Converting Bond Price Options to Yield Options
Summary
The document asks how to convert premiums for options on bond prices into premiums for options on bond yields, assuming normally distributed yields and Bachelier pricing. It describes a Taylor expansion of the yield as a function of bond price, expressing a nonlinear yield payoff through price-option payoffs across strikes. A second answer gives a local linear approximation: the bond price change is related to the yield change through the price sensitivity to yield, commonly represented by DV01 or BPV.
The conversion depends on the derivative’s sign and where the sensitivity is evaluated. The discussion notes that the choice of evaluation point is unresolved and may be absorbed into volatility inferred from market prices. The Taylor approach accounts for curvature in principle, while the sensitivity approximation is simpler and local; neither answer supplies a full calibration procedure or numerical example.
Key ideas
- Bond yield is a decreasing function of bond price, so price and yield option payoffs have opposite directional relationships.
- A Taylor expansion can represent a nonlinear yield payoff using a continuum of bond price option payoffs.
- A local conversion scales yield exposure by bond price sensitivity to yield, often expressed as DV01 or BPV.
- The sensitivity evaluation point is left open, and the approximation may depend on market-implied volatility.
Tags
Full text
# Bond option on price vs bond option on yield
# Bond option on price vs bond option on yield
Let the bond option paying $(K_P-P_T)^{+}$ at maturity $T$, where $K_P$ is the strike on the price of the bond and $P_T$ the price of the bond at maturity of the option. Suppose also that quoted premiums for this option today for different values of the strike are given.
I would like to price an option on the yield instead of the price. This option pays $(K_y - y_T)^{+}$, where $K_y$ is the strike in terms of yield and $y_T$ is the yield of the bond at the maturity of the option.
I would like to assume that yield is normal and use Bachelier to price this option. My question is, how does one translate the resulting premium of an option on the price of the bond into the premium of an option on the yield
## Answer by Robert (score 1)
https://quant.stackexchange.com/a/80771
I believe it should be possible to use a Taylor polynomial here:
Let's write $f(B_T) = y(K)-y(B_T)$, where $y$ denotes the (decreasing) function converting bond prices to yields.
Then $f(B_T) = f(K) + f'(K)\cdot(B_T-K)+\int_K^{B_T} f''(t)\cdot(B_T-t) dt$ and $f(B_T) = f'(K)\cdot(B_T-K)^{+}+\int_K^\infty f''(x)\cdot(B_T-x)^{+}dx$ for $B_T>K$ (note that $f(K)=0$).
As such the pay-off $[y(K)−y(B_T)]^{+}$can be written as $-y'(K)\cdot[B_T-K]^{+}-\int_K^\infty y''(x)\cdot(B_T-x)^{+}dx$.
## Answer by Andrea (score 0)
https://quant.stackexchange.com/a/81057
The previous answer is basically correct, but needs to be applied the other way round
$K_P-P_T$ can be written in terms of yields as $P(Y_K)-P(Y_T) \sim (Y_K - Y_T) \cdot P'_y$ where $P'_y$ is negative.
So $(K_P-P_T)^+ = (Y_T - Y_K)^+ \cdot (-P'_y)$
$P'_y$ is traditionally called DV01 or BPV which is quoted without the minus sign, so be careful. You need a positive value as a factor.
The question at which point to compute the derivative is left unanswered and ultimately absorbed into the volatility implied from market prices. But its impact will be anyway limited.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.