# S2000magician : Why "Z-spread" is called ?

s2000magician,

I must say “Thank you so much” for your contributions on this forum. I really benefited very much from you.

I know it is stuipd for me to raise this question after the exam as I forgot to raise this issue before the exam whereas my confusion is not clarified.

I know that Z-spread is the spread to be added to the Treasury spot rates so that the PV of the cash flows of the non-Treasury bond equal to the price of the non-Treasury bond.

My understanding is that the value of the embedded option must be accounted if the non-Treasury bond is an option-embedded non-Treasury bond.

Also my understanding is that in valuation of an option-embedded non-Treasury bond, we first calculated the price (value) of the option-embedded non-Treasury bond by dicounting the cash flows using the Z-spread plus the corresponding Treasury spot rates, then we add (in case of putable bond) the value of the embedded option to or deduct (in case of callable bond) the value of the embedded option from the calculated value mentioned above.

On the other hand, I know that the value of the embedded option is dependent on the level of the interest rates (yield) during the life of the bond because the cash flows are affected by the level of the interest rates (in the future).

My questions are :

Does “Zero-volatility” mean that we assume that the level of the interest rates remain unchanged (from the time point of the valuation to the maturity of the bond) ?

Or the meaning of “level of the interest rates remain unchanged” is different from that of “volatility is zero” ?

When you try to value a bond with an embedded option – say, a call option, or a put option – you create an interest-rate tree, where future interest rates might be higher or lower than their anticipated values, then trace all of the paths through the tree, with some rule for the conditions under which the option will be exercised. The “volatility” of this tree is a measure of how far the future interest rates can stray from their anticipated values, where the anticipated values are those contained in today’s yield curve (par curve, spot curve, or forward curve). (Note: this is a Level II idea; it’s not in the Level I curriculum.)

The “zero-volatility” in the Z-spread (zero-volatility spread) simply says that future interest rates cannot stray from today’s yield curve; thus, the Z-spread is a (constant) spread added to every point on today’s (spot) yield curve.

One of the reasons that this can be confusing is that most authors, when describing OAS and option value, (correctly) draw the Z-spread over the spot-curve, draw the OAS as if it were a spread over the (zero-volatility) spot curve, and then label the difference as the option value. Unfortunately, while this is useful for visualizing the option value, it’s inaccurate, because the OAS is a spread over an interest-rate tree: a nonzero-volatility spot curve.

S2000magician,

Thank you very much for your quick response but can you answer my question raised above ?

It seems to me that “the level of I/R (yield) remains unchanged” is different from “the volatility is zero”.

Sorry, alpha.

Yes, saying that the level of interest rates is unchanged is different from saying that the interest rate volatility is zero. Think of the former as a statement about the mean, the latter as a statement about the standard deviation.

S2000magician,

Can I say that the “option cost (spread)” is assumed to be a constant in measuring the Z-spread ?

The option cost will be a constant at that instant, but as the interest rate environment changes, so will the value of the option.

OK, I now understand that the option cost is a constant, at least at the moment when the valuation is done.

Thanks again !

My pleasure.

(This junk will come in handy next year . . . when you’re studying for Level II.)

can we have some good video on this topic.

still not 100% clear on it

If I can pass the L1 exam.

I need this as well

• For the above Q, Is it correct to say that in simple terms, z spread represents the extra yield for risk that a non treasury bond bears. Since this risk comes from the fundamental characteristics of the non treasury bond. It is mostly likely to remain constant in the future except if credit quality deteriorates or appreciates.
Thus, this is called zero volatility spread as it calculated using current spot rates and current market price so more fair representation

It is teh z-spread / zero volaitility spread as ti assumes no volatility in the yield curve.

When value bonds with optons, ie. callable and putable, we assume a distribution of interest rates, that there is volaitility in future rates. The z-spread assumes the current yield curve will predict future rates.

Understood. Thank you

But What I mentioned was for straight bonds without options. Is it correct still?

Becuase it assume no volatility in future rates.

Thanks. I am sorry if there was any confusion but how does it prove that there is no volatility in future rates?

z-spread does not prove it assumes

Not really.

It’s called zero volatility because it corresponds to a binomial interest rate tree whose volatility is zero; i.e., the unchanged (spot) yield curve.

Thank you Sir, although I couldn’t get it completely but I understood that - it is an idealized scenario but in reality the spot rates are continuously changing.

True that.