Pricing and Applications of Digital Installment Options

Abstract

For its theoretical interest and strong impact on financial markets, option valuation is considered one of the cornerstones of contemporary mathematical finance. This paper specifically studies the valuation of exotic options with digital payoff and flexible payment plan. By means of the Incomplete Fourier Transform, the pricing problem is solved in order to find integral representations of the upfront price for European call and put options. Several applications in the areas of corporate finance, insurance, and real options are discussed. Finally, a new type of digital derivative named supercash option is introduced and some payment schemes are also presented.

1. Introduction

Digital options, also referred to as binary options, are derivatives contracts whose value derives from the value of the so-called underlying asset (e.g., a financial index or a real estate asset). Digital options were first introduced in the 90s by Rubinstein and Reiner [1] and Turnbull [2], and since then their popularity has grown enormously in the derivatives market. In July 2008 they also reached the Chicago Board Options Exchange (CBOE), where digital option contracts on the S&P 500 Index and on the CBOE Volatility Index are traded. Given the peculiar payoff structure, they are embedded in many financial and insurance products and their valuation is crucial for corporate finance and real option problems as well.

In this paper we investigate the digital options world from a fresh perspective. In fact, in the literature, there are several articles devoted to digital options valuation, including [3–5], but none, as far as the authors know, has tried to solve the pricing problem considered here. Our aim is to recover pricing formulas for a wide variety of European installment derivatives with digital-type and path-independent payoffs. In contrast to the smooth payoff patterns of standard options, digital options have discontinuous payoffs, switched completely one way or the other depending on whether the terminal price of the underlying asset satisfies an exercise condition: if such condition occurs, the option pays out a predetermined amount dependent on the terms of the contract; otherwise, the option expires worthless. Furthermore, we consider that the premium can also be paid in installments over the contract lifetime and that the digital option can be lapsed at any payment date before maturity (for more details on the installment feature, see [6–9]).

The rest of the paper is organized as follows. In Section 2 a free boundary problem for the upfront price function and optimal stopping boundary is solved by the Fourier transform method. A general decomposition formula for European-style options with digital payoff structure and flexible payment plan is also derived. Using this approach, several applications in the areas of corporate finance, insurance, and real options are discussed in Section 3. In Section 4 the conclusions are drawn.

2. Problem Formulation

In the standard Black-Scholes-Merton (BSM) model [10, 11], in which there exists a unique risk-neutral probability measure ℚ such that any discounted price process is a martingale and the price process of the risky asset 𝑆=(𝑆_𝑡)_𝑡≥0 follows a geometric Brownian motion 𝑑𝑆_𝑡=𝜇𝑆_𝑡𝑑𝑡+𝜎𝑆_𝑡𝑑𝑊_𝑡, (2.1) where 𝑊=(𝑊_𝑡)_𝑡≥0 is a standard Brownian motion, 𝜎>0 the volatility, and 𝜇=(𝑟−𝛿) the drift, with 𝑟≥0 the risk-free rate and 𝛿≥0 the dividend yield, let us consider the European installment options written on 𝑆 with maturity date 𝑇<∞, payoff 𝐻(𝑆_𝑇), and installment rate 𝐿_𝑡=𝐿(𝑡). Let 𝐻∶ℝ⁺→[𝑦_𝐻,∞) be a bounded below and left-continuous (resp., right-continuous) function on ℝ⁺ and that 𝐻(𝑆_𝑇)→0 as 𝑆_𝑇→0 (resp., 𝐻(𝑆_𝑇)→0 as 𝑆_𝑇→∞) for call (resp., put) options. Assume that 𝐿∶[0,𝑇]→ℝ⁺ is a nonnegative real-valued function of bounded variation on [0,𝑇]. Let 𝑉^𝐸_𝑡=𝑉^𝐸(𝑆_𝑡,𝑡;𝐿_𝑡) be the initial premium function of the option at the time of purchase 𝑡∈[0,𝑇], defined on the domain 𝒟={(𝑆_𝑡,𝑡)∈ℝ⁺×[0,𝑇]}. Let us denote by 𝐴_𝑡=𝐴(𝑡;𝐿_𝑡) and 𝐺_𝑡=𝐺(𝑡;𝐿_𝑡), for 𝑡∈[0,𝑇], the optimal stopping (or free) boundaries of call and put options, respectively, such that the domain 𝒟 is divided into a stopping region 𝒮 and a continuation region 𝒞, that is, 𝑆ForcalloptionsForputoptions𝒮=_𝑡,𝑡∈0,𝐴_𝑡×[0,𝑇],𝒮=𝑆_𝑡,𝑡∈𝐺_𝑡×[;𝑆,∞0,𝑇]𝒞=_𝑡,𝑡∈𝐴_𝑡,∞×[0,𝑇],𝒞=𝑆_𝑡,𝑡∈0,𝐺_𝑡×[0,𝑇]. (2.2) In order to ensure that the fundamental constraint 𝑉^𝐸_𝑡≥0 is satisfied in the domain 𝒟, it is necessary to impose the following conditions: 𝑉^𝐸_𝑡𝑉=0,on𝒮;_𝐸𝑡>0,on𝒞; (2.3)(2.4) since the option is worth more alive than dead only in the continuation region 𝒞, where it is then optimal to continue with the installment payments until the maturity date. The initial premium is given by (2.3) if the asset price starts in the stopping region 𝒮, so we assume that the option is alive at the time 𝑡≥0 of entering into the contract.

The initial premium function 𝑉^𝐸_𝑡 and the optimal stopping boundary, {𝐴_𝑡}_{𝑡∈[0,𝑇]} for call options and {𝐺_𝑡}_{𝑡∈[0,𝑇]} for put options, jointly solve a free boundary problem consisting of the inhomogeneous BSM partial differential equation (PDE) in 𝒞, that is, 𝜕𝑉^𝐸_𝑡𝜕𝑡+𝜇𝑆𝜕𝑉^𝐸_𝑡+1𝜕𝑆2𝜎²𝑆²𝜕²𝑉^𝐸_𝑡𝜕𝑆²−𝑟𝑉^𝐸_𝑡=𝐿_𝑡,on𝒞, (2.5) subject to the following final and boundary conditions: 𝑉ForcalloptionsForputoptions_𝐸𝑡=𝐻𝑆_𝑇,𝑉^𝐸_𝑡=𝐻𝑆_𝑇,0≤𝑆_𝑇<∞;lim_{𝑆𝑡↓𝐴𝑡}𝑉^𝐸_𝑡=0,lim_{𝑆𝑡↑𝐺𝑡}𝑉^𝐸_𝑡=0,0≤𝑡<𝑇;lim_{𝑆𝑡↓𝐴𝑡}𝜕𝑉^𝐸_𝑡𝜕𝑆=0,lim_{𝑆𝑡↑𝐺𝑡}𝜕𝑉^𝐸_𝑡𝜕𝑆=0,0≤𝑡<𝑇. (2.6)

The Incomplete Fourier Transform (IFT) is used to solve the problem specified by (2.5)–(2.6) in order to obtain an integral representation of the upfront price for the class of European contingent claims with generic payoff and installment rate.

Using the change of variables 𝑆_𝑡=𝑒^𝑥 and 𝑡=𝑇−𝜏, we get the transformed function 𝑣_𝜏=𝑣(𝑥,𝜏)≡𝑉(𝑒^𝑥,𝑇−𝜏;𝐿_𝑡) with the continuation region 𝒞_𝑣 defined by 𝒞ForcalloptionsForputoptions_𝑣=(𝑥,𝜏)∈ln𝑎_𝜏,∞×[0,𝑇]𝒞_𝑣=(𝑥,𝜏)∈−∞,ln𝑔_𝜏×[0,𝑇], (2.7) where 𝑎_𝜏=𝑎(𝜏)≡𝐴(𝑇−𝜏;𝐿_𝑡) and 𝑔_𝜏=𝑔(𝜏)≡𝐺(𝑇−𝜏;𝐿_𝑡), respectively, denote the transformed free boundaries of call and put options. It follows that (2.5) reduces to a PDE with constant coefficients 𝜕𝑣_𝜏=1𝜕𝜏2𝜎²𝜕²𝑣_𝜏𝜕𝑥²+𝜌𝜕𝑣_𝜏𝜕𝑥−𝑟𝑣_𝜏−𝑙_𝜏,on𝒞_𝑣, (2.8) with 𝜌=(𝑟−𝛿−(1/2)𝜎²), 𝑙_𝜏=𝑙(𝜏)≡𝐿(𝑇−𝜏), ℎ(𝑥)≡𝐻(𝑒^𝑥) and the associated initial and boundary conditions given by 𝑣ForcalloptionsForputoptions₀=ℎ(𝑥),𝑣₀=ℎ(𝑥),−∞<𝑥<∞;lim_{𝑥↓ln𝑎𝜏}𝑣_𝜏=0,lim_{𝑥↑𝑔𝜏}𝑣_𝜏=0,0<𝜏≤𝑇;lim_{𝑥↓ln𝑎𝜏}𝜕𝑣_𝜏𝜕𝜏=0,lim_{𝑥↑𝑔𝜏}𝜕𝑣_𝜏𝜕𝜏=0,0<𝜏≤𝑇. (2.9)

In order to be able to apply the IFT to solve PDE (2.8) for 𝑣(𝑥,𝜏), we will consider the domain 𝒟_𝑣={(𝑥,𝜏)∈ℝ×[0,𝑇]} by expressing (2.8) as follows: ℋForcalloptionsForputoptions𝑥−ln𝑎_𝜏ℒ_BSM𝑣_𝜏+𝑙_𝜏=0,ℋln𝑔_𝜏−𝑥ℒ_BSM𝑣_𝜏+𝑙_𝜏=0, (2.10) where ℒ_BSM∶=(𝜕/𝜕𝜏)−(1/2)𝜎²(𝜕²/𝜕𝑥²)−𝜌(𝜕/𝜕𝑥)+𝑟 is the BSM differential operator and ℋ(𝑥) the Heaviside step function defined as ℋ(𝑥)=11,𝑥>0,2,𝑥=0,0,𝑥<0, (2.11) with the initial and boundary conditions that remain unchanged.

Theorem 2.1. Let 𝑣(𝑥,𝜏) be the solution of the PDE (2.10) satisfying the initial and boundary conditions (2.9). Then, the initial premiums of the European installment call and put options with maturity date 𝑇, payoff ℎ(𝑥), and payment plan 𝑙(𝜏) are given, respectively, by:  For  call  options 𝑣()=𝑒𝑥,𝜏^−𝑟𝜏𝜎√2𝜋𝜏^∞ln𝑎₀₊𝑒^{−(𝑥−𝑢+𝜌𝜏)2/2𝜎2𝜏}ℎ(𝑢)−𝑑𝑢_{𝜏0∞ln𝑎𝜉}𝑒^{−𝑟(𝜏−𝜉)}𝜎()𝑒2𝜋𝜏−𝜉^{−[𝑥−𝑢+𝜌(𝜏−𝜉)]2/2𝜎2(𝜏−𝜉)}𝑙(𝜉)𝑑𝑢𝑑𝜉,𝑓𝑜𝑟(𝑥,𝜏)∈ln𝑎_𝜏,∞×[0,𝑇). (2.12) For  put  options 𝑣()=𝑒𝑥,𝜏^−𝑟𝜏𝜎√2𝜋𝜏^{ln𝑔0+−∞}𝑒^{−(𝑥−𝑢+𝜌𝜏)2/2𝜎2𝜏}ℎ(𝑢)−𝑑𝑢_{𝜏0ln𝑔𝜉−∞}𝑒^{−𝑟(𝜏−𝜉)}𝜎()𝑒2𝜋𝜏−𝜉^{−[𝑥−𝑢+𝜌(𝜏−𝜉)]2/2𝜎2(𝜏−𝜉)}𝑙(𝜉)𝑑𝑢𝑑𝜉,𝑓𝑜𝑟(𝑥,𝜏)∈−∞,ln𝑔_𝜏×[0,𝑇). (2.13)Furthermore, the optimal stopping boundaries 𝑎(𝜏) and 𝑔(𝜏), respectively, for call and put options, satisfy the following relationships: 𝐹𝑜𝑟𝑐𝑎𝑙𝑙𝑜𝑝𝑡𝑖𝑜𝑛𝑠𝐹𝑜𝑟𝑝𝑢𝑡𝑜𝑝𝑡𝑖𝑜𝑛𝑠0=𝑣ln𝑎_𝜏,𝜏0=𝑣ln𝑔_𝜏,𝜏. (2.14)

In Appendix A, it is shown how the free boundary problem defined by (2.9)–(2.10) can be solved with the aid of the IFT. Therefore, to price a European installment option for any given payoff pattern ℎ(𝑥) and payment plan 𝑙(𝜏), (2.14) must be solved using numerical methods to find the optimal stopping boundary. Once this is found, the function 𝑣(𝑥,𝜏) can be evaluated via numerical integration. Existence and uniqueness of the solution to the pricing problem of European installment call and put options, as well as the regularity properties of the free boundary, are proved in [12, 13], respectively.

In order to enhance the mathematical tractability and economical meaning, we give a parametric representation of the European installment option price as function of the current asset price 𝑆_𝑡 and time to maturity 𝜏. This allows to deal with a wide range of payoff structures and payment schemes.

Theorem 2.2. Let 𝐻∶ℝ⁺→[𝑦_𝐻,∞) be differentiable, except for at most a finite number of points, and let 𝐿∶[0,𝑇]→ℝ⁺ be piecewise continuous over [0,𝑇]. Then, the initial premium functions 𝐶(𝑆_𝑡,𝜏) and 𝑃(𝑆_𝑡,𝜏) of the European installment call and put options can be expressed, respectively, as ℋ𝑆_𝑡−𝑎_𝜏𝐶𝑆_𝑡,𝜏=𝑐_𝐸^𝐵𝑆𝑀𝑆_𝑡,𝜏−Λ^C𝑆_𝑡,𝜏;𝑎(⋅),𝑆_𝑡,𝜏∈𝑎_𝜏×(];ℋ𝑔,∞0,𝑇_𝜏−𝑆_𝑡𝑃𝑆_𝑡,𝜏=𝑝_𝐸^𝐵𝑆𝑀𝑆_𝑡,𝜏−Λ^P𝑆_𝑡,𝜏;𝑔(⋅),𝑆_𝑡,𝜏∈0,𝑔_𝜏×(0,𝑇]; (2.15)(2.16) with 𝑐_𝐸^𝐵𝑆𝑀(⋅,⋅) and 𝑝_𝐸^𝐵𝑆𝑀(⋅,⋅) the generalized BSM European call and put option pricing formulas 𝑐_𝐸^𝐵𝑆𝑀𝑆_𝑡,𝜏∶=𝑆_𝑡𝑒^{−𝛿𝜏𝑑1(𝑆𝑡,𝑎0+,𝜏)}_−∞ℎ^′𝑧⁻¹𝑧_𝑢𝑒^{𝑧−1(𝑧𝑢)}𝑒^{−𝑧2𝑢/2}√2𝜋𝑑𝑧_𝑢+𝑒^{−𝑟𝜏𝑑2(𝑆𝑡,𝑎0+,𝜏)}_−∞ℎ𝜁⁻¹𝜁_𝑢−ℎ^′𝜁⁻¹𝜁_𝑢𝑒^{−𝜁2𝑢/2}√2𝜋𝑑𝜁_𝑢;𝑝_𝐸^𝐵𝑆𝑀𝑆_𝑡,𝜏∶=𝑆_𝑡𝑒^{−𝛿𝜏∞}_{𝑑1𝑆𝑡,𝑔0+,𝜏}ℎ^′𝑧⁻¹𝑧_𝑢𝑒^{𝑧−1(𝑧𝑢)}𝑒^{−𝑧2𝑢/2}√2𝜋𝑑𝑧_𝑢+𝑒^{−𝑟𝜏∞}_{𝑑2(𝑆𝑡,𝑔0+,𝜏)}ℎ𝜁⁻¹𝜁_𝑢−ℎ^′𝜁⁻¹𝜁_𝑢𝑒^{−𝜁2𝑢/2}√2𝜋𝑑𝜁_𝑢; (2.17)(2.18) and where Λ^C𝑆_𝑡,𝜏;𝑎(⋅)∶=^𝜏₀𝑒^{−𝑟(𝜏−𝜉)}𝑙(𝜉)𝑁𝑑₂𝑆_𝑡,𝑎_𝜉Λ,𝜏−𝜉𝑑𝜉;_P𝑆_𝑡,𝜏;𝑔(⋅)∶=^𝜏₀𝑒^{−𝑟(𝜏−𝜉)}𝑙(𝜉)𝑁−𝑑₂𝑆_𝑡,𝑔_𝜉,𝜏−𝜉𝑑𝜉; (2.19)(2.20) are the Discounted Expected Payment Streams (DEPS) of call and put options, respectively. Furthermore, the optimal stopping boundaries 𝑎_𝜏 and 𝑔_𝜏 are given by 0=𝑐_𝐸^𝐵𝑆𝑀𝑎_𝜏,𝜏−Λ^C𝑎_𝜏,𝜏;𝑎(⋅);0=𝑝_𝐸^𝐵𝑆𝑀𝑔_𝜏,𝜏−Λ^P𝑔_𝜏,𝜏;𝑔(⋅). (2.21)

Theorem 2.2, whose proof is reported in Appendix B, shows that the European installment option price can be divided into two parts: the value of an associated European option with the same characteristics (underlying asset, maturity, and payoff structure) and the DEPS component, which is the present value of all future installment payments along the optimal stopping boundary.

3. Valuation of Digital Installment Options

Now we consider some applications in order to illustrate the flexibility and generality of Theorem 2.2. They cover the fundamental class of digital-type and path-independent payoffs which form the building blocks for pricing a wide range of European derivative securities. In the subsequent propositions, which share a single proof given in Appendix C, we will provide explicit formulas for the initial premium and free boundary of the option corresponding to a certain digital payoff pattern and with a generic payment plan.

3.1. Cash-or-Nothing Payoff

The simplest option with binary payoff profile is the cash-or-nothing call (resp., put) option which pays off nothing if the underlying asset price 𝑆_𝑇∈[0,∞) finishes below (resp., above) the strike price 𝐾≥0, or pays out a predetermined constant amount 𝑋≥0 if the underlying asset finishes above (resp., below) the strike price. Then, the payoff function 𝐻^CoN(𝑆_𝑇) of this option contract is given by 𝐻^CoN𝑆_𝑇∶=𝑋ℋ₀𝑆_𝑇−𝐾,foracalloption,𝑋ℋ₀𝐾−𝑆_𝑇,foraputoption, (3.1) where ℋ₀(⋅) is the unit step function, defined as ℋ₀(𝑥)∶=0,𝑥≤0,1,𝑥>0. (3.2)

Proposition 3.1. Let 𝐻∶ℝ⁺→[𝑦_𝐻,∞) be defined by (3.1). Then, the initial premiums 𝐶^𝐶𝑜𝑁(𝑆_𝑡,𝜏) and 𝑃^𝐶𝑜𝑁(𝑆_𝑡,𝜏) of the European cash-or-nothing installment call and put options are given, respectively, by ℋ𝑆_𝑡−𝑎_𝜏𝐶^CoN𝑆_𝑡,𝜏=𝑋𝑒^−𝑟𝜏𝑁𝑑₂𝑆_𝑡,𝐾,𝜏−Λ^𝐶𝑆_𝑡(⋅),𝑆,𝜏;𝑎𝑓𝑜𝑟_𝑡,𝜏∈𝑎_𝜏×(];ℋ𝑔,∞0,𝑇_𝑡−𝑆_𝑡𝑃^CoN𝑆_𝑡,𝜏=𝑋𝑒^−𝑟𝜏𝑁−𝑑₂𝑆_𝑡,𝐾,𝜏−Λ^𝑃𝑆_𝑡(⋅),𝑆,𝜏;𝑔𝑓𝑜𝑟_𝑡,𝜏∈0,𝑔_𝜏×(0,𝑇]. (3.3)(3.4)

3.1.1. Application: Corporate Finance

The cash-or-nothing payoff structure is very versatile and apt to represent a wide variety of situations, such as a very interesting corporate finance problem: the executive incentive compensation evaluation. Suppose that a company is considering to introduce an incentive program for its employees such that, if at the end of the year a given productivity index is greater or equal to a contractually specified threshold 𝐾, each employee will receive a monetary incentive 𝑋. The employee payoff profile as function of the final productivity index is shown in Figure 1. From the company's perspective, it is very important to set 𝐾 and 𝑋 at the “right” level in terms of the cost benefit analysis. In order to perform such research, the valuation of the option given to the employees is crucial and to this extent the analysis here proposed can be very helpful.

Figure 1 

Corporate finance: executive incentive compensation.

3.2. Asset-or-Nothing Payoff

The binary asset-or-nothing payoff is characterized by a function 𝐻^AoN(𝑆_𝑇) of the form 𝐻^AoN𝑆_𝑇𝑆∶=^𝑇ℋ₀𝑆_𝑇𝑆−𝐾,foracalloption,_𝑇ℋ₀𝐾−𝑆_𝑇,foraputoption, (3.5) with 𝑆_𝑇∈[0,∞) the underlying asset price at maturity 𝑇 and 𝐾≥0 the strike price. From (3.5), we see that an asset-or-nothing call (resp., put) option pays off one unit of the underlying asset if and only if 𝑆_𝑇>𝐾 (resp., 𝑆_𝑇<𝐾), otherwise the payoff is zero.

Proposition 3.2. Let 𝐻∶ℝ⁺→[𝑦_𝐻,∞) be defined by (3.5). Then, the initial premiums 𝐶^𝐴𝑜𝑁(𝑆_𝑡,𝜏) and 𝑃^𝐴𝑜𝑁(𝑆_𝑡,𝜏) of the European asset-or-nothing installment call and put options are given, respectively, by ℋ𝑆_𝑡−𝑎_𝜏𝐶^𝐴𝑜𝑁𝑆_𝑡,𝜏=𝑆_𝑡𝑒^−𝛿𝜏𝑁𝑑₁𝑆_𝑡,𝐾,𝜏−Λ^C𝑆_𝑡(⋅),𝑆,𝜏;𝑎𝑓𝑜𝑟_𝑡,𝜏∈𝑎_𝜏×(];ℋ𝑔,∞0,𝑇_𝑡−𝑆_𝑡𝑃^𝐴𝑜𝑁𝑆_𝑡,𝜏=𝑆_𝑡𝑒^−𝛿𝜏𝑁−𝑑₁𝑆_𝑡,𝐾,𝜏−Λ^P𝑆_𝑡(⋅),𝑆,𝜏;𝑔𝑓𝑜𝑟_𝑡,𝜏∈0,𝑔_𝜏×(0,𝑇]. (3.6)(3.7)

3.2.1. Application: Insurance

An insurance contract provides protection against loss for which periodically a premium is paid in exchange for a guarantee that there will be a compensation, under stipulated conditions, for a loss caused by a specified event. Given the nature of the risk insured and the payment structure, installment options are very suitable for valuation purposes and asset-or-nothing payoffs in particular are often embedded in insurance products. Let us consider for instance an earthquake insurance that provides a coverage proportional to the earthquake magnitude. Suppose that the insurance is provided only above a certain earthquake magnitude level 𝐾 and that the resulting payoff grows linearly. The resulting insurance payoff as function of the earthquake magnitude is depicted in Figure 2.

Figure 2 

Insurance: earthquake.

3.3. Gap Payoff

The third derivative built from a digital payoff is the gap option. The payoff function 𝐻^G(𝑆_𝑇), for 𝑆_𝑇∈[0,∞), of this option contract is defined by 𝐻^G𝑆_𝑇𝑆∶=^𝑇−𝑋ℋ₀𝑆_𝑇−𝐾,foracalloption,𝑋−𝑆_𝑇ℋ₀𝐾−𝑆_𝑇,foraputoption, (3.8) that is, the holder of a call (resp., put) option receives nothing if the underlying asset price 𝑆_𝑇 finishes below (resp., above) the trigger price 𝐾≥0, or she gets a payout of 𝑆_𝑇−𝑋 (resp., 𝑋−𝑆_𝑇), with 𝑋≥0 the strike price, if and only if 𝑆_𝑇>𝐾 (resp., 𝑆_𝑇<𝐾). The gap payoff reverts to the standard one when trigger and strike prices coincide, that is, 𝐾≡𝑋. Notice that the trigger price 𝐾 determines whether or not the option contract will be exercised at expiry, while the strike price 𝑋 determines the amount of the nonzero payoff, which may be positive or negative depending on the settings of 𝑋 and 𝐾.

Proposition 3.3. Let 𝐻∶ℝ⁺→[𝑦_𝐻,∞) be defined by (3.8). Then, the initial premiums 𝐶^G(𝑆_𝑡,𝜏) and 𝑃^G(𝑆_𝑡,𝜏) of the European gap installment call and put options are given, respectively, by ℋ𝑆_𝑡−𝑎_𝜏𝐶^G𝑆_𝑡,𝜏=𝑆_𝑡𝑒^−𝛿𝜏𝑁𝑑₁𝑆_𝑡,𝐾,𝜏−𝑋𝑒^−𝑟𝜏𝑁𝑑₂𝑆_𝑡,𝐾,𝜏−Λ^C𝑆_𝑡,𝜏;𝑎(⋅),𝑓𝑜𝑟𝑆_𝑡,𝜏∈a_𝜏×(];ℋ𝑔,∞0,T_𝜏−𝑆_𝑡𝑃^G𝑆_𝑡,𝜏=𝑋𝑒^−𝑟𝜏𝑁−𝑑₂𝑆_𝑡,𝐾,𝜏−𝑆_𝑡𝑒^−𝛿𝜏𝑁−𝑑₁𝑆_𝑡,𝐾,𝜏,−Λ^P𝑆_𝑡,𝜏;𝑔(⋅),𝑓𝑜𝑟𝑆_𝑡,𝜏∈0,g_𝜏×(0,T]. (3.9)(3.10)

3.3.1. Application: Real Options

A rather common situation where gap option evaluation is relevant is the following real option case. Let us consider a request for tender by an agency for the supply of a given maximum quantity of goods of value 𝐾 in exchange of a fixed amount of money 𝑋, with 0<𝑋<𝐾. The quantity of goods supplied is a stochastic variable. It is known only at maturity and matches the demand at that date. Thus, the maximum quantity of goods is chosen by the agency large enough so that the demand does not exceed the level 𝐾. The cost of the goods is known, linear and independent from the quantity supplied (i.e., there are no economies of scale). The supplier's profit and loss as function of the value of the goods supplied is represented in Figure 3. It is interesting to note that if the actual demand exceeds the level 𝑋, the supplier will incur in a loss, unlike most options final payoffs.

Figure 3 

Real option: request for tender.

3.4. Range Binary Payoffs

So far we have considered options with payoffs having only one singularity. Range binary payoffs present two points of discontinuity in a given bounded interval within which must lie the terminal price of the underlying asset so that the option expires in-the-money. This feature makes range binary options cheaper than their plain vanilla counterpart.

Following the original idea in [14], we consider an option with an asset-or-nothing range binary payoff. This option contract, which is referred to as supershare option, entitles its owner to a given monetary unit value proportion of the underlying asset, provided that its price 𝑆_𝑇∈[0,∞) finishes within a lower value 𝐾_𝑙 and an upper value 𝐾_𝑢; otherwise, the option expires worthless. Then, the payoff function 𝐻^SS(𝑆_𝑇) of a supershare option is defined as follows: 𝐻^SS𝑆_𝑇𝑆∶=^𝑇𝐾_𝑙ℋ₀𝑆_𝑇−𝐾_𝑙−ℋ₀𝑆_𝑇−𝐾_𝑢,foracalloption,𝑆𝐾−^𝑇𝐾_𝑙ℋ₀𝐾_𝑢−𝑆_𝑇−ℋ₀𝐾_𝑙−𝑆_𝑇,foraputoption, (3.11) where 𝐾∶=(𝐾_𝑙+𝐾_𝑢)/𝐾_𝑙, with 0<𝐾_𝑙<𝐾_𝑢.

Proposition 3.4. Let 𝐻∶ℝ⁺→[𝑦_𝐻,∞) be defined by (3.11). Then, the initial premiums 𝐶^SS(𝑆_𝑡,𝜏) and 𝑃^SS(𝑆_𝑡,𝜏) of the European supershare installment call and put options are given, respectively, by ℋ𝑆_𝑡−𝑎_𝜏𝐶^SS𝑆_𝑡=𝑆,𝜏^𝑡𝐾_𝑙𝑒^−𝛿𝜏𝑁𝑑₁𝑆_𝑡,𝐾_𝑙,𝜏−𝑁𝑑₁𝑆_𝑡,𝐾_𝑢,𝜏−Λ^C𝑆_𝑡,𝜏;𝑎(⋅),𝑓𝑜𝑟𝑆_𝑡,𝜏∈a_𝜏×(];ℋ𝑔,∞0,T_𝜏−𝑆_𝑡𝑃^SS𝑆_𝑡,𝜏=𝑆𝐾−^𝑡𝐾_𝑙𝑒^−𝛿𝜏𝑁−𝑑₁𝑆_𝑡,𝐾_𝑢,𝜏−𝑁−𝑑₁𝑆_𝑡,𝐾_𝑙,𝜏−Λ^P𝑆_𝑡,𝜏;𝑔(⋅),𝑓𝑜𝑟𝑆_𝑡,𝜏∈0,g_𝜏×(0,T]. (3.12)(3.13)

We now introduce a new type of option contract that entitles the holder to receive a fixed cash amount 𝑋≥0 if the underlying asset price 𝑆_𝑇∈[0,∞) lies between a lower limit 𝐾_𝑙 and an upper limit 𝐾_𝑢, and zero otherwise. A cash-or-nothing range binary option, hereafter called supercash call option, has a payoff function 𝐻^SC(𝑆_𝑇) of the form 𝐻^SC𝑆_𝑇∶=𝑋ℋ₀𝑆_𝑇−𝐾_𝑙−ℋ₀𝑆_𝑇−𝐾_𝑢, (3.14) that is, the option finishes in-the-money if and only if 𝑆_𝑇∈(𝐾_𝑙,𝐾_𝑢].

Proposition 3.5. Let 𝐻∶ℝ⁺→[𝑦_𝐻,∞) be defined by (3.14). Then, the initial premium 𝐶^SC(𝑆_𝑡,𝜏) of the European supercash installment call option is given by ℋ𝑆_𝑡−𝑎_𝜏𝐶^SC𝑆_𝑡,𝜏=𝑋𝑒^−𝑟𝜏𝑁𝑑₂𝑆_𝑡,𝐾_𝑙,𝜏−𝑁𝑑₂𝑆_𝑡,𝐾_𝑢,𝜏−Λ^C𝑆_𝑡,𝜏;𝑎(⋅),𝑓𝑜𝑟𝑆_𝑡,𝜏∈a_𝜏,∞×(0,T]. (3.15)

3.4.1. Application: Structured Products

Range binary options are very attractive for creating structured financial products. In fact their peculiar payoff pattern allows to tailor made contracts meeting the expectations of the potential buyer. More importantly, by doing so it is possible to cut the costs relative to those scenarios that the buyer considers uninteresting. Therefore, the pricing framework of the range binary options is very useful for evaluation purposes in case of setting up investment strategies or when analyzing insurance products which are effective only within a certain range.

For instance, let us consider at time 𝑡≥0 an individual investor who holds a risky portfolio and needs to withdraw a fixed amount of money 𝑋>0 at a prespecified time 𝑇>𝑡 to meet an obligation. In this case, in order to hedge the market risk, the investor could buy a cash-or-nothing installment put option with maturity date 𝑇 which pays the constant amount 𝑋 if an appropriate index that mimics his/her portfolio value is below a certain threshold 𝐾. Alternatively, excluding some tail scenarios, he/she could choose a cheaper option, a supercash installment call option, that guarantees the same amount 𝑋 if at maturity 𝑇 the index value, as shown in Figure 4, lies within a certain range (𝐾_𝑙,𝐾_𝑢], with 𝐾_𝑢=𝐾. Furthermore, the installment feature allows the investor to drop the option as soon as the market conditions turn out to be reasonably favorable.

Figure 4 

Structured products: portfolio insurance.

3.5. Flexible Payment Plans

In this paragraph, we analyze in greater detail the wide class of time-varying payment schemes, specifying two different functional forms of the installment rate 𝐿(𝑡) and focusing on the DEPS representation for call options.

Assume an installment payment function 𝐿∶[0,𝑇]→ℝ⁺ of the form 𝐿(𝑡)𝑞∶=¹−𝑞₀𝑇𝑡+𝑞₀,𝑞₀,𝑞₁∈ℝ⁺₀. (3.16) Clearly the condition 𝑞₀<𝑞₁ (resp., 𝑞₀>𝑞₁) leads to a monotonically increasing (resp., decreasing) function of 𝑡. It is also straightforward to obtain the constant payment stream by setting 𝑞₀=𝑞₁≡𝑞, with 𝑞∈ℝ⁺₀. Figure 5(a) shows the plot of the installment rate 𝐿(𝑡) as a linear function of the time 𝑡 with a positive (solid line) and negative (dashed line) slope; the special case 𝐿(𝑡)≡𝑞 is represented by the horizontal dotted line. Three possible payment schemes can be outlined in this case: (1) it increases at a constant rate from 𝑞₀ at 𝑡=0 to 𝑞₁ at 𝑡=𝑇; (2) it decreases at a constant rate from 𝑞₀ to 𝑞₁; (3) it is constant for all times. Then, from (2.19), we get Λ^C_lin𝑆_𝑡,𝜏;𝑎(⋅)=𝛼₀^𝜏₀𝑒^{−𝑟(𝜏−𝜉)}𝑁𝑑₂𝑆_𝑡,𝑎_𝜉,𝜏−𝜉𝑑𝜉−𝛼₁^𝜏₀𝜉𝑒^{−𝑟(𝜏−𝜉)}𝑁𝑑₂𝑆_𝑡,𝑎_𝜉,𝜏−𝜉𝑑𝜉, (3.17) with 𝛼₀=𝑞₁ and 𝛼₁=(𝑞₁−𝑞₀)/(𝑇).

(a) 𝐿(𝑡) linear function of 𝑡

(b) 𝐿(𝑡) staircase function of 𝑡

Figure 5 

Examples of time-varying payment schemes.

Finally, the installment rate 𝐿(𝑡) is supposed to be a linear combination of characteristic functions of disjoint bounded intervals, that is, 𝐿(𝑡)∶=^𝑛−1_𝑘=0𝑞_𝑘𝒳_𝐼𝑘(𝑡),𝑞_𝑘∈ℝ⁺₀,∀𝑘, (3.18) where 𝑛∈ℕ and {𝐼_𝑘=[𝑡_𝑘,𝑡_𝑘+1),𝑘=0,1,…,𝑛−1} is a partition 𝒫 of the time interval [0,𝑇], with 𝒳_𝐼 the indicator function defined as 𝒳_𝐼(𝑡)=1,if𝑡∈𝐼,0,otherwise. (3.19) The function 𝐿∶[0,𝑇]→ℝ⁺ defined by (3.18) is called step function and it is continuous from the right: the size of the step at point 𝑡_𝑘 is |𝑞_𝑘−𝑞_𝑘−1|, for all 𝑘=1,2,…,𝑛−1. By definition, 𝐿(𝑡) is a piecewise constant function having at most a countable number of steps since it either remains the same or changes value going from one interval to the next. It follows that in this case, the payment scheme can be monotonic or not over time depending on whether {𝑞₀,𝑞₁,…,𝑞_𝑛−1} is a real sequence. Therefore, as shown in Figure 5(b), if, for 𝑘=1,…,𝑛−1, either condition 𝑞_𝑘−1≤𝑞_𝑘 or condition 𝑞_𝑘−1≥𝑞_𝑘 holds, then the installment rate 𝐿(𝑡) is either increasing (solid segments) or decreasing (dashed segments) with respect to 𝑡; while it reduces to a constant (horizontal dotted line) if and only if, for all 𝑘, 𝑞_𝑘≡𝑞, with 𝑞∈ℝ⁺₀. Note that, when 𝐿(𝑡) is strictly monotonic, then it is often referred to as the staircase function. Then, from (2.19), we have Λ^C_step𝑆_𝑡,𝜏;𝑎(⋅)=^𝑛−1_𝑘=0𝛼_𝑘^𝜏𝑘+1_𝜏𝑘𝑒^{−𝑟(𝜏𝑘+1−𝜉)}𝑁𝑑₂𝑆_𝑡,𝑎_𝜉,𝜏_𝑘+1−𝜉𝑑𝜉, (3.20) where 𝜏_𝑘=𝑡_𝑛−𝑡_𝑛−𝑘 is the length of the interval (𝑡_𝑛,𝑡_𝑛−𝑘] and 𝛼_𝑘=𝑞_{𝑛−(𝑘+1)}, for 𝑘=0,1,…,𝑛−1, are the constant coefficients associated with the partition 𝒫([0,𝑇]).

4. Conclusions

In this paper, we have studied the valuation of exotic options with digital payoff and flexible payment plan. Taking the free boundary problem formulation, we have used the Fourier transform method to solve the inhomogeneous Black-Scholes-Merton equation subject to the appropriate boundary and terminal conditions. An integral representation of the upfront price and its decomposition formula has been proposed.

The class of path-independent options with discontinuous payoffs forms the building blocks for pricing a wide range of securities such as barrier options, structured convertible bonds, and mortgage-backed securities. This framework has been usefully applied to solve problems in different areas, including corporate finance, insurance, and real options.

Appendices

A. Proof of Theorem 2.1

Focusing on call options, we reduce (2.10) to an ordinary differential equation (ODE) by using the auxiliary function 𝑦_𝜏=𝑦(𝑥,𝜏)∶=𝑒^−𝑥𝑣(𝑥,𝜏), for which the condition lim_𝑥→∞𝑦(𝑥,𝜏)=0 does hold. Writing 𝑣_𝜏 and its derivatives in terms of 𝑦_𝜏 and substituting into (2.10) yield ℋ𝑥−ln𝑎_𝜏𝜕𝑦_𝜏−1𝜕𝜏2𝜎²𝜕²𝑦_𝜏𝜕𝑥²−𝜃𝜕𝑦_𝜏𝜕𝑥+𝛿𝑦_𝜏+𝑚_𝜏=0, (A.1) with 𝜃=(𝜌+𝜎²), 𝑚_𝜏=𝑚(𝑥,𝜏)∶=𝑒^−𝑥𝑙(𝜏), and where the associated initial and boundary conditions (2.9) are expressed by 𝑦₀=𝑒^−𝑥ℎ(𝑥),−∞<𝑥<∞;lim_{𝑥↓ln𝑎𝜏}𝑦_𝜏=0,0<𝜏≤𝑇;lim_{𝑥↓ln𝑎𝜏}𝜕𝑦_𝜏𝜕𝑥=0,0<𝜏≤𝑇. (A.2)(A.3)(A.4) By applying the definition of the IFT to (A.1), we have ℱ^𝑎𝜏,∞𝜕𝑦_𝜏=1𝜕𝜏2𝜎²ℱ^𝑎𝜏,∞𝜕²𝑦_𝜏𝜕𝑥²+𝜃ℱ^𝑎𝜏,∞𝜕𝑦_𝜏𝜕𝑥−𝛿ℱ^𝑎𝜏,∞𝑦_𝜏−ℱ^𝑎𝜏,∞𝑚_𝜏, (A.5) where ℱ^𝑎𝜏,∞ is the IFT applied to the functions 𝑦(𝑥,𝜏) and 𝑚(𝑥,𝜏) in the continuation region 𝒞_𝑣.

Using the properties of the Fourier transform of the derivatives of 𝑦_𝜏 along with the boundary conditions (A.3)-(A.4), the following three identities are obtained ℱ^𝑎𝜏,∞𝜕𝑦_𝜏𝑦();ℱ𝜕𝑥=𝑖𝜔𝜔,𝜏_𝑎𝜏,∞𝜕²𝑦_𝜏𝜕𝑥²=−𝜔²𝑦();ℱ𝜔,𝜏_𝑎𝜏,∞𝜕𝑦_𝜏=𝜕𝑦()𝜕𝜏𝜔,𝜏;𝜕𝜏 (A.6) and substituting into (A.5) yields 𝑑𝑦(𝜔,𝜏)+𝜎𝑑𝜏2²𝜔²−𝜃𝑖𝜔+𝛿𝑦(𝜔,𝜏)=−𝑚(𝜔,𝜏), (A.7) where 𝑦(𝜔,𝜏)=ℱ^𝑎𝜏,∞{𝑦_𝜏}, 𝑚(𝜔,𝜏)=ℱ^𝑎𝜏,∞{𝑚_𝜏}, and with the initial condition 𝑦(𝜔,0)=ℱ^𝑎𝜏,∞{𝑦(𝑥,0)} calculated from (A.2). Using the integrating factor method, we find that the solution for 𝑦(𝜔,𝜏) is given by 𝑦(𝜔,𝜏)=𝑦(𝜔,0)𝑒^{−((𝜎2/2)𝜔2−𝜃𝑖𝜔+𝛿)𝜏}−^𝜏₀𝑒^{−((𝜎2/2)𝜔2−𝜃𝑖𝜔+𝛿)(𝜏−𝜉)}𝑚(𝜔,𝜉)𝑑𝜉. (A.8) Applying the inverse Fourier transform ℱ⁻¹, we obtain for the function 𝑦(𝑥,𝜏) the following representation: 𝑦(𝑥,𝜏)≡𝑦₁(𝑥,𝜏)−𝑦₂(𝑥,𝜏),(𝑥,𝜏)∈ln𝑎_𝜏,∞×(0,𝑇], (A.9) with 𝑦₁(𝑥,𝜏)∶=ℱ⁻¹𝑦(𝜔,0)𝑒^{−((𝜎2/2)𝜔2−𝜃𝑖𝜔+𝛿)𝜏};𝑦₂(𝑥,𝜏)∶=ℱ^−1𝜏₀𝑒^{−((𝜎2/2)𝜔2−𝜃𝑖𝜔+𝛿)(𝜏−𝜉)}𝑚(𝜔,𝜉)𝑑𝜉. (A.10)

To determine explicit expressions for 𝑦₁(𝑥,𝜏) and 𝑦₂(𝑥,𝜏), we will use the Convolution Theorem for Fourier transforms, which, for arbitrary functions 𝑓(𝑥,𝜏₁) and 𝑔(𝑥,𝜏₂), is written as ℱ⁻¹𝐹𝜔,𝜏₁𝐺𝜔,𝜏₂=1√2𝜋^∞−∞𝑓𝑥−𝑢,𝜏₁𝑔𝑢,𝜏₂𝑑𝑢. (A.11) In order to apply this result to the definition of 𝑦₁(𝑥,𝜏), we let 𝐹𝜔,𝜏₁=𝑒^{−((𝜎2/2)𝜔2−𝜃𝑖𝜔+𝛿)𝜏}, (A.12) and then using the inverse Fourier transform, we get 𝑓𝑥,𝜏₁=1√2𝜋^∞−∞𝑒^{−(𝜎2/2)𝜔2𝜏−𝑖𝜔[−(𝑥+𝜃𝜏)]}𝑒^−𝛿𝜏𝑒𝑑𝜔=^{−𝛿𝜏−(𝑥+𝜃𝜏)2/2𝜎2𝜏}𝜎√𝜏, (A.13) where the last expression is obtained by setting 𝜆₁=(𝜎²/2)𝜏, 𝜆₂=𝑖[−(𝑥+𝜃𝜏)], 𝑛=0, and making use of the following identity: ^∞_−∞𝑒^{−𝜆1𝜔2−𝜆2𝜔}𝜔^𝑛𝑑𝜔=(−1)^𝑛𝜋𝜆₁𝜕^𝑛𝜕𝜆^𝑛₂𝑒^{(𝜆22/4𝜆1)}, (A.14) in which 𝜆₁ and 𝜆₂ are any complex functions not involving the integration variable 𝜔, with Re(𝜆₁)≥0 and 𝑛∈ℕ₀. By rearranging terms of (𝑥+𝜃𝜏)², the function 𝑓(𝑥,𝜏₁) can be simplified as follows: 𝑓𝑥,𝜏₁=𝑒^{−𝛿𝜏−{(𝑥2+𝜌2𝜏2+2𝑥𝜌𝜏)+[(𝜎2+2𝜌)𝜏2𝜎2+2𝑥𝜎2𝜏]}/2𝜎2𝜏}𝜎√𝜏=𝑒^{−𝑟𝜏−𝑥−(𝑥+𝜌𝜏)2/2𝜎2𝜏}𝜎√𝜏. (A.15) Letting now 𝐺(𝜔,𝜏₂)=𝑦(𝜔,0) yields 𝑔𝑥,𝜏₂=ℋ𝑥−ln𝑎₀⁺𝑒^−𝑥ℎ(𝑥). (A.16) Then, substituting for 𝑓(𝑥−𝑢,𝜏₁) and 𝑔(𝑢,𝜏₂) into (A.11), we obtain 𝑦₁()=𝑒𝑥,𝜏^{−𝑟𝜏−(𝑥−𝑢)}𝜎√2𝜋𝜏^∞−∞𝑒^{−(𝑥−𝑢+𝜌𝜏)2/2𝜎2𝜏}ℋ𝑢−ln𝑎₀⁺𝑒^−𝑢ℎ(𝑢)𝑑𝑢. (A.17) Next, we consider the definition of 𝑦₂(𝑥,𝜏) which can be written as 𝑦₂(𝑥,𝜏)=^𝜏₀ℱ⁻¹𝑒^{−((𝜎2/2)𝜔2−𝜃𝑖𝜔+𝛿)(𝜏−𝜉)}𝑚(𝜔,𝜉)𝑑𝜉. (A.18) To evaluate the above integral by the Convolution Theorem, we let 𝐹𝜔,𝜏₁=𝑒^{−((𝜎2/2)𝜔2−𝜃𝑖𝜔+𝛿)(𝜏−𝜉)};𝐺𝜔,𝜏₂=𝑚(𝜔,𝜉); (A.19) then applying the inverse Fourier transform to the functions 𝐹(𝜔,𝜏₁) and 𝐺(𝜔,𝜏₂), we get, respectively, 𝑓𝑥,𝜏₁=1√2𝜋^∞−∞𝑒^{−(𝜎2/2)𝜔2(𝜏−𝜉)−𝑖𝜔[−(𝑥+𝜃(𝜏−𝜉))]−𝛿(𝜏−𝜉)}𝑒𝑑𝜔=^{−𝛿(𝜏−𝜉)−[𝑥+𝜃(𝜏−𝜉)]2/2𝜎2(𝜏−𝜉)}𝜎;𝑔𝜏−𝜉𝑥,𝜏₂=ℋ𝑥−ln𝑎_𝜉𝑒^−𝑥𝑙(𝜉), (A.20) where the last expression of 𝑓(𝑥,𝜏₁) is obtained by (A.14) setting 𝜆₁=(𝜎²/2)(𝜏−𝜉), 𝜆₂=𝑖[−(𝑥+𝜃(𝜏−𝜉))] and 𝑛=0. Hence, it is quite straightforward to express the function 𝑓(𝑥,𝜏₁) as follows: 𝑓𝑥,𝜏₁=𝑒^{−𝛿(𝜏−𝜉)−{[𝑥+𝜌(𝜏−𝜉)]2+[(𝜎2+2𝜌)𝜎2(𝜏−𝜉)2+2𝑥𝜎2(𝜏−𝜉)]}/2𝜎2(𝜏−𝜉)}𝜎=𝑒𝜏−𝜉𝜎^{−𝑟(𝜏−𝜉)−𝑥−[𝑥+𝜌(𝜏−𝜉)]2/2𝜎2(𝜏−𝜉)}.𝜏−𝜉 (A.21) Then, substituting for 𝑓(𝑥−𝑢,𝜏₁) and 𝑔(𝑢,𝜏₂) into (A.11) yields 𝑦₂(𝑥,𝜏)=^𝜏₀^∞_−∞𝑒^{−𝑟(𝜏−𝜉)−(𝑥−𝑢)}𝜎()𝑒2𝜋𝜏−𝜉^{−[𝑥−𝑢+𝜌(𝜏−𝜉)]2/2𝜎2(𝜏−𝜉)}ℋ𝑢−ln𝑎_𝜉𝑒^−𝑢𝑙(𝜉)𝑑𝑢𝑑𝜉. (A.22) Finally, replacing the expressions for 𝑦₁(𝑥,𝜏) and 𝑦₂(𝑥,𝜏) into (A.9), making use of the Heaviside step function on the continuation region 𝒞_𝑣, and recovering 𝑣(𝑥,𝜏) as 𝑒^𝑥𝑦(𝑥,𝜏), we obtain the integral representation (2.12) defining the initial premium 𝑣(𝑥,𝜏) of the European installment call option.

Similarly, for put options, by expressing PDE (2.10) and (2.9) in terms of the newly defined auxiliary function 𝑦(𝑥,𝜏)∶=𝑒^𝑥𝑣(𝑥,𝜏), we are able to apply the IFT (since the required condition lim_{𝑥→−∞}𝑦(𝑥,𝜏)=0 is fulfilled) and then obtain the solution 𝑦(𝜔,𝜏) for the resulting ODE. Taking the inverse Fourier transform of this solution in conjunction with the Convolution Theorem and then recovering the original function 𝑣(𝑥,𝜏), we derive the integral representation (2.13) for the initial premium 𝑣(𝑥,𝜏) of the European installment put option.

Now, applying the second condition in (2.9), that is, imposing that the function 𝑣(𝑥,𝜏) is equal to zero as 𝑥 tends to the optimal stopping boundary, we get (2.14) which completes the proof.

B. Proof of Theorem 2.2

Focusing on call options, we consider the first integral in the right-hand side of (2.12), which can be written as follows: 𝑣₁()𝑒𝑥,𝜏∶=^−𝑟𝜏𝜎√2𝜋𝜏^∞ln𝑎₀⁺𝑒^{−(𝑥−𝑢+𝜌𝜏)2/2𝜎2𝜏}ℎ^′(𝑢)+𝑒𝑑𝑢_−𝑟𝜏𝜎√2𝜋𝜏^∞ln𝑎₀⁺𝑒^{−(𝑥−𝑢+𝜌𝜏)2/2𝜎2𝜏}ℎ(𝑢)−ℎ^′(𝑢)𝑑𝑢, (B.1) with ℎ^′(𝑥)=𝑑ℎ(𝑥)/𝑑𝑥. Multiplying and dividing by 𝑒^𝑢 in the first term of 𝑣₁(𝑥,𝜏) yield 𝐼₁()𝑒𝑥,𝜏∶=^−𝑟𝜏𝜎√2𝜋𝜏^∞ln𝑎₀⁺ℎ^′(𝑢)𝑒^𝑢𝑒^{−((𝑥−𝑢+𝜌𝜏)2−2𝑢𝜎2𝜏)/2𝜎2𝜏}𝑑𝑢. (B.2) Adding and subtracting appropriate terms, we have 𝑒^{−((𝑥−𝑢+𝜌𝜏)2−2𝑢𝜎2𝜏)/2𝜎2𝜏}=𝑒^{−((𝑥−𝑢+𝜌𝜏)2−2𝑢𝜎2𝜏+2𝑥𝜎2𝜏+𝜎4𝜏2+2𝜌𝜎2𝜏2)/2𝜎2𝜏}𝑒^{(2𝑥𝜎2𝜏+2𝜌𝜎2𝜏2+𝜎4𝜏2)/2𝜎2𝜏}=𝑒^{−[𝑢−(𝑥+(𝜌+𝜎2)𝜏)]2/2𝜎2𝜏}𝑒^{𝑥+(𝑟−𝛿)𝜏}, (B.3) and substituting it into 𝐼₁(𝑥,𝜏) yields 𝐼₁(𝑥,𝜏)=𝑒^𝑥𝑒^−𝛿𝜏lim_𝑐→∞^𝑐_ln𝑎0+ℎ^′(𝑢)𝑒^𝑢𝑒^{−(1/2)[(𝑢−(𝑥+(𝜌+𝜎2√)𝜏))/𝜎𝜏]2}𝜎√2𝜋𝜏𝑑𝑢. (B.4) Setting 𝑧_𝑢=𝑧(𝑢)∶=𝑥−𝑢+(𝜌+𝜎²)𝜏/𝜎√𝜏 and defining 𝑑₁(𝑥,𝑦,𝜏)∶=(ln(𝑥/𝑦)+(𝜌+𝜎²)𝜏)/𝜎√𝜏, we get 𝐼₁(𝑥,𝜏)=𝑒^𝑥𝑒^{−𝛿𝜏𝑑1(𝑒𝑥,𝑎0+,𝜏)}_−∞ℎ^′𝑧⁻¹𝑧_𝑢𝑒^{𝑧−1(𝑧𝑢)}𝑒^{−𝑧2𝑢/2}√2𝜋𝑑𝑧_𝑢. (B.5) Similarly, the second term of 𝑣₁(𝑥,𝜏) can be written as 𝐼₂(𝑥,𝜏)∶=𝑒^−𝑟𝜏lim_𝑐→∞^𝑐_ln𝑎0+ℎ(𝑢)−ℎ^′(𝑢𝑒)^{√−(1/2)[(𝑥−𝑢+𝜌𝜏)/𝜎𝜏]2}𝜎√2𝜋𝜏𝑑𝑢. (B.6) Setting 𝜁_𝑢=𝜁(𝑢)∶=(𝑥−𝑢+𝜌𝜏)/𝜎√𝜏 and defining 𝑑₂(𝑥,𝑦,𝜏)∶=(ln(𝑥/𝑦)+𝜌𝜏)/𝜎√𝜏, we obtain 𝐼₂(𝑥,𝜏)=𝑒^{−𝑟𝜏𝑑2(𝑒𝑥,𝑎0+,𝜏)}_−∞ℎ𝜁⁻¹𝜁_𝑢−ℎ^′𝜁⁻¹𝜁_𝑢𝑒^{−𝜁2𝑢/2}√2𝜋𝑑𝜁_𝑢. (B.7) Furthermore, the second integral in the right-hand side of (2.12) can be simplified as follows: 𝑣₂(𝑥,𝜏)∶=^𝜏₀lim_𝑐→∞^𝑐_ln𝑎𝜉𝑒^{−𝑟(𝜏−𝜉)}𝑒^{−(1/2)[(𝑥−𝑢+𝜌(𝜏−𝜉))/𝜎𝜏−𝜉]2}𝜎()𝑙(𝜉)2𝜋𝜏−𝜉𝑑𝑢𝑑𝜉. (B.8) Performing the change of variable 𝜂_𝑢=𝜂(𝑢)∶=(𝑥−𝑢+𝜌(𝜏−𝜉))/𝜎𝜏−𝜉 and using the above definition of 𝑑₂(⋅,⋅,⋅), yields 𝑣₂(𝑥,𝜏)=^𝜏₀𝑒^{−𝑟(𝜏−𝜉)}𝑙(𝜉)^{𝑑2(𝑒𝑥,𝑎𝜉,𝜏−𝜉)}_−∞𝑒^{−𝜂2𝑢/2}√2𝜋𝑑𝜂_𝑢=𝑑𝜉_𝜏0𝑒^{−𝑟(𝜏−𝜉)}𝑙(𝜉)𝑁𝑑₂𝑆_𝑡,𝑎_𝜉,𝜏−𝜉𝑑𝜉, (B.9) where 𝑁(⋅) is the standard normal cumulative distribution function given by the formula 𝑁𝑦1∶=√2𝜋^𝑦−∞𝑒^{−(1/2)𝜐2}𝑑𝜐. (B.10)

Substituting the expressions of 𝑣₁(𝑥,𝜏) and 𝑣₂(𝑥,𝜏) into (2.12) and reverting back to the original variable via 𝑆_𝑡=𝑒^𝑥, we obtain (2.15) in conjunction with the expressions (2.17) and (2.19) defining 𝑐^BSM_𝐸(𝑆_𝑡,𝜏) and Λ^C(𝑆_𝑡,𝜏;𝑎(⋅)), respectively. Finally, because the early stopping decision is optimal when 𝑆_𝑡=𝑎_𝜏, by applying the value matching condition (2.14), we get (2.21).

C. Proof of Propositions 3.1–3.5

In this appendix, we consider the European installment call option whose payoff function 𝐻∶ℝ⁺→[𝑦_𝐻,∞) is defined by 𝐻𝑆_𝑇=𝜙𝑆_𝑇ℋ₀𝑆_𝑇−𝐾_𝑙−ℋ₀𝑆_𝑇−𝐾_𝑢, (C.1) where 𝜙∶ℝ⁺→ℝ is a linear function of the terminal asset price 𝑆_𝑇, that is, 𝜙𝑆_𝑇∶=𝜙₀+𝜙₁𝑆_𝑇,𝜙₀,𝜙₁∈ℝ, (C.2)𝐾_𝑙 and 𝐾_𝑢, with 0<𝐾_𝑙<𝐾_𝑢, are, respectively, a lower and an upper limit to the values of 𝑆_𝑇 for which the option is in-the-money. Performing the change of variable 𝑆_𝑇=𝑒^𝑢, the function ℎ(𝑢)≡𝐻(𝑒^𝑢) and its first derivative can be written, respectively, as ℎ(𝑢)=𝜙𝑒^𝑢ℋ₀𝑒^𝑢−𝐾_𝑙−ℋ₀𝑒^𝑢−𝐾_𝑢;ℎ^′(𝑢)=𝜙₁𝑒^𝑢ℋ₀𝑒^𝑢−𝐾_𝑙−ℋ₀𝑒^𝑢−𝐾_𝑢+𝜙𝑒^𝑢𝛿₀𝑒^𝑢−𝐾_𝑙−𝛿₀𝑒^𝑢−𝐾_𝑢, (C.3) where 𝛿₀(⋅) is the Dirac delta function, representing the derivative of ℋ₀(⋅) and defined by the following two properties: (1)𝛿₀(𝑥)∶=(2)0,𝑥≠0,∞,𝑥=0,^∞_−∞𝛿₀(𝑥)𝑑𝑥=1. (C.4)

From (B.5)–(B.7) in the proof of Theorem 2.2, it follows that 𝑐^BSM_𝐸(⋅,⋅) is the sum of two integrals, 𝐼₁(⋅,⋅) and 𝐼₂(⋅,⋅), as expressed by the general formula (2.17). Substituting the expression for ℎ^′(⋅) into (B.5) and then rearranging terms yield 𝐼₁(𝑥,𝜏)=𝑒^𝑥𝑒^{−𝛿𝜏𝑑1(𝑒𝑥,𝑎0+,𝜏)}_−∞𝜙₁ℋ₀𝑒^{𝑧−1(𝑧𝑢)}−𝐾_𝑙−ℋ₀𝑒^{𝑧−1(𝑧𝑢)}−𝐾_𝑢𝑒^{−𝑧2𝑢/2}√2𝜋𝑑𝑧_𝑢+𝑒^𝑥𝑒^{−𝛿𝜏𝑑1(𝑒𝑥,𝑎0+,𝜏)}_−∞𝜙𝑧⁻¹𝑧_𝑢𝑒^{𝑧−1(𝑧𝑢)}𝛿₀𝑒^{𝑧−1(𝑧𝑢)}−𝐾_𝑙−𝛿₀𝑒^{𝑧−1(𝑧𝑢)}−𝐾_𝑢𝑒^{−𝑧2𝑢/2}√2𝜋𝑑𝑧_𝑢, (C.5) with 𝑧⁻¹(𝑧_𝑢)=𝑥−𝑧_𝑢𝜎√𝜏+(𝜌+𝜎²)𝜏. It is easy to see (e.g., by integration by parts) that the second term vanishes and then, splitting the first one in two parts, the above equation becomes 𝐼₁(𝑥,𝜏)=𝜙₁𝑒^𝑥𝑒^{−𝛿𝜏𝑑1(𝑒𝑥,𝑎0+,𝜏)}_−∞𝑒^{−𝑧2𝑢/2}√ℋ2𝜋⁰𝑒^{𝑧−1(𝑧𝑢)}−𝐾_𝑙𝑑𝑧_𝑢−^{𝑑1(𝑒𝑥,𝑎0+,𝜏)}_−∞𝑒^{−𝑧2𝑢/2}√ℋ2𝜋⁰𝑒^{𝑧−1(𝑧𝑢)}−𝐾_𝑢𝑑𝑧_𝑢. (C.6) Solving the inequalities 𝑒^{𝑧−1(𝑧𝑢)}−𝐾_𝑙>0 and 𝑒^{𝑧−1(𝑧𝑢)}−𝐾_𝑢>0 for 𝑧_𝑢 and using the indicator function 𝟙_{𝑥>0}=ℋ₀(𝑥) yield 𝐼₁(𝑥,𝜏)=𝜙₁𝑒^𝑥𝑒^{−𝛿𝜏𝑑1(𝑒𝑥,𝑎0+,𝜏)}_−∞𝑒^{−𝑧2𝑢/2}√𝟙2𝜋^{{𝑧𝑢<𝑑1(𝑒𝑥,𝐾𝑙,𝜏)}}𝑑𝑧_𝑢−^{𝑑1(𝑒𝑥,𝑎0+,𝜏)}_−∞𝑒^{−𝑧2𝑢/2}√𝟙2𝜋^{{𝑧𝑢<𝑑1(𝑒𝑥,𝐾𝑢,𝜏)}}𝑑𝑧_𝑢. (C.7) Since it can be shown that 𝑎₀⁺∶=lim_𝜏→0⁺𝑎_𝜏, that is, the optimal stopping boundary at expiry, is equal to the trigger prices 𝐾_𝑙 and 𝐾_𝑢 for the first and the second integrals, respectively, the above equation becomes 𝐼₁(𝑥,𝜏)=𝜙₁𝑒^𝑥𝑒^{−𝛿𝜏𝑑1(𝑒𝑥,𝐾𝑙,𝜏)}_−∞𝑒^{−𝑧2𝑢/2}√2𝜋𝑑𝑧_𝑢−^{𝑑1(𝑒𝑥,𝐾𝑢,𝜏)}_−∞𝑒^{−𝑧2𝑢/2}√2𝜋𝑑𝑧_𝑢, (C.8) and using the definition of the cumulative distribution function 𝑁(⋅) given in (B.10) yields 𝐼₁(𝑥,𝜏)=𝜙₁𝑒^𝑥𝑒^−𝛿𝜏𝑁𝑑₁𝑒^𝑥,𝐾_𝑙,𝜏−𝑁𝑑₁𝑒^𝑥,𝐾_𝑢,𝜏. (C.9) Plugging the expressions for ℎ(⋅) and ℎ^′(⋅) into (B.7), it follows that 𝐼₂(𝑥,𝜏)=𝑒^{−𝑟𝜏𝑑2(𝑒𝑥,𝑎0+,𝜏)}_−∞𝜙₀ℋ₀𝑒^{𝜁−1(𝜁𝑢)}−𝐾_𝑙−ℋ₀𝑒^{𝜁−1(𝜁𝑢)}−𝐾_𝑢𝑒^{−𝜁2𝑢/2}√2𝜋𝑑𝜁_𝑢+𝑒^{−𝑟𝜏𝑑2(𝑒𝑥,𝑎0+,𝜏)}_−∞𝜙𝑒^{𝜁−1(𝜁𝑢)}𝛿₀𝑒^{𝜁−1(𝜁𝑢)}−𝐾_𝑙−𝛿₀𝑒^{𝜁−1(𝜁𝑢)}−𝐾_𝑢𝑒^{−𝜁2𝑢/2}√2𝜋𝑑𝜁_𝑢, (C.10) with 𝜁⁻¹(𝜁_𝑢)=𝑥−𝜁_𝑢𝜎√𝜏+𝜌𝜏. However, the second term into 𝐼₂(𝑥,𝜏) vanishes and the first one can be split in two parts. Solving the inequalities 𝑒^{𝜁−1(𝜁𝑢)}−𝐾_𝑙 and 𝑒^{𝜁−1(𝜁𝑢)}−𝐾_𝑢 for 𝜁_𝑢 and using the same arguments as before, we get 𝐼₂(𝑥,𝜏)=𝜙₀𝑒^−𝑟𝜏𝑁𝑑₂𝑒^𝑥,𝐾_𝑙,𝜏−𝑁𝑑₂𝑒^𝑥,𝐾_𝑢,𝜏. (C.11) Going back to 𝑆_𝑡=𝑒^𝑥 and substituting for 𝐼₁(⋅,⋅) and 𝐼₂(⋅,⋅) from (C.9) and (C.11) into (2.17) yield 𝑐^BSM_𝐸𝑆_𝑡,𝜏=𝜙₁𝑆_𝑡𝑒^−𝛿𝜏𝑁𝑑₁𝑆_𝑡,𝐾_𝑙,𝜏−𝑁𝑑₁𝑆_𝑡,𝐾_𝑢,𝜏+𝜙₀𝑒^−𝑟𝜏𝑁𝑑₂𝑆_𝑡,𝐾_𝑙,𝜏−𝑁𝑑₂𝑆_𝑡,𝐾_𝑢,𝜏. (C.12)

Propositions 3.4-3.5 are easily proved using (C.12) since the in-the-money region is the bounded interval (𝐾_𝑙,𝐾_𝑢]. Plugging {(𝜙₀=0)∧(𝜙₁=1/𝐾_𝑙)} into (C.12), we have the explicit expression for 𝑐^BSM_𝐸(𝑆_𝑡,𝜏) and substituting it into (2.15) yields (3.12). Similarly, setting {(𝜙₀=𝑋)∧(𝜙₁=0)} in (C.12) and substituting the resulting expression into (2.15), we get (3.15).

The proofs of Propositions 3.1–3.3 can be obtained by setting 𝐾_𝑙=𝐾 and taking the limit of the right-hand side of (C.12) as 𝐾_𝑢 goes to infinity, that is, using the following result: 𝑐^BSM_𝐸𝑆_𝑡,𝜏=𝜙₁𝑆_𝑡𝑒^−𝛿𝜏𝑁𝑑₁𝑆_𝑡,𝐾,𝜏+𝜙₀𝑒^−𝑟𝜏𝑁𝑑₂𝑆_𝑡,𝐾,𝜏, (C.13) since the option is in-the-money on the unbounded interval (𝐾,∞). Equations (3.3) and (3.6) are derived from (2.15) by setting the couple of parameters values {(𝜙₀=𝑋)∧(𝜙₁=0)} and {(𝜙₀=0)∧(𝜙₁=1), respectively, and plugging it into (C.13). In a similar way, setting {(𝜙₀=−𝑋)∧(𝜙₁=1)} in (C.13) and substituting the expression for 𝑐^BSM_𝐸(𝑆_𝑡,𝜏) into (2.15) yield (3.9).