Monat: Dezember 2024

The previous post about parity relations between installment calls and American puts has not taken into account dividends. This is due to the fact that $$\lim_{q_n \rightarrow 0} c(t, S_t; t_n, q_n) \neq S_t$$ whenever $\delta \neq 0$.

However, numerical experiments show that

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]:

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).