Choosing Black or Bachelier for a European Bond Option
Summary
The document examines whether Black’s lognormal model or Bachelier’s normal model is more suitable for pricing a European option embedded in a zero-coupon bond. It sets out a Black-style call valuation using the forward bond price, the discount factor to option expiry, and the option strike. The question arises from the link between bond prices and interest rates: rates are sometimes modeled as normal, while asset prices are often modeled as lognormal.
The responses explain that a normally distributed yield can imply a lognormally distributed zero-coupon bond price, since bond price depends exponentially on yield. They also say that Black and Bachelier prices can agree when their volatility inputs are calibrated appropriately, particularly for a single at-the-money option. Model choice depends on whether volatility is quoted on price or yield; Black cannot use negative yield as its underlying, though it can be applied to positive bond prices. The discussion cautions that bond options may be illiquid, making direct market quotes scarce and volatility estimates difficult to obtain.
Key ideas
- A zero-coupon bond price depends exponentially on its yield, so a normal yield assumption can imply lognormal bond prices.
- Black and Bachelier can produce matching prices when their volatility inputs are calibrated consistently.
- For a single at-the-money option, the model choice may matter less than using an appropriate volatility quote.
- Black’s model cannot directly use a negative yield as its underlying, but it can model a positive bond price.
- Illiquid bond options can make market prices and implied volatility difficult to observe.
Tags
Full text
# Appropriate Model for Pricing European Bond Option, Black or Bachelier?
# Appropriate Model for Pricing European Bond Option, Black or Bachelier?
Suppose an European embedded option zero coupon bond with par value $L$, strike price $X$, the maturities for embeded option and bond are $T_o$ and $T_b$, $T_o<T_b$, respectively. If we assume that the forward bond prices follows log-normal distribution, then by using the Black model, the price of call option at $t$, $t<T_o$, will be \begin{align*} C(t,P_F(T_o,T_o,T_b),X) &= P(t,T_o)\mathbb{E}_{T_o}\left\{\max(P_F(T_o,T_o,T_b) - X,0)\right\}\\ &= P(t,T_o)\left[P_F(t,T_o,T_b)\Phi\left(d_1\right) - X\Phi\left(d_2\right)\right]\\ &= LP(t,T_b)\Phi\left(d_1\right) - P(t,T_o) X\Phi\left(d_2\right) \end{align*} where $P_F(t,T_o,T_b)$ is the forward bond price such that \begin{equation*} P_F(t,T_o,T_b) = L\frac{P(t,T_b)}{P(t,T_o)} \end{equation*} $\mathbb{E}_{T_o}\{\cdot\}$ denotes expectation in a world that is forward risk neutral with respect to $P(t,T_o)$ such that \begin{equation*} \mathbb{E}_{T_o}\{P_F(T_o,T_o,T_b)\} = P_F(t,T_o,T_b) \end{equation*}
I am currently reading Howard Corb's book "Interest Rate Swaps and other Derivatives", on Chapter 5.3.1 on page 181, it discusses that
- the lognormal distribution is more appropriate for modeling stock prices, because stocks with higher price tend to have greater price fluctuations.
- the normal distribution is good assumption for interest rate, because there is no significant difference on volatility between higher rates and lower rates.
Because forward bond price is closely related to the interest rate, I am wondering, should I use the Bachelier model rather than Black model to valuate the embeded option price in a zero coupon bond?
P.S. The Bachelier model assumes that the T-forward price of an asset at time $t$, $P_F(t)$ follows an arithmetic Brownian motion with volatility $\sigma_N$, such that \begin{equation*} dP_F(t) = \sigma_N dW(t) \end{equation*}
## Answer by dm63 (score 3)
https://quant.stackexchange.com/a/80992
Note that we have the zero coupon price $$P = e^{-yT_b}$$, so that $$lnP = -yT_b$$, where $y$ is the bond yield . So if you believe yield follow a normal distribution , the zero coupon bond is lognormally distributed. Now as @AKDemy says, it doesn’t much matter what model you use for a single ATM option, but it seems that the lognormal assumption may be better if you are pricing options across a range of strikes.
## Answer by AKdemy (score 1)
https://quant.stackexchange.com/a/80990
In theory it doesn't matter. You can use Black or Bachelier and the price will agree with each other, provided the Vols are appropriate (see this answer with Bloomberg screenshots).
It will depend if you use price or yield vol though, because yield can be negative, in which case Black is undefined. If you use price vol, black is perfectly fine and what Bloomberg OVME defaults to. That price will also match quotes you get from market makers (provided you get an IV and price quote).
The main problem you will have is that the market is very illiquid, and you will not get quotes for IV or prices on vendor platforms like Bloomberg. You can only infer IVs from related ETFs with "liquid" options.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.