Pricing European Bond Options Requires Volatility Beyond the Yield Curve
Summary
The document explains why a yield curve alone cannot determine the market price of a European call on a zero-coupon bond. The curve supplies discount factors and forward rates, which are enough to value fixed cash flows and instruments built from them. An option's payoff depends on the future bond price, so its valuation also requires assumptions about that price's dynamics and volatility.
The answers point to Black's model as one possible framework, with bond-price volatility as an additional input. They compare this to pricing an equity index option, where the underlying level, rates, and dividend assumptions do not replace implied volatility. The discussion gives no calibration procedure, numerical example, or model comparison. The resulting price therefore depends on the chosen volatility and model assumptions; a yield curve by itself is insufficient to recover a unique option value.
Key ideas
- A yield curve determines discount factors and forward rates used to value cash flows.
- A bond option price also depends on uncertainty in the bond price at expiry.
- Black's model is mentioned as one possible approach when bond-price volatility is supplied.
- Different volatility assumptions can produce different option values.
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Full text
# How to price an European call on zero-coupon from the yield curve?
# How to price an European call on zero-coupon from the yield curve?
It is known that the price of an European call of maturity $T^*$ on zero-coupon of maturity $T$ is given by
$$p(0,T)= B(0,T^*)\mathbb E ^{\mathbb Q_{T^*}}\left[ (B(T^*,T)-K)^+\right]$$
where $B(0,T)$ is the zero-coupon value at time $0 $ of maturity $T$ and $\mathbb Q_{T^*}$ is forward risk neutral measure. It's also known that $B(t_1,T_2)= e^{-(t_2-t_1)R_{t_2}(t_1)}$ what let me to the question:
How to calculate this price having the yield curve as the only input data ?
## Answer by Richi Wa (score 1)
https://quant.stackexchange.com/a/16266
The yield curve gives you the tools to calculate everyhing that is derived from it. Derivatives from the yield curve only are e.g. - Fixed rate bonds - Forward Rate agreements - Floaters - Swaps.
All these are discounted cashflows or portfolios based on discounted cashflows and forward rates (which you can calculated from the yield curve).
If you calculate options (swaptions, options on fixed-rate bonds) then you will need the volatiltiy from the market. You need additional data.
It is just as with a stock-index option. I need the stock price, risk-free rate, dividend-yield estimate and (!) the implied vol. Any other vol different than the implied vol with give me a different price (different than the traded market price).
## Answer by SRKX (score 0)
https://quant.stackexchange.com/a/15541
I think the yield curve is not what you need here. The idea is to have a model for the dynamics of the bond process $dB(t,T)$ (which you can compute by having dynamics for short-term interest rate $dr_t$.
A common assumption is to use Black 76 model with $F = B(0,T)$ if I remember well. You will also need to know the volatility $\sigma$ of your bond prices.
Filipovic's book is an excellent reference for this (and much more).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.