Installment Options

Introduction An installment option is an option where the holder makes a down payment at the beginning and then pays additional amounts at specific dates during the option’s lifetime to keep it active. If any payment is missed, the option immediately expires and all previous payments are lost forever.

Understanding Compound Options The simplest installment option is actually something called a „compound option“ – which is simply an option that gives you the right to buy another option.

Think of it like this: You first buy an option (let’s call it the „parent option“) that expires at some date. When that parent option expires, you have the choice to pay an additional amount to receive a second option (the „child option“).

This two-step payment structure is exactly like an installment option with one installment payment.

Black and Scholes‘ Original Insight Interestingly, Black and Scholes themselves viewed regular stock options as compound options. They reasoned that stock prices are really just options on a company’s value: if a company’s assets (A) minus its liabilities (L) are positive, stockholders get that value – otherwise they get nothing. Mathematically: St=e−rTE((A−L)+), where the (.)+ means „positive values only, zero if negative.“

The Mathematical Framework More formally, an n-installment option works like this: You have a payment schedule with n payments of amounts q1, q2, …, qn due at times t1, t2, …, tn respectively. We write this as q=(ti,qi) where the final time tn equals T (the option’s expiration).

Strictly speaking:

\[
c_{n}(t, S_t; \mathbf{q}) = c(…. c(t, S_t; t_n q_n); t_{n-1}, q_{n-1}) …,), t_1, q_1).
\]
The discrete n-installment call \( c_n(t, S_t, \mathbf{q}) \) was valued analytically by Griebsch, Wystup and Kühn in their paper from 2007 [GWK07] which involves the n-variate cumulative normal distribution.

A Bermudan put is a put on an underlying \(S_t\) with discrete exercise dates \(t_1, . . . , t_n\). It is the the discretization of the American put.

Let us assume that the strike may be variable, hence we define a floating strike plan \(\mathbf{K} = (t_i, K_i)^n \in (\mathbb{R}^2)^n\). Its value is denoted by \(p_n^{Ber}(t, S_t; \mathbf{K})\).

Essential for the further consideration is the following theorem from Davis, Schachermayer, and Tompkins [DST01]:

The net present value of all payments for the installment call corresponds to the value of the underlying call plus a Bermudan put on this call, where the strikes correspond to the net present values of all future installments.

The rather unhelpful Bermudan put on a standard call in the previous theorem \( p_{n-1}^{Ber}(t, c(t, S_t; q_n, t_n); \mathbf{K}) \) seems overly complicated. Can this be simplified?

One trick is to just let \(q_n\) vanish.

1. On the left hand side of the equation, the n-installment call converges to the (n-1)-installment call when the last strike disappears: \[ \lim_{q_n \rightarrow 0} c_n(t, S_t; \mathbf{q}) = c_{n-1}(t, S_t; \mathbf{q_1^{n-1}}), \]. where \(\mathbf{q_1^{n-1}}\) is the vector \(\mathbf{q}\) without the \(n\)-th column, i.e., \[ \mathbf{q_1^{n-1}} = ((t_1, q_1), \ldots, (t_{n-1}, q_{n-1})) \in (\mathbb{R}^2)^{n-1}. \]
2. On the right hand side, the value of the plain vanilla option approximates its underlying: \[ \lim_{q_n \rightarrow 0} c(t, S_t; t_n, q_n) = S_t. \]
3. The Bermudan put on the call converges to the Bermudan put on the underlying \(S_t\): \[ \lim_{q_n \rightarrow 0} p_{n-1}^{Ber}(t, c(t, S_t; t_n, q_n); \mathbf{K}) = p_{n-1}^{Ber}(t, S_t; \mathbf{K}) . \]

Clealy, when we put \( n = n+1 \) and start from there, this will happen.

Hence, by replacing \(n\) by \((n+1)\) in the theorem above, we arrive at the following simplification:

Surprisingly, this corollary verifies Geske/Johnson’s [GJ84] result about Bermudan puts with Griebsch, Wystup, Kühn’s [GWK07] result about the discrete \(n\)-installment call within the Black-Scholes framework – and vice versa.

The Continuous Case

How do installment options behave in the infinitesimal case?

A continuous-time installment option on an underlying \(S_t\) is an installment option with a continuous installment rate \(q\), such that the holder of the option pays the issuer exactly \(q · dt\) in the infinitesimal interval \(dt\). The value of this option is denoted by \(C(t, S_t; q)\).
A very useful result has once again been elaborated by [DST01]:

Given the corollary about Bermudan options before, and the result about how you connect discrete and continuous installment options, we can choose installment rates

\[ q_i^{(n)} = \left\{ \begin{array}{ll} K, & i = n \\ K\left( 1-e^{-r T/n} \right), & i < n \end{array} \right. \]

Then, the discrete installment call converges to the continuous installment call with rate \( q = rK \). The sum on the left hand side of the corollary converges to \( K \).

The Bermudan put on the right hand side converges to the American put \( P(t, S_t) \).

Hence,

„Proof“: So, once we define \( q_i^{(n)} \) according to the equation, the first term clearly converges to the installment call with rate \( r K \) .

The Bermuda Put clearly converges to the American put.

\( S_t \) does not converge to anything but itself.

So what’s left is the sum:

\[
\sum_{i = 1}^n q_i e^{-rt_i} = \sum_{i = 1}^{n-1} q_i e^{-rt_i} + K e^{- rT }.
\]

Given that \( q_n=K \), but \( q_i = K(1-e^{-irT/n}) \) for i < n,

we will find that

\[
\begin{eqnarray}
\sum_{i = 1}^{n-1} q_i e^{-rt_i} & = & K \sum_{i = 1}^{n-1} (1-e^{-rT/n})e^{-rt_i} \\
& = & K \left( \sum_{i = 1}^{n-1} e^{-rt_i} – \sum_{i = 2}^n e^{-rt_i} \right) \\
& = & K ( e^{-r t_1} – e^{-rT} ).
\end{eqnarray}
\]

For \( n \rightarrow \infty \) the first term goes to 1. Hence, the total sum converges to \( K \).

References

• [DST01] Davis, M., Schachermayer, W., & Tompkins, R. (2001). Pricing, No-arbitrage Bounds and Robust Hedging of Installment Options, Quantitative Finance 1 (2001), 567-610.

• [GJ84] Robert Geske and H. E. Johnson (1984). The American Put Option Valued Analytically, The Journal of Finance 39.5 (1984), 1511-1524.

• [GWK07] Susanne Griebsch, Christoph Kühn, and Uwe Wystup (2007). Instalment options: a closed-form solution and the limiting case, CPQF Working Paper Series 5 (2007).