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Journal ArticleDOI

A dynamic programming approach to price installment options

01 Mar 2006-European Journal of Operational Research (North-Holland)-Vol. 169, Iss: 2, pp 667-676

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.

AbstractInstallment options are Bermudan-style options where the holder periodically decides whether to exercise or not and then to keep the option alive or not (by paying the installment). We develop a dynamic programming procedure to price installment options. We study in particular the geometric Brownian motion case and derive some theoretical properties of the IO contract within this framework. We also characterize the range of installments within which the installment option is not redundant with the European contract. Numerical experiments show the method yields monotonically converging prices, and satisfactory trade-offs between accuracy and computational time. Our approach is finally applied to installment warrants, which are actively traded on the Australian Stock Exchange. Numerical investigation shows the various capital dilution effects resulting from different installment warrant designs.

Topics: Valuation of options (58%)

Summary (2 min read)

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.

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A Dy n a mic P r o g ra mmin g A p p ro a ch to P ric e
In sta llment O p t ion s
Hatem Ben-Ameur
Michèle Breton
Pascal Fran çois
April 2004
Corresponding author: Michèle Breton, Centre for Research on e-nance, 3000 Côte-
Ste-Catherine, Montreal H3T 2A7, Canada. M ail to: michele.breton@hec.ca.
CREF, GERAD and HEC Montréal
CREF and H EC Montréal.
1

A Dy n a mic P r o g ra mmin g A p p ro a ch to P ric e
In s ta llment O p t io n s
Abstract
Installment options are Bermudan-style options where the holder period-
ically decides whether to exercise or not and then to keep the option alive or
not (by paying the installmen t). We develop a dynamic programming pro-
cedure to price installment options. We study in particular the Geometric
Brownian Motion case and derive some theoretical properties of the IO con-
tract within this framework. We also characterize the range of installments
within which the installment option is not redundant with the European
contract. Numerical experiments show the method yields monotonically
converging prices, and satisfactory trade-os between accuracy and com-
putational time. Our approach is nally applied to installment warrants,
which are actively traded on the Australian Stock Exchange. Numerical in-
vestigation shows the various capital dilution eects resulting from dierent
installment warrant designs.
Keywords: Finance, Dynamic Programming, Option pricing, Install-
ment Option, Installment W arrant.
Acknowledgements: We acknowledge nancial support from NSERC,
SSHR C, IFM
2
and HEC Montréal.
2

1Introduction
Installment Options (IO) are akin to Bermudan options except that the
holder must regularly pay a premium (the “installment”) to keep the option
alive. The pre-specied dates (thereafter “decision dates”) at which the IO
may be striked correspond to the installment schedule. Therefore, at each
decision date, the holder of the IO must choose between the follo wing
1. to exercise the option, which puts an end to the contract;
2. not to exercise the option and to pay the installment, which keeps the
option alive until the next decision date;
3. not to exercise the option and not to pay the installment, which puts
an end to the con tract.
Among the most actively traded installment options throughout the
world presently are the installment warrants on Australian stocks listed
on the Australian Stoc k Exchange (ASX). Installment options are a recent
nancial innovation that introduces some exibility in the liquidity manage-
ment of portfolio strategies. 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 consider-
ably reduces the cost of entering into a hedging strategy.
1
In addition, the
1
Risk managers may e nter the IO contract at a low initial cost and adjust the install-
ment schedule with respect to their cash forecasts and liquidity constraints. This feature is
particularly attractive for corporations which massively hedge interest rate and currency
risks with forwards, futures or swaps because standard option contracts imply a cost at
entry that m ay be incompatible with a temp orary cash shortage.
3

non-payment of an installment suces to close the position at no transac-
tion cost. This reduces the liquidity risk typically associated with other
over-the-counter derivatives.
The aim of this paper is twofold. First, we tackle the problem of pricing
IOs using Dynamic Programming (DP) in a general setting. Second, we
investigate the properties of IOs through theoretical and numerical analysis
in the Black and Scholes (1973) setting.
Literature on IOs is scarce. Davis, Schachermayer and Tompkins (2001,
2002) derive no-arbitrage bounds for the price of the IO and study static
versus dynamic hedging strategies within a Black-Scholes framework with
stochastic volatility. Their analysis however is restricted to European-style
IOs, which allows for an analogy with compound options. Davis, Schacher-
mayer and Tompkins (2003) value venture capital using an analogy with
IO.
Algorithms based on nite dierences have been widely used for pricing
options with no known closed-form solution (see e.g. Wilmott, Dewynne
and Howison (1993) for a survey). Dynamic programming stands as an al-
ternative for low dimensional option pricing. By contrast to nite dierence
methods, DP does not require time discretization. A DP formulation for
pricing American options can be traced back to Chen (1970). He was able
to generate theoretical prices directly for a limited number of decision dates.
Note however that his paper appeared before the seminal Black and Sc holes
(1973) contribution and therefore does not apply risk neutral pricing.
Ben Ameur, Breton and L’Écuyer (2002) show that DP combined with
nite elements is particularly well suited for options involving decisions at
a limited number of distant dates during the life of the contract. Examples
include Bermudan-style options, callables, and convertibles. By construc-
4

tion, IOs allow for both early exercise and installment payment decisions
periodically.
The rest of the paper is organized as follows. In Section 2, we develop
the model. In Section 3, we solve the Bellman equation and show ho w the
discretization and approximation are made. Section 4 presents the special
case of the Geometric Brownian Motion and derives properties of the value
function in this setting. We present numerical illustrations in Section 5.
In Section 6 we show how to adapt our approach to the pricing of install-
ment warran ts, which are actively traded on the Australian Stock Exchange
(ASX). Section 7 concludes.
2 The m odel
Let the price of the underlying asset {S} be a Markov process that veries
the fundamental no-arbitrage property. Let t
0
=0be the installment option
(IO) inception date and t
1
,t
2
,...,t
n
(t
n
= T ) a collection of decision dates
scheduled in the contract. An installment design is characterized by the
vector of premia π =(π
1
,...,π
n1
) that are to be paid by the holder at
dates t
1
,...,t
n1
to keep the IO alive. The price of the IO is the upfront
payment v
0
required at t
0
to enter the contract.
The exercise value oftheIOatanydecisiondatet
m
,form =1,...,n,
is explicit in the contract and given by
v
e
(s)=
(s K)
+
, for the installment call option
(K s )
+
, for the installment put option
,(1)
where s = S
t
m
is the price of the underlying asset at t
m
and (x)
+
denotes
the function max {0,x}. By the risk-neutral principle, the holding value of
5

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Abstract: In Foreign Exchange Markets Compound options (options on options) are traded frequently. Instalment options generalize the concept of Compound options as they allow the holder to prolong a Vanilla Call or Put option by paying instalments of a discrete payment plan. We derive a closed-form solution to the value of such an option in the Black-Scholes model and prove that the limiting case of an Instalment option with a continuous payment plan is equivalent to a portfolio consisting of a European Vanilla option and an American Put on this Vanilla option with a time-dependent strike.

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  • ...ncum[x ] := 1/2*(Erf[x/Sqrt[2]] + 1); (*cumulative standard normal*) ndf[x ] := Evaluate[D[ncum[x], x]]; (*standard normal density*)...

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  • ...[2] lie in the middle field of all observed computation times....

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  • ...Ben-Ameur, Breton and François [2] develop a dynamic programming procedure to compute the value of Instalment options and investigate the properties of Instalment options through theoretical and numerical analysis....

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Abstract: We present three approaches to value American continuous-installment options written on assets without dividends or with continuous dividend yield. In an American continuous-installment option, the premium is paid continuously instead of up-front. At or before maturity, the holder may terminate payments by either exercising the option or stopping the option contract. Under the usual assumptions, we are able to construct an instantaneous riskless dynamic hedging portfolio and derive an inhomogeneous Black--Scholes partial differential equation for the initial value of this option. This key result allows us to derive valuation formulas for American continuous-installment options using the integral representation method and consequently to obtain closed-form formulas by approximating the optimal stopping and exercise boundaries as multipiece exponential functions. This process is compared to the finite difference method to solve the inhomogeneous Black--Scholes PDE and a Monte Carlo approach.

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  • ...Ben-Ameur et al. (2004) develops a dynamic-programming procedure to price American-style installment options and derive some theoretical properties of the installment option contract within the geometric Brownian motion framework....

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TL;DR: Valuing European continuous-installment options written on dividend-paying assets in the standard Black-Scholes-Merton framework using the Laplace transform approach, which results in explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping boundary in a closed form.
Abstract: Installment options are path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing European continuous-installment options written on dividend-paying assets in the standard Black–Scholes–Merton framework. The valuation of installment options can be formulated as a free boundary problem, due to the flexibility of continuing or stopping to pay installments. On the basis of a PDE for the initial premium, we derive an integral representation for the initial premium, being expressed as a difference of the corresponding European vanilla value and the expected present value of installment payments along the optimal stopping boundary. Applying the Laplace transform approach to this PDE, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping boundary in a closed form. Abelian theorems of Laplace transforms enable us to characterize asymptotic behaviors of the stopping boundary close and at infinite time to expiry. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the optimal stopping boundary.

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  • ...In actual markets, installment options have been traded actively, e.g., installment warrants on Australian stocks listed on the Australian Stock Exchange (ASX) (Ben-Ameur et al., 2005, 2006), a 10-year warrant with 9 annual payments offered by Deutsche Bank (Davis et al., 2001) and so on....

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  • ...For American-style discrete-installment options, Ben-Ameur et al. (2006) developed a DP algorithm for computing the option value approximated by a piecewise-linear interpolation, which is applied to valuing ASX installment warrants with dilution effects....

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References
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Abstract: If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options, the formula and the analysis that led to it are also applicable to corporate liabilities such as common stock, corporate bonds, and warrants. In particular, the formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default.

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"A dynamic programming approach to p..." refers background or methods in this paper

  • ...Second, we investigate the properties of IOs through theoretical and numerical analysis in the Black and Scholes (1973) setting....

    [...]

  • ...Second, we investigate the properties of IOs through theoretical and numerical analysis in the Black and Scholes (1973) setting. Literature on IOs is scarce. Davis et al. (2001, 2002) derive no-arbitrage bounds for the price of the IO and study static versus dynamic hedging strategies within a Black–Scholes framework with stochastic volatility. Their analysis however is restricted to European-style IOs, which allows for an analogy with compound options. Davis et al. (2003) value venture capital using an analogy with IO....

    [...]

  • ...Black and Scholes (1973) suggest to price warrants as an option on the issuer s equity (i.e. stocks plus warrants). For so doing, the valuation formula must be adjusted for dilution. Specifically, let M, N, and c respectively denote the number of outstanding warrants, the number of outstanding shares, and the conversion ratio. Extending the approach by Lauterbach and Schultz (1990), the installment warrant in this context is interpreted as––a fraction of––an IO issued by the firm....

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  • ...Black and Scholes (1973) suggest to price warrants as an option on the issuer s equity (i....

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Abstract: This paper applies the technique for valuing compound options to the risky coupon, bond problem. A formula is derived which contains n-dimensional multivariate normal intecjrals. It is shown that, for some compound option problems, the special correlation structure allows an application of an integral reduction which may simplify the numerical evaluation. The effects of various indenture restrictions on the formula are discussed, and a new formula for evaluating subordinated debt is presented.

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"A dynamic programming approach to p..." refers background in this paper

  • ...…attractive for corporations which massively hedge interest rate and currency risks with forwards, futures or swaps because standard option contracts imply a cost at entry that may be incompatible with a temporary cash shortage. closed-form solution (see e.g. Wilmott et al. (1993) for a survey)....

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