication of a stochastic versus deterministic model is that a deterministic model cannot properly model options.Interest rate volatility is a required input for option pricing.Stochastic modeling is rapidly becoming the norm and is frequently used in financial modeling, considering financial markets exhibit random variability.
To model interest rate volatility, a Monte Carlo simulation creates many specific interest rate paths, with each path producing its own unique set of cash flows.The value of an instrument using the Monte Carlo approach is the average of the present values calculated over the many paths.The Monte Carlo approach is an iterative process that repeatedly generates interest rate paths using the inputs employed in the model.These inputs determine the number and dispersion of the paths.The basic inputs for a Monte Carlo simulation include the interest rate model, number of paths and, in some cases, a mean reversion speed.Also required is an assumption for interest rate volatility.
The interest rate model defines the behavior of interest rates and how they evolve over time.The most basic of such is a one-factor interest rate model, also known as a short-rate model, so named because future interest rates are determined by the variability of one rate: the short rate.The short rate is a theoretical, meaning not directly observable, monthly rate, and it is derived from the current-term structure of interest rates.Once the short rate is derived, the variability of all interest rates is driven by the variability of the short rate.Short-rate models thus suffer from the drawback that there is only one factor driving interest rate variability.Multi-factor interest rate models are more advanced and allow for multiple sources of interest rate variability from which future rates evolve. Exhibit2 illustrates a 100-path Monte Carlo simulation of one-month LIBOR over 60 months, using ZM Financial Systems’ proprietary short-rate model and a mean reversion speed of 0.12.In this case, the deterministic path is determined by implied forward rates and is emboldened.Remember, if a stochastic process is not used, then the evolution of interest rates has no variability within the model and is assumed to follow a single path with certainty.
Exhibit2: 60-Month Monte Carlo Simulation of One-Month LIBOR, 100 Paths
Interest rate volatility is a multi-dimensional, complex topic.There are many ways to model it,with varying degrees of complexity and accuracy.Given the current state of technology, modeling interest rate volatility is no longer a luxury; it is a requirement.Proper risk modeling allows for better decision-making capabilities, and deterministic models are rapidly becoming outdated as they give way to better techniques for modeling risk.
The author would like to thank Robert Perry, Managing Director, Investment Management Group, and David Montgomery, Managing Director, Analytics and Development, at ALM First Financial Advisors for their contribution and directional insight.
Alec Hollis joined ALM First Financial Advisors in 2012. Mr. Hollis performs asset liability analyses for financial institutions, various ‘what-if’ analyses, budget forecasting, liquidity forecasting and any other modeling requirement to fit the needs of ALM’s clients. As an Associate, Mr. Hollis’s additional responsibilities include the presentation of results to client ALCOs and senior management as well as mentoring new financial analyst team members.
Mr. Hollis holds a bachelor’s degree in finance from the University of Notre Dame in South Bend, Indiana.
ZM Financial Systems, Inc. (2013). Formulas and Methodologies.