1.Book says, if interest rates are volatile, Z-spread not appropriate to use to value bonds with embedded options since Z-spread includes cost of embedded option. So i understand that if the security does not have an embedded option, then we use Z-spread, but what does the “if interest rates are volatile part mean”?
can someone please explain how the difference between z-spread and nominal spread increases as slope of yield curve increases. 3. when they say “z-spread only considers one path of interest rates: the current Treasury spot rate curve. While OAS is added to the spot rates along each and every path in an interest rate tree”? So in an interest rate tree, for every up and down state, OAS is added to each rate, while z-spread only deals with discounting using current Treasury spot rate?
Recall from Derivatives that as the volatility of the price of the underlying increases, the value of both call options and put options increases. Further, if the volatility is zero, the only time value is discounting at the risk-free rate: option values remain relatively unchanged. Thus, if interest rates are volatile, the value of embedded options can change a lot, while if they’re not volatile, the value of those options remains relatively stable.
If you recall from Level I, when you have a bunch of numbers (_x_1, _x_2, . . ., xn) and you compute the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H), then,
A ≥ G ≥ H,
and they’re equal only when all of the numbers are the same; i.e., the more the variation in the numbers, the greater the difference between A and G, the greater the difference between G and H, and, therefore, the greater the difference between A and H.
The nominal spread – the difference between the YTM for the given bond and the YTM for a comparable risk-free bond (a Treasury, say) – is somewhat like an arithmetic mean. The Z-spread is somewhat like an harmonic mean. Thus, if the yield curve is flat (all of the numbers are equal), then they’re equal; as the slope of the yield curve steepens (whether normal or inverted), they diverge more and more.
Concerning that, there is something I really don’t get it.
Z-spread = OAS + option cost. So if OAS increases, Z-spread increases holding option costs constant. Further, Z-spread is added to each rate on the spot rate curve and makes PV of bond’s CF equal to market price. OAS is also added to each spot rate (in the binomial interest rate tree) and makes the bond value equal to its market price. So, where is the difference of both rates referring to the “adding”-fact, because both makes the value of the bond equal to its market price?
The Z-spread is added to each point on the zero-volatility (static) spot curve, and assumes that the option will not be exercised. Because the price assumes that the option might be exercised (so the price includes the value of the option-free bond plus the value of the option), the Z-spread includes the value of the option-free bond plus the value of the option.
The OAS is added to each point on a binomial tree (a non-zero-volatility spot curve), and assumes that the option will be exercised under appropriate circumstances. (The rule for when it will be exercised is at the discretion of the analyst.) Because the price assumes that the option might be exercised, and the OAS assumes that the option might be exercised, the value of the option cancels, leaving only the value of the option-free bond.
