Novel Analytic Representations for Caps, Floors, Collars, and Exchange Options on Continuous Flows, Arbitrage‐Free Relations, and Optimal Investments

• Published date: 01 December 2024
• Author: José Carlos Dias,João Pedro Vidal Nunes,Fernando Correia da Silva
• Date: 01 December 2024
• DOI: http://doi.org/10.1002/fut.22549

Journal of Futures Markets

RESEARCH ARTICLE

Novel Analytic Representations for Caps, Floors, Collars,

and Exchange Options on Continuous Flows,

Arbitrage‐Free Relations, and Optimal Investments

José Carlos Dias

1,2

| João Pedro Vidal Nunes

1,2

| Fernando Correia da Silva

1

1

Department of Finance, Iscte—Instituto Universitário de Lisboa, Lisbon, Portugal |

2

Business Research Unit (BRU‐IUL), Lisbon, Portugal

Correspondence: José Carlos Dias (jose.carlos.dias@iscte.pt)

Received: 9 February 2024 | Revised: 14 June 2024 | Accepted: 6 August 2024

Funding: This work was supported by the Fundação para a Ciência e a Tecnologia (grant number UIDB/00315/2020).

Keywords: arbitrage‐free | caps | continuous flows | exchange options| floors and collars | investment opportunities

ABSTRACT

We offer analytic formulae for valuing finite maturity profit caps and floors that are contingent on continuous flows without the

need for subtracting the risk‐neutral expectation of the forward starting perpetual solution from the corresponding perpetual

solution. The related price caps, floors, and collars are easily obtained from any analytic representation of profit caps and floors

using some arbitrage‐free relations. Finally, we offer two novel methods for calculating the optimal triggers of investment

projects in the presence of price floors and collars regimes in a way that is much simpler than the ones currently used.

JEL Classification: G13, G31

1 | Introduction

Caps and floors are contractual agreements that are commonly

used in interest rate and commodity products aiming to provide

upper and lower bounds to the possible ranges of outcomes.

These contracts are usually used in a multi‐period setting, so

that each element of a cap or a floor—known as caplet and

floorlet, respectively—provides limits to the payoff values. In

this context, a cap or floor is the sum of a series of caplets or

floorlets (with each caplet or floorlet operating over 3 or

6 months). Several applications of these contracts can be found,

for example, in Black (1976), Boyle and Turnbull (1989), and

Hull (2018).

The real options literature adapted the rational of caps and

floors by showing that the cash flows from a project can also be

viewed as a set of options on continuous flows. In particular,

McDonald and Siegel (1985) argue that the functional form



Jvtdt=()

T

00can be understood as the (present) value of an

investment project producing a continuous cash flow vt()

0that

captures the positive part of a stochastic net profit on time

t

while avoiding losses. Nevertheless, McDonald and Siegel

(1985) focus their discussion on the derivation and properties of

each caplet vt()

0and do not present the analytical solution for

the functional form J.

Shackleton and Wojakowski (2007) provide analytic formulae

for valuing finite maturity (profit) caps and floors that are

contingent on continuous flows following a lognormal distri-

bution. Therefore, they offer an efficient closed‐form solution

for determining the value of the finite‐lived investment project

of McDonald and Siegel (1985). To accomplish this purpose,

they use perpetual methods inspired in the real options litera-

ture and evaluate the finite‐lived case using the risk‐neutral

expectation of a forward start perpetuity. More specifically, and

considering, for instance, the case of a finite‐lived profit cap

expiring in

T

years, its value is decomposed (or replicated) by a

portfolio that includes a long position in a perpetual profit cap

© 2024 Wiley Periodicals LLC.

1869 of 1922Journal of Futures Markets, 2024; 44:1869–1887

https://doi.org/10.1002/fut.22549

and a short position in a forward start perpetual profit cap that

begins after

T

years. The use of this replicating portfolio is

feasible because the individual caplets contained within each

integral are independent. This time decomposition technique

has emerged in the literature as the stepping stone to many

other recent applications of real options models with related

caps, floors, and collars—for example, Barbosa et al. (2018),

Adkins and Paxson (2019), Adkins et al. (2019), Barbosa et al.

(2020), Barbosa, Rodrigues, and Sardinha (2022), and Paxson,

Pereira, and Rodrigues (2022).

In a nutshell, we offer three main contributions to the litera-

ture. First, we produce novel analytic representations for valu-

ing finite‐lived profit caps and profit floors that are contingent

on continuous flows, but without the need for using the repli-

cating portfolio approach offered by Shackleton and

Wojakowski (2007), that is, that do not require the computation

of forward starting perpetuities. By doing so, and following the

guidelines of Ju (1998), we are able to directly tackle the valu-

ation of finite‐lived profit caps and profit floors and, hence,

provide a closed‐form solution of the integral that is required

for determining the value of the finite‐lived investment project

of McDonald and Siegel (1985). Moreover, we show that the

usual put–call duality in the spirit of McDonald and Schroder

(1998) is still valid for a sum of finite or infinite continuum of

European‐style caplets and floorlets, which implies that any

finite maturity (resp., perpetual) profit cap solution can be used

to evaluate the corresponding finite‐lived (resp., perpetual)

profit floor (or vice versa), with a simple and appropriate

switching of model parameters.

As Shackleton and Wojakowski (2007) have already produced

analytical formulae for valuing finite maturity profit caps and

profit floors, our first contribution might be seen as just an

alternative way for evaluating such contingent claims. Even

though this is true, we show, to the best of our knowledge, for

the first time as to how the corresponding integrals with respect

to time can be directly tackled analytically. Our novel equiva-

lent analytical representation is feasible under the lognormal

diffusion because it is possible to solve the integrals with respect

to time using integration by parts, as in this case, the problem

involves only the cumulative density function of the standard

univariate normal distribution.

However, the time decomposition approach of Shackleton and

Wojakowski (2007) is very general because the replicating

portfolio is valid regardless of the underlying stochastic process

of the continuous flows. For example, the constant elasticity of

variance (CEV) diffusion considered in Dias and Nunes (2011)

and Dias, Nunes, and Silva (2024) involves noncentral χ

2

dis-

tributions laws for which integrals with respect to time cannot

be solved analytically using integration by parts. Fortunately, it

is still possible to apply the replicating portfolio technique of

Shackleton and Wojakowski (2007) because the risk‐neutral

expectations of forward start perpetuities under the CEV pro-

cess can be calculated. Nevertheless, our analytical formulae

might be considered, at least, an equivalent and viable alter-

native methodology for the case where the underlying process is

assumed to follow a geometric Brownian motion, which is

actually the most common assumption that is used in the vast

majority of real options applications.

More importantly, we demonstrate that our analytical approach

can be easily extended to price (and hedge) profit caps and

floors on continuous exchange flows under the more general

two‐dimensional modeling framework considered in Fischer

(1978), Margrabe (1978), and McDonald and Siegel (1985,

section 6), for which no closed‐form solutions exist in the lit-

erature. Even though the valuation of these contingent claims

involves two stochastic variables, we show that the level of

complexity of our approach is similar to the one associated to a

one‐dimensional problem, that is, to the case where the exercise

price is assumed to be constant. In addition, our analytical

representation for the aforementioned two‐factor model yields

all the (numerical) results of Shackleton and Wojakowski

(2007) as special cases of our more general modeling setup.

We recall that different, but related, contractual price arrange-

ments with caps, floors, and collars clauses are often used by

policy makers as incentive vehicles for promoting investment

decisions of private firms—see, for example, Barbosa et al.

(2018), Adkins and Paxson (2019), Adkins et al. (2019), Barbosa

et al. (2020), Barbosa, Rodrigues, and Sardinha (2022), Paxson,

Pereira, and Rodrigues (2022) and the references contained

therein. For instance, a price floor is a contract in which there is

a floor level (or a minimum price guarantee) that protects the

investor against the downside risk of adverse cash flows. By

contrast, a price cap arrangement includes a cap level (or a

maximum price guarantee) aiming that the investor foregoes

abnormally favorable cash flows. A price collar arrangement

contains both cap and floor clauses.

Even though price caps, price floors, and price collars can be

evaluated analytically using also a replicating portfolio

approach, their direct link with the profit caps and profits floors

studied in Shackleton and Wojakowski (2007) has not been

properly acknowledged in the literature. To fill this gap, and as

our second contribution, we show that such price caps, price

floors, and price collars are easily obtained from any analytic

representation of profit caps and profit floors—for example, the

closed‐form solution offered in Shackleton and Wojakowski

(2007) or the alternative analytical methodology that is pro-

posed in this paper—using some arbitrage‐free relations. If any

of these relationships is not obeyed, then an arbitrage oppor-

tunity emerges instantaneously. Moreover, we also show how to

derive the Greeks of all these finite‐lived contractual

agreements.

As our last contribution, we offer two new methods to calculate

the optimal triggers of investment in a way that is much simpler

than the ones offered in Barbosa et al. (2018), Barbosa et al.

(2020), and Barbosa, Rodrigues, and Sardinha (2022)—for the

case of projects with floor‐style clauses—and Adkins et al.

(2019), Barbosa et al. (2020), and Paxson, Pereira, and

Rodrigues (2022)—for the case of projects with collar‐style ar-

rangements. The first method is based on the novel analytical

representation proposed in this paper, whereas the second

method uses the original formulae of finite‐lived caps offered by

Shackleton and Wojakowski (2007). Independently of the ana-

lytical representation that is chosen, we show that the compu-

tation complexity of the thresholds of investment is reduced to

solving only a single nonlinear equation instead of two or three

nonlinear equations that are presently required in the literature

1870 of 1922 Journal of Futures Markets, 2024