Pricing Bond Options with Credit Risk and Default Conventions

# The bond option formula in Hull cannot be used for corporate bonds


# The bond option formula in Hull cannot be used for corporate bonds












John C. Hull gives the standard Black-76 formula for bond options:

$C = e^{-rt}(FN(d_1)-KN(d_2))$.

This formula cannot be used for corporate (defaultable) bonds.

Let's consider the following example. We have a call option on a bond with excercise in 1Y. The bond is a zero coupon corporate bond, which matures in 1Y immidiately after the option excercise time. Flat risk free rate is $r=0.05$, risky rate (bond yield) is $y=0.08%$. The notional is $N=100$, the strike is $K=0$ (we get the bond for free). Let's assume that the volatility is close to zero either.

According to Hull $C=e^{-rt}(F*1-0*0)=100*e^{-0.05*1}$.

This is obviously wrong for corporate bonds, as we carry the credit risk and the price of this call should equal the price of the risky bond itself. No one is going to magically replace this bond with a good one. If the risky bond defaults, we get the defaulted bond.

So the correct price seems to be $C = e^{-yt}FN(d_1)-e^{-rt}KN(d_2) = 100*e^{-0.08}$ (my guess).

Does anybody know more about the Black-76 bond option forumla for defaultable (corporate) bonds and its greeks?

## Answer by Andrea (score 2, accepted)

https://quant.stackexchange.com/a/81490

In a world with multiple discounting, you need to be careful how you discount each payment.

For instance: is the option cash settled or physically settled. Is it collateralised or not? What do you mean by risk free rate?

For the BS formula, what really matters is the Forward of the asset, as you can read here: https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model#Alternative_formulation

Then, the problem of how to discount is not BS-specific, and it is very possible that you will have to discount the bond part (with $y$) differently than the cash part (with $r$).

On the other hand, it might be interesting to analyse the default model, you mentioned, where default affetcs everything.

In this model, there is a single discount rate ($r$) and a default intensity ($\lambda=y-r$).

You can split every contract as

$call(T) = \mathbb{E}[0 \, if \, \tau \le T] + \mathbb{E}[call \, if \, \tau > T]$

Where $\tau$ is the default time, which is exponential and independent, so

$call(T) = \mathbb{E}[call \, | \, \tau > T] \mathbb{P}[\tau > T]$

and since $\mathbb{P}[\tau > T] = e^{-\lambda \, T}$

this acts like an extra uniform discounting applied to everything.

But, before going into options, make sure you are happy with pure discounting, so start with contracts like $(S-K)$, not $(S-K)^+$, and only then move to options.

## Answer by Adam N. (score 3)

https://quant.stackexchange.com/a/81493

We already have an appropriate accepted answer. Still, I would like to reformulate it in my own words.

The question seems to assume that $F=N=100$. This can't be correct (unless the terms are such that a credit event invalidates the option), nobody would buy this bond on a forward basis at par because then they would expose themselves to an unacceptable credit risk. Current market price of the bond is $P=N\cdot e^{-yt}$, and the forward price on a no-arbitrage basis will be $F=P\cdot e^{rt}$. Then the Black formula checks out.

## Answer by Sentinel (score 0)

https://quant.stackexchange.com/a/81529

After some research, I would like to share my own answer. Please, let me know, if I missed something.

$F = Ne^{-y(T-t)} = Ne^{-(r+\lambda)(T-t)}$

Then I substitute $F$ in the equations [1] and [2].

[1] Knock-out

$C=e^{-(r+λ)t} (FN(d_1 )-KN(d_2 ))=Ne^{-yT} N(d_1 )-Ke^{-yt} N(d_2)$

[2] No Knock-out

$C=e^{-rt} (FN(d_1 )-KN(d_2 ))=Ne^{\lambda t-yT} N(d_1 )-Ke^{-rt} N(d_2)$

The reasoning behind the math: in [2] the call option holder needs a compensation for the possibility of receiving the defaulted bond, which is equal to $e^{\lambda t}$.