There are some probs I came across concerning about bond options pricing. I felt embarrassed to ask but they really confused me Here is the hot potato Suppose the interest rate follows that Vasicek model: dr=a(b-r)dt+sigma*dz To price the options on zero-coupon bonds, in Jamshidian's approach, c=P(t,s)N(d1)-KP(t,T)N(d1-sigma(p)), in which s is the maturity of the zero-coupon bond and T is the maturity of the bond options. -----I am OK with that so far. In defining d1 and sigma(p), Jamshidian argued that d1=ln(P(t,s)/P(t,T)K)/sigma(p)+sigma(p)/2, in which sigma(p) is defined as a whole junk w.r.t. sigma, a, T, t, s. Hull also used his formula in his famous book. But in Rebonato's book and Haug's formula book, they said that the sigma(p) term in d1 is like sigma*sqrt(t) in stead. My guess is the notation of sigma(p) is something related to the volatility of zero-coupon bond, which is the underlying asset in this case. Any one could explain the discrepancy for me? Or are they actually identical? Any thoughts will be most appreciated.

Hi,what is happening in the vasicek /hw models is the following. You know that the returns of the zero bonds are normally distributed, i..e. the bond prices follow lognormal processes. Consider two bonds with prices B_1 and B_2:we havedB_i = r * B_i dt + sigma_i B_i dW(t), where sigma_i is a function of the model parameters (can be easily derived if you apply Ito's lemma on the closed form solution of the bond prices).Next consider a new variable Z = B_2 / B_1. Apply Ito's lemma and change into the forward measure which is associated with B_1 being the numéraire. You obtain a process of the formdZ = Z * sigma_Z dW.The Value of an option under the forward measure is given by V = E[Payoff / B_1(T)) * B_1(0). If the maturity of the option is T_1 than B_1(T_1) = 1, thusV = E[Payoff] * B_1(0).The payoff of our bond option is max(B_2(T1) - K,0) = max(Z(T1) - X,0).You have to determine the distribution of Z to evaluate the expectation. It turns out that Z is lognormal. The total variance of the log of Z is i given byvar = int_0^T_1(sigma_Z^2)ds.Therfore the sigma that you find in the formulas is just the total variance of the log of Z until expiry of the option. You can consider the variance as "anualized" number (that's what Rebonato seems to do): Then you have to divide the total variance by the time until expiry T_1. Or you can take the number just as it is and you have the sigma of Jamshidian. Hope that this helps.

Hi mucki,Thank you very mcuhThats very informative. Cheers,sam