Issuer-default knockout bond options

Contract definition

In this lesson, “knockout” has one trigger: if the issuer defaults before the exercise time, the option is extinguished and pays no rebate. This knockout is not a barrier on the market price of the bond.

The underlying bond is a different claim. The bond can still pay recovery after default. So the recovery is part of the bond value, and it is never part of the option value in the default state.

Value the underlying bond first

An alive node is a node at which the issuer has not defaulted. At an alive node with lattice-time-index $\explain{lattice-time-index}{i}$ and lattice-state-index $\explain{lattice-state-index}{j}$, use the local node discount factor $\explain{lattice-node-discount-factor}{d_{\explain{lattice-time-index}{i},\explain{lattice-state-index}{j}}}$, the local risk-neutral up weight $\explain{lattice-node-up-weight}{q_{\explain{lattice-time-index}{i},\explain{lattice-state-index}{j}}}$, the conditional-node-survival-probability $\explain{conditional-node-survival-probability}{s_{\explain{lattice-time-index}{i},\explain{lattice-state-index}{j}}}$, the next surviving cash flow $\explain{surviving-bond-next-cash-flow}{CF^{\explain{alive-bond-value}{B}}_{\explain{lattice-time-index}{i}+1}}$, and the local bond recovery $\explain{bond-default-recovery-cash}{R^{\explain{alive-bond-value}{B}}_{\explain{lattice-time-index}{i},\explain{lattice-state-index}{j}}}$. The local surviving-bond-continuation-value is the bond value at the next time, conditional on survival. It is the next surviving cash flow plus the expected alive bond value:

$$\explain{surviving-bond-continuation-value}{H^B_{i,j}} =\explain{surviving-bond-next-cash-flow}{CF^B_{i+1}} +(1-\explain{lattice-node-up-weight}{q_{i,j}}) \explain{alive-bond-value}{B^{\mathrm{alive}}_{\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j}}} +\explain{lattice-node-up-weight}{q_{i,j}} \explain{alive-bond-value}{B^{\mathrm{alive}}_{\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j}+1}}$$

(5.2.1)

The Alive-state bond value is the discounted expectation over survival and default. Survival gives the continuation value from (5.2.1), and default gives the recovery:

$$\explain{alive-bond-value}{B^{\mathrm{alive}}_{i,j}} =\explain{lattice-node-discount-factor}{d_{i,j}} \left[ \explain{conditional-node-survival-probability}{s_{i,j}} \explain{surviving-bond-continuation-value}{H^B_{i,j}} +\left(1-\explain{conditional-node-survival-probability}{s_{i,j}}\right) \explain{bond-default-recovery-cash}{R^B_{i,j}} \right]$$

(5.2.2)

The values at each node are ex-cash-flow. The model assumes that recovery is paid at the next lattice date. This timing is a simplification.

Apply the knockout to option continuation

At call expiry, if the issuer has not defaulted, the option value is the call payoff:

$$\explain{knockout-bond-option-value}{O^{\mathrm{KO}}_{T,\explain{lattice-state-index}{j}}} =\max\left( \explain{alive-bond-value}{B^{\mathrm{alive}}_{T,\explain{lattice-state-index}{j}}} -\explain{option-strike-price}{K},0\right)$$

(5.2.3)

Before expiry, the option value in the default state is zero. So the option value at a node is the discounted, survival-weighted expectation of the option values at the two successor nodes:

$$\explain{knockout-bond-option-value}{O^{\mathrm{KO}}_{i,j}} =\explain{lattice-node-discount-factor}{d_{i,j}} \explain{conditional-node-survival-probability}{s_{i,j}} \left[ \left(1-\explain{lattice-node-up-weight}{q_{i,j}}\right) \explain{knockout-bond-option-value}{O^{\mathrm{KO}}_{\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j}}} +\explain{lattice-node-up-weight}{q_{i,j}} \explain{knockout-bond-option-value}{O^{\mathrm{KO}}_{\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j}+1}} \right]$$

(5.2.4)

(5.2.4) has no recovery term, unlike (5.2.2). If the conditional survival probability at the current node is zero, the option value is zero, also when the bond recovery is positive.

Diagram 5.2.1Issuer-default knockout valuation flow. Value the underlying bond first, including its separate input recovery. At option expiry, apply the payoff only in alive states. Before expiry, discount only the survival-weighted option values, because the option value at default is zero.

Worked example

Consider a lattice with two annual steps. At option expiry, the input values of the final period give alive bond values of USD 82.80 and USD 88.65. With a call strike of USD 85, (5.2.3) gives terminal option values of USD 0 and USD 3.65. At valuation time, the discount factor is 0.95, the conditional survival probability is 0.90, and the up weight is 0.50. The option values are:

$$\begin{aligned} \explain{knockout-bond-option-value}{O}^{\mathrm{KO}}_{1,0}&=\max(82.80-85,0)=0\\ \explain{knockout-bond-option-value}{O}^{\mathrm{KO}}_{1,1}&=\max(88.65-85,0)=3.65\\ \explain{knockout-bond-option-value}{O}^{\mathrm{KO}}_{0,0} &=0.95(0.90)\left[\left(1-0.50\right)\left(0\right)+0.50\left(3.65\right)\right]\\ &=1.560375\ \text{USD per option} \end{aligned}$$

(5.2.5)

The domain function valueDefaultKnockoutBondOption performs the same bond recursion and option recursion, in the same order.

Put-call parity with a default knockout

Put-call parity for options without a knockout does not hold unchanged for this contract. In a default state before expiry, the call value minus the put value is zero. So the matching synthetic forward must also be extinguished by the same default event. A default-free forward does not match the options in every state.

Model and source boundary

Hull describes European bond options. Separately, Hull describes CDS forwards and CDS options that cease to exist if the reference entity defaults. This lesson combines the two descriptions for teaching: an option on the issuer’s bond that is knocked out by issuer default. The lesson implements the stated contract. It does not claim that all bond options use this convention. The joint lattice for rates and credit is an input; it is not calibrated.

Knowledge check 5.2.1 Issuer-default knockout bond option

1. DirectQuestion 1Not answered

2. TransferQuestion 2Not answered

3. DirectQuestion 3Not answered

4. TransferQuestion 4Not answered

Sources

The bond-option cash-price conventions follow Hull, Chapter 28 §28.1. Hull describes the analogous extinguishment at default for CDS forwards and CDS options in Chapter 24 §24.5. Backward induction follows Chapter 12 [1]. A recovery convention must state its base and its timing, as the fixed-income treatment emphasizes [2].

References

1. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). draft ↩
2. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). draft ↩