5 July 2018 CFA Institute Journal Review

# Pricing Coupon Bond Options and Swaptions under the Two-Factor Hull–White Model (Digest Summary)

1. Servaas Houben, CFA
Interest rate models are used to evaluate such interest rate–sensitive market instruments as bonds and swaptions. These models depend both on the model approach and the value of the model parameters. The authors suggest a new method for pricing bonds and options that can also be used for calibrating parameters in the Hull–White model.

## How Is This Research Useful to Practitioners?

In general, interest rate modeling can be applied using either a one- or two-factor model, whereby a two-factor model also takes into account the correlation between different rates.

In the past, Hull and White (Review of Financial Studies 1994) created a two-factor model that allows a closed-form solution for zero-coupon bonds. For European bonds and swaptions, a scenario analysis approach is still required. An analytical formula for pricing European swaptions was created by Brigo and Mercurio (Springer 2006). One main drawback of their method is that the integral to be solved does not have clear boundaries. Schrager and Pelsser (Mathematical Finance 2006) created a more efficient method in which the error term increases with tenor and maturity.

Overall, the challenge remains the choice of model parameters, such that the outcome of the model is in line with market pricing (market consistency).

The solution the authors suggest requires a numerical integration like that of Brigo and Mercurio but with clear boundaries. The suggested approach also has a good tradeoff between computational effort and results quality to deliver a better fit.

Pricing of bonds and swaptions is an area of interest for investment professionals working in risk management and fixed income. In this short but concise article, the authors give a good overview of the different methods available and the strengths and weaknesses for each.

## How Did the Authors Conduct This Research?

The authors first create their model setup and then use market prices of swaptions to estimate the model parameters. They derive functions for pricing zero-coupon bonds and for pricing bond options and swaptions. They use the Martingale property, that the future value of a sequence of random variables (a stochastic process) is equal to the current value, and Itô’s lemma, which relates the change in the function of a random variable to the change in that random variable itself, to derive a formula for the volatility of the coupon bond.

Because the swaption and bond option payoff are similar, they can be priced in a similar way, and only the bond’s cash flows after the option’s maturity have to be taken into account. The authors assume that the coupon price of the bond is lognormal and that interest rates behave in line with the Hull–White process. Using the assumption of no arbitrage, they obtain prices for a coupon bond call and put option and for a payer and receiver swaption.

Thereafter, the authors use at-the-money swaptions to derive market consistent volatility parameters such that the square error between market and model swaption prices is minimized. They then use last-business-day prices in the period 2011–2014 for 10-year swaptions to calibrate the model parameters. They compare their results with those of the Schrager and Pelsser model. The authors derive results similar to those of Brigo and Mercurio and conclude that their new model provides a good tradeoff between computational effort and results quality.

## Abstractor’s Viewpoint

The authors provide a very concise analysis of interest rate modeling by referring to previous research, mentioning both strengths and weaknesses, which allows the reader to get an idea of the tradeoffs to consider when modeling interest rates. Next, they derive formulas for bond option and swaption pricing. Last, they use market data to assess their proposed model assumptions while comparing them with previous work.

One potential drawback of the parameter calibration is the seemingly limited data the authors use (i.e., only the last business day of 2011–2014). Furthermore, only 10-year swaptions have been taken into account, which makes it unclear whether the method proposed also works with swaptions of different maturities and tenor values. Nevertheless, the tradeoff between computational requirements and accuracy of results seems well balanced in the new model, making it an interesting alternative for practitioners.

Although the research might scare some off by the application of formulas, readers should be able to grasp the underlying ideas and challenges that interest rate modeling entails.

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