Why Coupon Bond Options Do Not Decompose into ZCB Options in Multifactor Models
Summary
The document explains a limitation in valuing a European option on a coupon-bearing bond as the sum of options on its individual cash flows. In a one-factor interest-rate model, the coupons and principal are driven by the same underlying factor and are treated as perfectly correlated. Under that special dependence structure, the option on the total bond value can be related to the sum of options on the individual zero-coupon bonds.
With two or more factors, different parts of the yield curve can respond differently, so the values of zero-coupon bonds at different maturities are not perfectly correlated. The option payoff depends on the combined bond value, and the option on that sum generally cannot be replaced by a sum of separate option values. The answer gives a conceptual explanation rather than a full derivation or model-specific pricing formula. Its decomposition claim relies on the one-factor correlation structure, and the document recommends consulting dedicated interest-rate modelling texts for technical details.
Key ideas
- A coupon bond’s value is the sum of the present values of its coupons and principal.
- In a one-factor model, cash flows are driven by the same factor and are treated as perfectly correlated.
- Multifactor models permit less than perfect correlation among zero-coupon bond values at different maturities.
- An option on the combined bond value generally differs from the sum of options on its individual cash flows.
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# Why can a two-factor interest rate model not be used to value a coupon bearing bond as the sum of options on ZCBs # Why can a two-factor interest rate model not be used to value a coupon bearing bond as the sum of options on ZCBs I am currently reading some notes which state that > For one-factor models, the value of a European option on a coupon bond can be calculated as the sum of European options on zero-coupon bonds (ZCBs). The process is described in Hull Section 31.4. For two-factor models, this doesn’t apply because yield curve changes could cause ZCBs to increase in value at some durations, whilst decreasing in value at other durations. If I am not mistaken (please correct me if I am), a one factor model cannot produce varying shapes of future yield curve (i.e. it gives rise to parallel shifts in the yield curve over time). By contrast, a two-factor model allows the shape of the yield curve to change over time. However, I am struggling to understany why this difference means that the value of a European option on a coupon bond can be calculated as the sum of European options on ZCBs for a two factor model. That is, why this would be the case on account of the yield curve being able to change in shape. Could someone please help me to see why this is the case? ## Answer by piterbarg (score 3, accepted) https://quant.stackexchange.com/a/61785 On a conceptual level an option on a coupon bonds is an option on a sum of the coupons (and principal), and we are comparing it to the sum of the options on coupons. In a one-factor model all coupons/individual options on coupons are essentially 100% correlated as driven by the same one factor. Hence, we can link an option on the sum to the sum of the options. This is not the case in a two- and more-factor models as the coupons are not 100% correlated. For a more technical explanation you should go beyond Hull and read some books on interest rate modelling specifically. Please let me know if you need any recommendations here.
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