Research — Nov 24, 2025
Beyond Interest Rates: Why Stochastic Credit is Key to Callable Bond Valuation
By Christoph Puetter, Loïc Tudela, and Stefano Renzitti
Overview
Callable coupon bonds have become increasingly common in debt markets over the past two decades, largely because the embedded call feature gives issuers flexibility to manage funding costs and liability maturities (Becker et al., 2024; California Debt & Investment Advisory Commission, 2020). For investors, the call option introduces reinvestment risk: if the issuer redeems early, proceeds must be reinvested, often at lower yields. Empirically, callable bonds issue at an average coupon premium of roughly 2.7% over comparable non-callable instruments, although the exact spread depends on the issuer and structure.
For fixed-to-variable callable bonds, it is common market practice when calculating analytics such as duration, to assume that the issuer will redeem at the first call date. These instruments are typically structured so that, after the switch, the floating coupon resets at a spread well above the issuer’s prevailing credit spread at bond issuance, making continuation prohibitively expensive and hence very unlikely under initial market conditions. The first-call assumption is therefore a convenient simplification and often not unreasonable.
However, relying solely on this assumption can mask important dynamics. If the model ignores volatility in the true driver of the call decision—most often credit spreads—the result can be unstable hedges and misleading risk measures near the call boundary. For example, if the credit spread has widened since issuance, continuation past the initial call date becomes more likely. Spread duration can then flip abruptly between first-call and maturity regimes, understating risk on one side of the boundary and overstating it once crossed. By contrast, models incorporating stochastic credit and calculating pricing measures based on a reasonable estimate of call dynamics produce smoother hedge adjustments and more realistic P&L distributions.
In this paper, we show that modeling interest-rate dynamics alone is insufficient. Adding credit-risk dynamics to the model leads to more stable sensitivities, highlighting that dynamic credit risk is crucial in modeling these bonds accurately, particularly the call features.
To demonstrate this, we analyze the impact of adding a stochastic credit component on both prices and sensitivities for a set of fixed-to-float callable bonds. For comparison, our standard model is a one-factor Hull-White model for the short rate associated with a discount curve and a deterministic basis for the forward curve. An optional static credit-spread curve can be defined on top of the calibrated interest-rate model.
Our proposed model extends this by introducing a correlated credit factor, also modeled as a one-factor Hull-White process.
Finally, for both models, when a market price is provided for the instrument, an adjustment is applied so that the model price matches the market price. In the standard model, this adjustment is a discounting margin. In the credit model, it corresponds to the level of a flat initial credit-spread curve used for calibration. The calibrated spread—often described as an option-adjusted spread (OAS)—can also serve as a relative-value measure, capturing liquidity, structural nuances, and any behavioral or regulatory incentives embedded in market pricing.
Our first test instrument is issued by an investment-grade firm. It pays fixed coupons every 6 months during the initial fixed-rate period, at which point the coupons become indexed to a 5-year resetting par-swap rate, still paying every 6 months. The issuer can call the bond at each reset date. A positive floating-rate spread is applied on top of the par-swap rate.
Because the call and reset frequencies are perfectly aligned, changes in interest rates have little effect on the call decision; higher rates increase both the floating coupons and the discount rate, largely offsetting one another. Credit spreads, however, influence only the discounting of cash flows, so changes in credit spreads directly affect the call decision. By modeling the risk factor that truly drives the option, we observe a gradual change between the two extremes of calling at the first opportunity and never calling the bond (see Figure 1).
Figure 1. Model price as a function of the floating-rate spreads of various models.
On the y-axis, we show the theoretical price for the various models:
- a static model of interest rates (no rate volatility)
- a standard stochastic interest-rate model
- the proposed composite model including both interest rates and credit
On the x-axis, we show the trade’s floating-rate spread. Increasing it leads to higher coupons and, as a result, motivates the issuer to call the bond earlier. This axis represents moneyness: the right side corresponds to the in-the-money region and the left side to the out-of-the-money region.
As expected, the two interest-rate models match perfectly, confirming that rate volatility is not relevant for this instrument. The plot clearly defines two regimes for the issuer as it moves rapidly from deep out-of-the-money to deep in-the-money. On the left side, the change in price can be attributed entirely to the change in the floating-rate spread for a fixed-to-float bond, without altering the call decision. On the right side, the flat section indicates that once the bond is expected to be called, further increases in the floating-rate spread have little or no impact on price.
The credit model, however, transitions smoothly along the moneyness axis. The curvature suggests that changes in the floating-rate spread affect the call decision, as well. This shows that we have captured the correct risk factor for this instrument and further confirms that the standard model is insufficient.
Credit-spread sensitivities and value at risk (VaR)
Now we examine how using the proposed model affects market-risk measures, specifically, sensitivities and VaR (see Figure 2).
Figure 2. Analysis of various relative credit scenario shifts and the resulting price relative to the baseline market price.
On the x-axis, we plot relative credit scenario shifts and, on the y-axis, the resulting price relative to the baseline market price. For the standard interest-rate models, a static credit curve is added on top of the calibrated discount curve, and a credit scenario represents a proportional shift of that static curve. A value of 0% corresponds to no change in the credit spread, -100% on the far left represents removing credit risk entirely, 100% represents doubling the current spread, and 200% on the far right represents adding twice the current spread (i.e., a total credit spread equal to three times the original level). For the credit model, the same relative shifts are applied to the initial calibrated credit-spread curve.
The earlier price observations are confirmed. The interest-rate model shifts rapidly from one sensitivity regime to another, one corresponding to a short-dated duration (deep in-the-money), the other to a long-dated duration (deep out-of-the-money). In summary, the interest-rate models offer two sensitivity regimes: either nearly flat when the bond is in-the-money, or steep when it is out-of-the-money. Depending on where the trade currently sits relative to moneyness, sensitivities can change sharply.
In contrast, the credit models produce more gradual sensitivities, with no abrupt transitions. Because sensitivities evolve progressively, they can be either lower or higher than in the interest-rate models depending on the trade’s position in moneyness.
For the instrument studied so far, the trade sits deep in-the-money under the interest-rate model. As shown in Figure 2, the sensitivities are therefore larger.
Now we consider a trade closer to the at-the-money level.
Figure 3. Analysis of various relative credit scenario shifts and the resulting price relative to the baseline market price.
As shown in Figure 3, the current level lies in the transition region for the interest-rate model. For an increase in credit spreads, the price drops sharply under the interest-rate model because the sensitivity corresponds to the duration of a long-maturity fixed-to-float bond, the highest possible duration for this trade. In contrast, the credit models are less affected by these proportional credit-spread shifts, transitioning more smoothly between regimes.
We now examine how the new model affects VaR. Based on the previous discussion, the outcome depends on where the moneyness lies under a pure interest-rate model. Figures 4 and 5 show the 95% 1-day VaR for the trades discussed so far.
Figure 4. The 95% 1-day VaR for Trade1.
Figure 5. The 95% 1-day VaR for Trade2.
As expected from the credit-spread analysis, the VaR on the first trade is higher under the credit model than under the interest-rate model. The opposite is true for the second trade. The difference arises from the abrupt regime change in the interest-rate model, which highlights its limitations as a single-factor representation for these trades.
The models best suited to these structures are therefore the credit-based ones. Switching to such a model can significantly affect both sensitivities and VaR. The most important feature when using a credit model is the stability and consistency of sensitivities. To illustrate this, we show how spread duration is affected on a day-to-day for the trades studied so far.
Hedge stability
Figures 6 and 7 show the spread durations over the 4 months leading up to the March 2025 valuation date. As expected, the credit-aware model produces higher spread durations for the deep in-the-money trade. Under the interest-rate model, the duration aligns with that of a vanilla fixed-rate bond maturing at the first call date, reflecting the assumption that the bond will be redeemed at that point. The credit model relaxes this assumption by incorporating the possibility of continuation, and the resulting extension risk is directly reflected in its higher and more stable duration profile.
Figure 6. In-the-money spread duration time-series.
For the second trade, which is close to the at-the-money level, spread duration under the interest-rate model displays jumps as it switches between regimes. The credit model results are more stable because it accurately captures extension risk.
Figure 7. At-the-money spread duration time-series.
Computational cost
One might reasonably expect a performance cost from adding a risk factor. A simple composite-factor approach allows the two drivers to be combined into a single effective process. The resulting model captures both rate and credit risk with minimal performance impact.
Key takeaway
Modeling credit dynamics explicitly removes the discontinuities inherent in interest-rate-only approaches, stabilizes hedges, and captures the behavior of fixed-to-variable callable bonds more accurately, without additional performance cost.
References
Becker B, Campello M, Thell V, Yan D. Credit risk, debt overhang, and the life cycle of callable bonds. Review of Finance, 2024, 28:945-984. doi: 10.1093/rof/rfae001
California Debt and Investment Advisory Commission. Benefits and limitations of option adjusted spread analysis. 2020.