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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?

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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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