Introduction
An installment option is an option where the holder makes a down payment at the beginning of the term and further installments at predetermined dates during its lifetime to maintain the option. Once a payment is missed, the option
lapses and all previous payments are non-refundable.
The simplest form of an installment option is a compound option. A compound option is an option on an option, for instance a call on a call. The compound call on a call can be viewed as an installment option with one installment: When the mother option expires, the option holder has the right, but not the obligation to buy (or sell) the daughter option for a given price \(q_1\). This is equivalent to paying the installment rate \(q_1\).
Interestingly, Black and Scholes themselves regarded stock options as „compound options“, because they viewed stock prices themselves as call options on the assets \(A \) minus the liabilities \( L \) without personal liability( the \( (.)^+ \) operator ), or
\[
S_t=e^{-rT} \mathbb{E}((A-L)^+ ).
\]
The Discrete Generalization
More formally, a discrete-time installment option or n-installment-option on an underlying \(S_t\) is an installment option with a countable, usually finite installment plan \[\mathbf{q} = (t_i, q_i) \in (\mathbb{R}^2)^n, t_1 < . . . < t_n = T\] where the payment \(q_i\) is due at time \(t_i\). Its value at time \(t\) is denoted by \(c_n(t, S_t; \mathbf{q})\) for the call. Hence, \(c_1(t, S_t; K, T)\) is the standard European call, and \( c_2(t, S_t; (t_1, q_1; K, T)) \) is the standard compound call on a call.
Furthermore, discrete n-installment options are deeply nested compound options in the sense that an n-installment option is actually an option on an option on… an option (n times).
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]:
Theorem (Davis/Schachermayer/Tompkins)
For the \(n\)-installment call \(c_n(t, S_t; \mathbf{q})\), one obtains the following identity for the net present value of all installment payments:
\[
\begin{eqnarray}
c_n(t, S_t; \mathbf{q}) + \sum_{i=1}^{n-1} q_i e^{-r t_i} & = & c(t,S_t; t_n, q_n)\\
& & + p_{n-1}^{\text{Ber}}(t, c(t, S_t; t_n, q_n); \mathbf{K}),
\end{eqnarray}
\]
where
\[
K_i = \sum_{j=i}^{n-1}q_i e^{-r(t_j-t_i)}.
\]
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.
- 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., \[ \mathbf{q_1^{n-1}} = ((t_1, q_1), \ldots, (t_{n-1}, q_{n-1})) \in (\mathbb{R}^2)^{n-1}. \]
- On the right hand side, the value of the plain vanilla option approximates its underlying: \[ \lim_{q_n \rightarrow 0} c(t, S_t; t_n, q_n) = S_t. \]
- 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:
Corollary
Let \( p_n^{Ber}( t, S_t; \mathbf{K} ) \) be a Bermuda put with strike vector \( \mathbf{K} \).
Under the condition that \( q_n := K_n \), and for all \( i < n \)
\[
q_i := K_i - \sum_{j = i+1}^n q_j e^{-r t_i} > 0,
\]
then the following identity holds:
\[
c_n(t, S_t; \mathbf{q}) + \sum_{i=1}^{n} q_i e^{-r t_i} = S_t + p_n^{\text{Ber}}(t, S_t; \mathbf{K}).
\]
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]:
Lemma
Let \( C(t, S_t; q) \) be a continuous installment call with rate \( q \).
Let \( c_n(t, S_t; \mathbf{q^{(n)}}) \) be an \(n\)-installment call with equidistant installment dates
\( t_i^{(n)} = iT/n \) and constant rates
\[
q_i^{(n)} = \frac{q}{r} \left( 1 – e^{-rT/n} \right).
\]
Then the following limiting identity holds:
\[
\lim_{n \to \infty} c_n(t, S_t; \mathbf{q^{(n)}}) = C(t, S_t; q).
\]
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,
Theorem
\[
C(t, S_t; rK) + K = P(t, S_t) + S_t.
\]
„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).