Callable bonds

In this blog post, we introduce the pricing of callable bonds using interest rate models. In our work across Derivatives Model Validation, Market Risk and IRRBB framework implementations, we frequently encounter material positions in callable bonds within client portfolios. The embedded optionality of these instruments adds significant complexity to their valuation and risk assessment compared with standard bonds. This overview offers practical guidance for modelling and managing that complexity.

We begin by explaining the principles behind pricing callable bonds, followed by an introduction to interest rate models, with a focus on the Hull-White one-factor model. We then demonstrate how to calibrate the model and use it to price a callable bond. Based on the resulting price and sensitivities, we highlight key characteristics of callable bonds. Finally, we discuss their treatment under IRRBB and FRTB regulatory frameworks.

For further reading on Market Risk, FRTB and IRRBB, you can refer to our blog posts on the latest reporting requirements for FRTB, our article about the IMA approach and methods to calculate Value-at-Risk and Expected Shortfall, our blogpost about IRRBB management, as well as our article on Non-Maturity Deposit modelling techniques.

A callable bond is a bond that can be reimbursed anticipatively by the issuer on a unilateral basis, for example when cheaper funding sources are available on the market. For pricing purposes, a callable bond can be considered as the combination of a plain bond and a call option (a cap) sold to the issuer (callable bond = plain bond + short option). One approach is to first calculate the option value using an analytical Black-76 model. This approach works well in cases where the bond is callable on a single date. But generally, there is a call schedule, for example yearly call dates starting 5 years after issuance. The bonds may also be callable continuously after the first call date. In both cases, the option’s expiry is of a non-European type (respectively Bermudian or American) and hence the options cannot be priced analytically. Although, the Black-76 model may be used as an approximation, more rigorous pricing requires the use of a numerical approach (trees, lattices, finite-difference methods or Monte Carlo simulations).

A bond’s price is dependent on the current yield curve, which has a risk-free and a credit or liquidity risk spread component (also known as z-spread). In this article, we follow the industry practice of modeling the bond call option using an interest rate model, assuming that the credit/liquidity spread is constant (non-stochastic). To price an option, we then need to define the probability distribution of the yield curve and the payoff of the option.

There are numerous models to model the interest rate movements, but the more common and straightforward approach is to model the instantaneous short rate. The forward rates and the yield curve are then implied from the drift (or trend), volatility and possibly mean reversion parameters assigned to the short rate. A consequence of this approach is that the different tenors of the yield curve are fully correlated to the short rate movement. This is acceptable for the pricing of bond options. Indeed, an analysis of yield curve movements using a Principal Component Analysis, shows that most of the movements across the term structure (~85%) can be explained by a single factor associated to changes in the level of the yield curve (i.e. parallel shifts)¹. Obviously, by modelling only the short rate, we cannot model changes in the yield curve shape (e.g., twists in the yield curve). For options where the payoff depends on the curvature or slope of the curve (e.g., CMS spread options), generally a 2-factor model is used.

In its most basic form, a short-rate model specifies the evolution of the short rate with an equation of the following form:

dr=b (r,t) dt+σ (r,t)dz 

Changes in the short rate (dr) depend on:

• a drift term (br,t ), which may follow a time structure and depend itself on the level of the short rate (r), introducing a mean-reversion effect;
• the volatility of the short rate (σ), which may be dependent on time and the level of the short rate.;
• a random “white noise” process, typically a Weiner process (normal distribution with a mean of zero and a standard deviation of one).

The initial suite of interest rate models – like Vasicek and CIR models were able to provide closed form solution for pricing zero-coup bonds (ZCB) and European-style options, but they lacked the ability to match the current yield curve. Such models where the term structure of interest rates is produced endogenously (“inside the model”) are called “equilibrium models”. In contrast, “no-arbitrage models” allow for the term structure of interest rates to be provided as an input as part of the drift term. Example of these are the Hull-White and the Black-Karasinski model.

In this article we assume that the callable bonds have an American exercise type and develop a trinomial tree or lattice approach to pricing. A trinomial tree is used to construct a discrete time approximation of the SDE for short rate.

In this approach, the entire price evolution is modelled as a tree of possible paths branching out and recombining, where at each discrete time step, the short rate can move to 3 possible states (up/down/stay the same). The probabilities are chosen so that the tree matches the drift and variance of the HW1F process, under the calibrated parameters.

Below is a high level illustration of the trinomial branching process. On X-axis is the future time, the Y-axis (j) signifies the change in short rate and each node represents the various states that the short rate can be in. From each node there is a positive probability of an up, down or no move. The probabilities are a function of α, j and change in time.

We worked through a short example, starting with the calibration of a Hull-White 1 factor model using Quantlib, an open-source pricing library. Post-calibration, we calculated the price of the bond and option and compared it with market quote. The calculation was also benchmarked against Finalyse’s internal pricing engine (Finalyse Valuation Services). The steps are summarized below:

We priced a listed Bond (ISIN: FR001400F067, CCY: EUR, Value date: 20/06/25, Close price: 107.05, Rate type: Fixed) which has a callable date 3.25 years from the value date and maturity at 3.75 years from the value date.

The Bond was rated as BBB and hence a credit spread was applicable on the Bond. Before calculating the θ(t) parameter, the credit spread applicable on the Bond should be added yield curve. Since this was a Callable Bond, the Option Adjusted Spread (OAS) was calculated. OAS is similar to the Z-spread (as defined earlier), but adjusted for the option price:

OAS=ZSpread-Option Price

The OAS can be calculated by using an interest rate model (like HW1F) by first calibrating it and then calculating the OAS using the market price of the Bond. For this exercise we used a similar approach by calibrating the HW1F model and then using the quoted price of the Bond and the HW1F tree approach, the OAS was calculated by recursively updating the spread component. Another (approximate) approach to calculate OAS is to use the market quoted Z-spread for similar rating and tenure of the Bond, although this approximation will not replicate the exact price of the Callable Bond.

Next, the OAS was added to the yield curve and this updated curve was used for the HW1F calibration and subsequently for the Callable Bond valuation.

Below are some plots which showcase the price of the callable bond, the option price, and the bond price without the option (stripped bond), with respect to various interest rate shocks. We see that the option price increases as the interest rates reduce and consequently the difference between the callable bond price and the stripped bond price increases. Intuitively, as rates decline, the issuer is able to access cheaper alternative funding and has a greater incentive to redeem the bond early.

However, in this example, the value of the call option is limited because the first call date is only 6 months before the final bond redemption and the benefits from calling the bond are limited. In order to understand dynamics of the callable bond pricing, we keep the bond maturity unchanged but reduce the first call date from 3.25 years to  1.25 years from value date. The results are displayed in the new plots below. As expected, due to the higher residual maturity of the bond on the first call date, the option price has increased and its sensitivity to interest rate shocks has also increased.

A comparison of the bond price and option price with respect to the 2 different residual maturities is presented below. We see that as the rates become higher (i.e., chances of call option being exercised becomes lower), the price of both options converges. Concurrently, the price of the 2 callable bonds are very different in a negative shock scenario and converges when the shocks are high.  

Finally, to illustrate the impact of callability on a bond’s duration and convexity, we plotted below the duration of 3 bonds against a range of interest rate shocks. The first bond (blue) is a plain bond with a maturity of 1.25Y from value date. In effect, this is the representation of a bond callable after 1.25Y when a bank assumes that the bond will be called immediately 100% of the time. The second bond (red) is a callable bond with the first call date in 1.25Y and final maturity in 3.75Y using our HW1F model. The third bond (green) is the actual bond quoted in the market (i.e., callable “lately” after 3.25Y  and maturing after 3.75Y), again priced using our HW1F model.

We see that the duration of the bond callable after 1.25Y with a final maturity after 3.75Y (red) is very sensitive to interest rate movements (high convexity). When rates are low, callability is certain and its duration converges to the duration of the plain bond with a maturity of 1.25Y (green). As interest rate increases, the gap with the plain bond increases and the duration eventually converges towards the bond that is callable “lately” (green). We also see that the duration of the bonds with no or very “late” callability (blue, green) declines slightly as interest rates increase, while the duration of the bond with a high option value (red) shows an opposing trend. These observation illustrate the importance of modelling callability correctly and the severe limitations of assuming that callable bonds always repay on the first or next call date.