# 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 ?

• What is $R_{t_2}(t_1)$? I guess there is a typo. – Richard Jan 19 '15 at 18:21

## 2 Answers

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).

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).

• @That Thanks for your comment. I know that. My question is all about how to calculate it just raving the Yield Curve as input and nothing else. – Paul Nov 20 '14 at 1:39
• @Paul arf. I understand now. I know it's somehow in there but maybe specifying this (i.e. that you know Black is a solution but you don't want it) in the question would have made it clearer. – SRKX Nov 20 '14 at 1:41