# [Help~~]Z-spread & OAS with 10% or 20% interest volatility

Hi, I got one question bothering me for a week: It is said that for non-callable bond, its OAS is equal to the Z-spread. Besides, while interest volatility changes, the OAS of option-free bonds stay the same. The statement does sound quite reasonable itself, but I’m confused by its calculations… Z-spread is like 0% interest volatility. The calculations of Z-spread, of OAS with 10% interest volatility, and of OAS with 20% interest volatitiliy for a non-callable bond are “trial and error method” while applying totally different set of yields as denominators (only the yield on the current 1-year spot on-the-run Treasury is used same for all 3 calculations), how can it be possible that the SPREAD (OAS or Z-spread) added on each denominator which makes the price equal to the market price be the same in all these three cases? Thanks in advance!

I don’t know what you are looking at exactly, but if the bond has no embedded options the interest rate volatility should not affect the price of the bond. That means that if you are creating a model for interest rates that you are going to use for valuing bonds with embedded options, it needs to give you the same price for an option-free bond regardless of interest rate volatility. It turns out that there are a few pretty significant issues in that.

Hi JoeyDVivre. Thanks for reply. I’m sorry to make it quite long. Let me try to use a modified example on the notes to express my confusion: if we know yield on 1-year on-the-run U.S. Treasury security: i0=4.5749% then we assume interest rate volatility, sigma=15% then we input the known market price=102.99 and coupon rate=7.0% into the binomial model, via trail and error, finally we got i1L=5.3210% & i1U=7.1826% (which is the procedure stated in the notes) and if we assume another volatility sigma=25%, apparently i1L will be lower and i1U will be higher. Or if we assume sigma=0, then i1L=i1U, so called zero volatility. The formula for above result is as follows: for sigma=15%, the formula to yield i1L and i1U is (also quoting from notes book 5, p65): Vo=1/2{[1/2*(100+7)/(1+i1U)+1/2*(100+7)/(1+i1U)+7]/(1+i0)+[1/2*(100+7)/(1+i1L)+1/2*(100+7)/(1+i1L)+7]/(1+i0)} =1/2[(107/(1+i1U)+7)/(1+i0)+(107/(1+i1L)+7)/(1+i0)] =7/(1+i0)+1/2[107/(1+i1U)+107/(1+i1L)]/(1+i0); for sigma =25%, same formula with 15% but with lower i1L, say i1L* and higher i1U, say i1U*; for sigma=0, i1L=i1U, say i1: Vo=[107/(1+i1)+7]/(1+i0); then I’d like to calculate a 2-year non-callable bond’s OAS spread. The market price of it is known, say Vb. We simply input Vb to all three cases and add an X on the basis of the abovementioned formulas and to apply the “trial and error”. Then we get X1, X2 and X3, which will make the both sides equal. sigma=15% Vb=7/(1+i0+X1)+1/2[107/(1+i1U+X1)+107/(1+i1L+X1)]/(1+i0+X1); sigma=25% Vb=7/(1+i0+X2)+1/2[107/(1+i1U*+X2)+107/(1+i1L*+X2)]/(1+i0+X2); sigma=0 Vb=107/(1+i1+X3)+7]/(1+i0+X3) =7/(1+i0+X3)+1/2[107/(1+i1+X3)+107/(1+i1+X3)]/(1+i0+X3); Even before trying to get the final results from trial and error, just looking at these formulas, the X is added to the denominators, I can’t figure out how can it be possible that X1=X2=X3, while i1L, i1U, i1L*, i1U* and i1 are totally different?

Because you’ve just gone in a circle. When you computed those interest rates, e.g., i1L and i1U, you did it so that they would satisfy the current price of the option free bond and the volatility assumption. Now you fix volatility and compute the OAS. The OAS of these bonds is constant because the price of the bonds is constant.

I totally agree and understand that “When you computed those interest rates, e.g., i1L and i1U, you did it so that they would satisfy the current price of the option free bond and the volatility assumption” But, why “The OAS of these bonds is constant because the price of the bonds is constant.” OAS was calculated via trial and error in formulas abovementioned, price is constant as Vb in the example, but how can X1=X2=X3? BTW may I ask, according to the claimed definition of OAS, which should correctly measure the spread of a callable bond, that X1 should equal to X2 and equal to X3?

OAS isn’t really calculated according to “trial-and-error” as if there is something haphazard about it. It’s that you can’t come up with ssome closed form solution for it so you have to use some numerical procedure to get it. They used “trial-and-error” to avoid using some phrase like “gradient methods” which would have just confused people.

Hi Brian, really thank you for your time. I actually also know that even if I leave this issue, the problems of the exam can still be solved cuz we’re not asked to calculate the OAS on the exam day. But I really think the definition on the notes (How OAS is calculated) is deficient, it’s trying to figure out an object or a methodology compatible with Z-spread definitions and correctly measure the option cost. However, it’s not what it claimed to be. Do you think that using the description of “How OAS is calculated” on the notes (Book5 P70&P63), we can deduct and prove that 1. an option-free bond’s OAS is equal to its Z-spread; 2. an option-free bond’s OAS remain the same when interest volatility assumption changes; or do you think that it won’t happen, cuz the two sides of the equation apparantly won’t be equal under the description of “How OAS is calculated”?

I don’t have the notes so I’m not sure exactly what they say but 1. is absolutely true and 2. is more or less true although if you add things like default risk there is no reason at all to believe that an option-free bond’s spread is unaffected by interest rate vol.