A dynamic programming approach to price installment options

TL;DR: A dynamic programming procedure to price installment options is developed that yields monotonically converging prices, and satisfactory trade-offs between accuracy and computational time.

About: This article is published in European Journal of Operational Research.The article was published on 2006-03-01 and is currently open access. It has received 33 citations till now. The article focuses on the topics: Valuation of options.

Summary

1 Introduction

• Installment Options (IO) are akin to Bermudan options except that the holder must regularly pay a premium (the “installment”) to keep the option alive.
• Instead of paying a lump sum for a derivative instrument, the holder of the IO will pay the installments as long as the need for being long in the option is present.
• In particular, this considerably reduces the cost of entering into a hedging strategy.
• Second, the authors investigate the properties of IOs through theoretical and numerical analysis in the Black and Scholes (1973) setting.
• Dynamic programming stands as an alternative for low dimensional option pricing.

2 The model

• Let the price of the underlying asset {S} be a Markov process that verifies the fundamental no-arbitrage property.
• Equation (3) models the choices that are available to the option holder: he will pay the installment and hold the option as long as the net holding value is larger than the exercise value.
• Otherwise, according to the exercise value, he will either exercise the option (when positive) or abandon the contract (when null).
• One way of pricing this IO is via backward induction using (1)-(3) from the known function vn = ve.

3 Solving the DP equation

• The idea is to partition the positive real axis into a collection of intervals and then to approximate the option value by a piecewise linear interpolation.
• (11) Key in the applicability of the DP procedure is how efficiently the integrals (9)-(10) can be computed.
• This is the well known problem of estimating the probability of rare events.
• The authors also derive some theoretical properties of the IO contract within this framework.

4 The Geometric Brownian Motion framework

• The authors now derive some theoretical properties related to the design of installment call options in the GBM framework.
• Symmetric results hold for installment put options.
• Obviously, this function is always strictly positive.
• The net holding value reaches 0 at a unique threshold xn−1, and the exercise value at a unique threshold yn−1, where xn−1 and yn−1 depend on the IO parameters.
• Figure 1 plots the curve representing the net holding value of the installment call option vhm (s)− πm for any decision date m.

5.1 Convergence speed and accuracy

• The model for the diffusion is the Geometric Brownian Motion with no dividend (Black-Scholes model).
• Matrices [Aki] and [Bki] are precomputed before doing the first iteration.
• Table 1 displays the main pricing properties of their approach.
• A four-digit accuracy can be obtained with a 1000-point grid, which implies a computational time that does not exceed two seconds.
• Third and most importantly, convergence to the “true” price is monotonic.

5.2 Non-redundant IO contracts

• Table 2 reports prices of installment calls for various levels of constant installments.
• Thus, for any installment greater than 5.076, the holding region vanishes, and the installment call is worth the European call expiring at the next decision date.
• Table 2 reports installment call upfront payments for various levels of installment and strikes.
• It is worth mentioning that the IO “greeks” may be readily obtained from the approximate value function, a piecewise linear function which is known at all dates for all possible values of the underlying asset.

6 Application to ASX installment warrants

• One of the most actively traded installment options throughout the world are currently the installment warrants on Australian stocks.
• Some of the ASX installment warrants (called rolling installment warrants) have several installments and their expiry date may be up to 10 years.
• Table 3 reports installment warrant upfront payments for various degrees of dilution.

7 Conclusion

• The authors have developed a pricing methodology for installment options using dynamic programming.
• Numerical experiments indicate that prices converge monotonically and quickly reach good levels of accuracy.
• The authors approach is flexible enough to be extended to other pricing issues involving installment options.
• Levered equity may be seen as a compound call on asset value when debt bears discrete coupons (see Geske (1977)).
• At each coupon date, shareholders decide whether or not to call the debt.

Figures

Table 3: Installment warrant prices and the dilution effect

Table 2: IO prices and installment level

Figure 2

Figure 3: Installment warrants listings and volume

Figure 1

Table 1: IO prices and computational time