Modeling exercise notice time using lattices?

Question

I am interested in modeling callable (say European) bonds which have a time gap between when the future call exercise is decided and when the call actually occurs (payoff) - say 7 business days. I am hoping to do this using a short rate lattice. However, the payoff of this option should be path dependent since the option can move in and out of the money between these two aforementioned days. So, some simplifying assumptions/estimates are going to need to be made, I believe?

My question: What is a good way to go about doing this?

Some ideas that come to mind:

1. Simply ignore the notice timing difference and just place the payoff at the exercise date and discount.

2. Compute the expected present value of the payoff at the (earlier) decision date and compute the max between that and zero as the "payoff", and discount from there.

3. Build a lattice which recombines except for in between those two dates to allow for path dependence of the payoff.

Answer 1

Consider the exercise decision is made one second before the payoff is determined. This has more value than if the time lag is a week because there is less risk of suboptimal exercise.

Put it another way, if the time.e lag is a year or more, it is random noise on exercise decision vs payoff. So early ex value is low.

Long story short, if you ignore the time lag, I think you should be conservative as you assume (implicitly) that perfectly rational exercise occurs in all states with forward sight (i.e. you avoid early ex if not optimal at payoff)

Answer 2

Well I think I may have figured this out? So here goes:

Assume that one is valuing the option at time $t$, with exercise notice time $t_N$ and exercise (payoff) date of $t_E > t_N$. Suppose one is interested in the value of a call option on some bond with value $B(t)$ and strike $K$. Let $D(t_1,t_2)$ denote the discount factor from $t_1$ to $t_2$.

Then the value of the option is given by:

$$V(t) = \widetilde{\mathbb{E}} \left[ D(t,t_E) P(t_E) | \mathcal{F}_t \right]$$

Where $P(t_E)$ is the payoff of the option at time $t_E$. This payoff is either equal to $(B(t_E) - K)$ or 0 depending on if one decided to exercise at time $t_N$. So, let $A$ denote the event that one decides to exercise the option at $t_N$. Assuming rational exercise, one would exercise precisely when the expected payoff is positive at $t_N$, i.e. when:

$$\widetilde{\mathbb{E}} \left[D(t_N, t_E) (B(t_E) - K) | \mathcal{F}_{t_N} \right] > 0$$

If we use the notation $F(t_N, B(t_E), K)$ to denote the value of the forward contract at time $t_N$ on the bond $B$ with strike $K$ and expiry $t_E$, this event is simply given by $F(t_N, B(t_E), K) > 0$. Thus we have:

$$\mathbb{I}_A = \mathbb{I}_{\left\{F(t_N, B(t_E), K) > 0\right\}}$$

Thus we may write the payoff in terms of this event $A$ via:

$$P(t_E) = (B(t_E) - K) \mathbb{I}_A$$

Note that this payoff may actually be negative if the option was (expected to be) in-the-money at $t_N$ but was eventually out-of-the-money at $t_E$. Plugging this value in for the price to solve for $V(t)$ one gets:

$$\begin{align*} V(t) &= \widetilde{\mathbb{E}} \left[ D(t,t_E) P(t_E) | \mathcal{F}_t \right] \\ &= \widetilde{\mathbb{E}} \left[ D(t,t_E) (B(t_E) - K) \mathbb{I}_A | \mathcal{F}_t \right] \\ &= \widetilde{\mathbb{E}} \left[ \widetilde{\mathbb{E}} \left[ D(t,t_E) (B(t_E) - K) \mathbb{I}_A | \mathcal{F}_{t_N} \right] | \mathcal{F}_t \right]\\ &= \widetilde{\mathbb{E}} \left[ D(t,t_N) \mathbb{I}_A \underbrace{\widetilde{\mathbb{E}} \left[ D(t_N,t_E) (B(t_E) - K) | \mathcal{F}_{t_N} \right]}_{F(t_N, B(t_E), K)} | \mathcal{F}_t \right]\\ &= \widetilde{\mathbb{E}} \left[ D(t,t_N) \mathbb{I}_{\left\{F(t_N, B(t_E), K) > 0\right\}} F(t_N, B(t_E), K) | \mathcal{F}_t \right]\\ &= \widetilde{\mathbb{E}} \left[ D(t,t_N) F(t_N, B(t_E), K)^+ | \mathcal{F}_t \right]\\ &= \widetilde{\mathbb{E}} \left[ D(t,t_N) \widetilde{\mathbb{E}} \left[D(t_N, t_E) (B(t_E) - K) | \mathcal{F}_{t_N} \right]^+ | \mathcal{F}_t \right]\\ \end{align*}$$

One can interpret this in the following way when utilizing a lattice for pricing:

1. Compute the payoffs at $t_E$ for exercising the call options whether or not they are in or out of the money.
2. Discount the payoffs to $t_N$ and take the expected value.
3. Take the maximum of the previous value and 0 and assign that value to the option for each node at $t_N$.
4. Discount the values from (3) to $t$ to obtain $V(t)$.