PRICING AND APPLICATION OF REAL R&DOPTIONS IN A TWO-FACTOR JUMP MODEL

Wilson Koh and Dean Paxson ∗

Manchester Business School, University of Manchester, UK

March 1, 2006

Abstract

We first derive explicit formulas for a real perpetual American call option

assuming the asset price follows a double-exponential jump diffusion process.

Similar analytical computation is extended and applied to a research and de-

velopment (R&D) effort with two stochastic factors. Both project value and

investment costs summarise our uncertainties in creating investment opportu-

nities. The presence of Levy jumps underline significant positive and negative

impacts on the project future cash flows and therefore on the investment de-

cision. We then apply the theoretical models to an empirical biotechnological

R&D case scenario. In our gene-based drug application, we find that a R&D

project that experiences mixed-exponential jumps encourages postponement of

optimal timing to entry compared to its counterpart that follows a Poisson jump

process.

Keywords: Real options; jump diffusion process; Levy process; double-exponential;two-factors.

JEL Classification Code: D81, G31, O32

∗Both authors are from Manchester Business School, University of Manchester, Booth Street West,M15 6PB, UK. Corresponding author: wkoh@man.mbs.ac.uk

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A-PDF MERGER DEMO

1 Introduction

Research and development (R&D) in pharmaceuticals and biotechnologies frequently

involves upward jumps or downward jumps, i.e. drugs can turn into mega-selling block-

buster products or alternatively, pipeline compounds suffer clinical trial failures and

withdrawal from the markets. Hence real R&D investment appraisal should rely on

a model focusing on these aspects, rather than geometric Brownian motion (gBm)

alone. We know that asset prices do not follow a Gaussian pattern1. Even the world of

real assets is not Gaussian2. For pricing simple financial options and path-dependent

options under a wide class of jump diffusion processes and Levy process, we refer to

Boyarchenko and Levendorskiı (2002) [12], Boyarchenko and Levendorskiı (2002a) [10],

Mordecki (1999) [42], Mordecki (2002) [43], Kou and Wang (2004) [35], Sepp (2004)

[51], Schoutens (2003) [48], Carr and Madan (1998) [15] and Carr and Wu (2003) [16].

They have led us to new theorems on the puzzles behind the imperfection of the Black-

Scholes model3. However, most of the real option models that apply a non-gBm process

do not extend beyond a mean-reversion technique or a log-normal jump diffusion. Most

have nothing to say about the apparent fat tails of probability distribution functions4.

This paper extends a class of exponential jump-diffusion process with stationary in-

dependent increments or infinitely divisible distribution i.d.d in continuous time to a

real option context. The most famous jump-diffusion framework applied to real op-

tion models is perhaps Merton (1976) [40]5. Although Merton (1976) [40] is good for

modeling events of surprises and a better option-pricing model than Black and Scholes

(1976) [9], it is by no means more realistic because of its Gaussian jumps. Exponen-

tial jump processes proposed by Boyarchenko and Levendorskiı (2002a) Boyarchenko

and Levendorskiı (2002a) [10], Mordecki (2002) [43] and Kou and Wang (2004)[35] are

not only applicable to real projects with sudden swings in returns but also do not as-

sume that returns conform to normal kurtosis and skewness. While Boyarchenko and

1 Mandelbrot (1963)[38] shows that stock price returns do not have a normal distribution. Thisbecame a precedent for subsequent research on fat tails, many which revolve around the imperfectionof Black-Scholes option pricing framework. Prime examples are Madan, Carr and Chang (1998) [37]and Carr and Madan (1998) [15].

2 Yang and Brosen (1992)[56] and Deaton and Laroque (1992) [23] show evidence that commodityprices exhibit significant skewness and kurtosis.

3Heston (1993)[30], Carr, Geman, Madan, Yor (2003)[14] and Sepp (2004) [51] model with sto-chastic jumps in option prices which may partially explain implied volatility smiles generated withthe Black-Scholes model.

4An exception is Boyarchenko and Levendorski (2002b) [11] who applied fat-tailed models to realoption cases.

5Several authors apply this jump-diffusion process in real options including Brach and Paxson(2001) [13], Clewlow and Strickland (2000) [20], Martzoukos (2003) [39] and Cartea and Figueroa(2005) [17].

2

Levendorskiı (2002a) [10] and Mordecki (1999) [42] models can only assume either up-

ward or downward jumps, Kou and Wang (2004) [35] propose log-double-exponential

jumps in both directions and provide analytical solutions for the optimal stopping time.

Our contribution in this paper is two-fold. Firstly, we derive an explicit formula of an

investment call option in a double-exponential jump-diffusion framework. Secondly,

we extend our solution to a real R&D investment option model with two stochastic

factors, namely asset value and investment cost. We then analyse the impact of such

exponential jumps on the optimal investment decision of a R&D effort to create new

drugs. We select a R&D option because R&D ventures are susceptible to sudden and

drastically-bad or extremely-good clinical outcomes6. Hence, they pose potential op-

portunities to make or break the firm’s fortune through unexpected changes in project

future cash flows. Two-factor real option models in a jump-diffusion setting can be

found in Villaplana (2003) [54], that considers a mean-reverting jump diffusion which

is non-Levy. Also, while Boyarchenko and Levendorskiı (2002a) [10] and Boyarchenko

and Levendorskiı (2002b) [11] apply Wiener-Hopf factorisation to attain closed-form

solutions for their Levy-driven jump model, we compute our analytical price formula

through assuming a Laplace transform time-domain.

The rest of the paper is organised as follows. In Section 2, we recall the idea and basic

properties of a double-exponential jump process. The derivation of an analytical form

of a perpetual call option with double-exponential jumps is presented in Section 3. We

then consider a real R&D option model where asset values and investment cost fol-

low different stochastic processes. This option model also includes double-exponential

jumps for possible up and down events. Closed-form solutions for value functions and

triggers are presented in Section 4. We consider some sensitivity analyses of the R&D

jump model in Section 5. In section 6, we provide a case study on investment options of

a biopharmaceutical company Human Genome Sciences (HGSI). We analyse potential

occurrence of events and investment opportunities in its subsequent phases of pipeline

projects. Section 7 concludes.

6Numerous statistical studies of clinical trials confirm this observation. Examples are DiMasi etal. (1995a) [25], DiMasi et al. (1995b) [26] and Tufts CSDD [53].

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2 Double-Exponential Jump Diffusion Process and

Properties

2.1 Levy processes and jump-diffusion processes.

A Levy process is a process with stationary and independent increments which is

based on a more general distribution than the normal distribution. In order to rep-

resent skewness and excess kurtosis, the distribution in a Levy process has to be infi-

nitely divisible 7. For every such infinitely divisible distribution, there is a stochastic

process X = Xt, t ≥ 0 called a Levy process, which starts at zero and has inde-

pendent and stationary increments such that the distribution of an increment over

[s, s + t], s, t ≥ 0.,i.e. Xt+s − Xs has (φ(u))t as its characteristic function. General

reference works on Levy processes are Bertoin (1996) [8], Sato (1999) [47] and Apple-

baum (2004) [3].

Generally a Levy process may consist of three independent components, namely a lin-

ear drift, a Brownian diffusion and a pure jump. The latter is characterised by the

density of jumps, which is called the Levy density. We denote this density by f(x).

The Levy density has the same mathematical requirements as a probability density

except that it does not need to be integrable and must be zero at the origin. Jumps

of sizes that occur according to a compound Poisson process are hence a type of Levy

process 8. The double-exponential jump diffusion process is a class of this type. We

discuss it in the next sub-section.

A Levy process can be completely specified by its moment generating function E[euXt ] =

eκ(u)t. κ is the Levy exponent. The equivalent description is in terms of the character-

istic exponent, ζ, definable from the representation E[eiξX(t)] = e−ζ(ξ)t, where i =√−1

and hence κ(u) = ζ(−iu). If the jump component is a compound Poisson process, then

the Levy exponent is

κ(u) = uγ2 +σ2

2u2 +

∫ +∞

−∞(euy − 1)f(dy) (1)

where σ2 and γ are the variance and drift coefficients from the Gaussian component

7Suppose φ(u) is the characteristic function of a distribution. If for every positive integer n, φ(u)is also the nth power of a characteristic function, the distribution is infinitely divisible.

8The proofs are shown in Cont and Tankov (2004) [21], pp.70-75.

4

and f(dy) satisfies ∫

<0min1, |y|f(dy) < +∞ (2)

The cumulant characteristic function ζ(u) is often called the characteristic exponent,

which satisfies the following Levy-Khintchine formula,

ζ(u) = iγu− 1

2σ2u2 +

∫ +∞

−∞(eiuy − 1− iuy1|y|<1)ν(dy) (3)

where the triplet of Levy characteristics is given by (γ,σ2,ν).

2.2 Double-Exponential Properties

Here we mainly consider the asymmetric double-exponential probability density func-

tion proposed by Kou and Wang (2004) [35]:

$(J) = $−(J) + $+(J) = pη+e−η+J1J≥0 + qη−eη−J1J<0 (4)

where 1 > η+ > 0 and η− > 0. So 1η+ and 1

η− are the means of the two exponential

distributions with positive and negative jump sizes, respectively. Constants p and q

behave as p > 0 and q > 0 and they represent the probabilities of positive and negative

jumps respectively, so p + q = 1. Note that η+ < 1 is imposed to ensure that the

underlying variable has finite expectation.

We note that

ε[eΦJ ] =

∫ +∞

−∞eΦJ$(J)dJ =

pη+

Φη+ − 1+

qη−

Φη− + 1− 1 (5)

provided that − 1η− < Φ < 1

η+ . A simple mathematical evaluation yields

% = E[eJ − 1] =pη+

η+ − 1+

qη−

η− + 1− 1 (6)

The double-exponential diffusion model works better with jumps in asset prices than a

log-normal type because it uses a double-exponential type distribution to chart the first-

passage time of a stock price to a flat boundary, b. Under a general jump diffusion model

like Merton (1976) [40], a process X(t) might hit the boundary exactly or overshoot

the boundary by X(τb) − b, where τb is the first time that the process crosses the

boundary. This is known as the first stopping time9. We do not know where X(t) will

9Concepts of stopping time for a continuous process can be found in Shreve (2004) [52],pp.340-

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end up; we need to get an exact distribution of the overshoot, X(τb)− b, in particular

P [X(τb) − b = 0] and P [X(τb) − b > 0]. Secondly, we also need to determine the

dependence structure between the overshoot and the first passage time, τb, as well

as the dependence structure between the overshoot and terminal value X(t) under

the reflection principle. Both concerns can be solved by using an exponential-type

distribution of a certain jump size, due to the memoryless property of the exponential

distribution10.

2.3 Double-Exponential Jump Diffusion Process

It is well-known that all jump diffusion models are incomplete. Because of this, we will

assume risk-neutrality throughout the paper. We will assign a risk-neutral probability

measure, following the approach of Kou and Wang (2004) [35]. Under the double-

exponential jump diffusion model, the asset prices behave according to the following

stochastic process:

dS(t) = µS(t)dt + σS(t)dW + S(t)d(

N(t)∑i=1

(Vi − 1)) (7)

where W is a standard Wiener process, N(t) is a Poisson process with rate λ andViare non-negative random variables with a double exponential distribution described in

(4). Under a particular risk-neutral measure P ∗, the new jump diffusion process is

given by

dS(t) = (r − λ∗%∗)S(t)dt + σS(t)dW ∗(t) + S(t)d(

N(t)∑i=1

(Vi − 1)) (8)

when we let S(t) → X(t) = S(t)S(0)

. X(t) is then the new return process expressed by

X(t) = (r − 1

2σ2 − λ∗%∗)t + σW ∗(t) +

N∗(t)∑i=1

Υ∗i (9)

345. A derivation of first passage time for double-exponential distribution can be found in Kouand Wang (2003) [34]. Alili and Kyprianou (2004) [2] have linked known identities for first passagetime and overshoot of a Levy process with solutions of perpetual American put and smooth-pastingprinciples posed by authors like Gerber and Shiu (1998) [29], Boyarchenko and Levendorskiı (2002a)[10], Chesney and Jeanblanc (2004) [19], Mordecki (1999) [42] and Chan (1999) [18].

10For proof on deriving the memoryless property of a exponential distribution, refer to Cont andTankov (2004) [21] pp.44-48 and Kou and Wang (2003) [34] pp.508-509. Both include the conversionto conditional memoryless property of jump diffusion from exponential random variables.

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where %∗ is defined like in (6), where risk-neutrality is not considered.

If X is a Levy process on < with the characteristic exponent given by (3), then we have

an infinitesimal generator, denoted as L defined for all twice continuously differentiable

functions ψ(x). This infinitesimal generator is an integro-differential operator which

acts as follows:

Lψ(x) =1

2σ2ψ

′′(x) + γψ

′(x) +

∫ +∞

−∞[ψ(x + y)− ψ(x)]f(dy) (10)

Because (10) applies to a general Levy process, we rewrite the equation for a more

specific infinitesimal generator of a double-exponential jump process11:

Lψ(x) =1

2σ2ψ

′′(x) + (r − 1

2σ2 − λ∗%∗)ψ

′(x) +

∫ +∞

−∞[ψ(x + J)− ψ(x)]$∗(dJ) (11)

where the process x follows (9), and $(dJ) as well as % are from (4) and (6) respectively

except that they now follow a risk-neutral probability measure P ∗.

In our study we will be solving for other partial integro differential equations or PIDEs

in Laplace domain. The Laplace transform is one of the classical methods for solving

ODEs, PDEs and integral equations. The idea is to transform the problem to a space

where the solution is relatively easy to obtain. Fortunately for our case of perpetual

American options, the option prices already exist in Laplace domain, as represented

by (11). Because we are not required to invert the Laplace transforms to obtain option

prices, we can attain closed-form solutions of the option values12.

3 Pricing Perpetual Vanilla Call Options under Dou-

ble Exponential Jumps

First we solve for the unbounded problem of vanilla options and next consider the

bounded problem of a perpetual vanilla call with dividends. To price vanilla options

under a double-exponential jump diffusion, we will need the following formula, pre-

sented in Laplace domain.

11The difference between this and (10) is that we are under a risk-neutral probability measure.12Note that the Laplace transform of ψ(x, τ) is defined by

U(x, ρ) = L[ψ(x, τ)] =∫ ∞

0

ψ(x, τ)eρτdτ (12)

where ρ is a transform variable with a positive real part, <ρ > 0. (11) is also in Laplace domain.

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Lemma 1 The equation

f(x) =1

2σ2x2 + µx + λ(

pη+

η+ − x+

qη−

η− + x− 1)− α = 0 (13)

for α > 0, has four real roots: β1, β2, -β3, -β4, such that

0 < β1 < η+ < β2 < ∞, 0 < β3 < η− < β4 < ∞ (14)

Proof is given in Appendix A.1.

Note that

κ(x) =1

2σ2x2 + µx + λ(

pη+

η+ − x+

qη−

η− + x− 1) (15)

in (13) is the Levy exponent described in Section 2.1. For call options with double-

exponential jumps, we will be particularly interested in β∗1 and β∗2 which are defined as

the unique roots of

f(β∗1) = 0, f(β∗2) = 0, 0 < β∗1 < η1 < β∗2 < ∞ (16)

as α → 0.

Taking our infinitesimal generator shown in (11) and equating to αψ(x), we obtain the

following ordinary integro differential equation or OIDE:

1

2σ2ψ

′′(x) + µψ

′(x)− αψ(x) +

∫ +∞

−∞[ψ(x + J)− ψ(x)]$∗(dJ)

= −maxψ[ex − 1], 0 (17)

where µ = r − δ − 12σ2 − λ% and δ is the dividend yield.

To simplify notation, we shall drop the superscript * in the parameters, i.e., using

λ, %, $, p, q, η+, η− instead of λ∗, %∗, $∗, p∗, q∗, η+∗, η−∗. The assumption is that all

the processes and parameters henceforth are under the risk-neutral probability mea-

sure P ∗. The solution to the above equation, under the boundaries of a call payoff,

i.e.ψ(x) = max(x− x∗, 0), is specified by the following:

Proposition 1 The value of a vanilla call option with dividends under a double-

exponential jump diffusion is given by

ψ(v) =

Avβ1 + Bvβ2 if v < v∗,

v −K if v ≥ v∗.(18)

8

where v, v∗, A and B are defined as

v = ex,

v∗ =Kβ1β2(η

+ − 1)

η+(β1 − 1)(β2 − 1),

A =(η+ − β1)

[Kβ2(β2 − 1) + η+β2

[v∗(β2 − 1)−Kβ2

]+ η+2(

v∗(1− β2) + Kβ2

)]

v∗β1η+(η+ − 1)(β1 − β2)(1 + η+ − β1 − β2)

B =(η+ − β2)

[Kβ1(β1 − 1) + η+β1

[v∗(β1 − 1)−Kβ1

]+ η+2(

v∗(β1 − 1)−Kβ1

)]

v∗β2η+(η+ − 1)(β2 − β1)(1 + η+ − β1 − β2)

The proof is given in Appendix A.213.

Figure 1 shows the graphs of the value of a perpetual American call option with divi-

dends versus its parameters S, K, λ, r, σ, p, q, 1/η+, 1/η−, δ and α which represent the

underlying stock or asset price, exercise price, jump rate, riskless interest rate, price

volatility, probability of upward jump, probability of downward jump, mean of positive

jump distribution, mean of negative jump distribution, dividend yield and probabil-

ity density value of the double-exponential distribution function respectively. We also

show the trigger, v∗ sensitivity to σ. The default parameter values are: S = 100,

K = 100, λ = 3, r = 0.06, σ = 0.5, p = 0.3, 1η+ = 0.03, 1

η− = 0.02, δ = 0.03 and

α = 0.5.

[Figure1]

Our initial results suggest that a call option with double exponential jump model

behaves like a normal vanilla call option. The option value increases as asset price in-

creases. Like a model with fundamental jump diffusion characteristics, our option value

increases as λ or intensity of ’sudden events’ increases. The option value also appears

13An explicit result similar to Eq.(18) is independently derived by Mordecki (2002) [43] for per-petual call options with double-exponential jumps. However the difference between our method andMordecki (2002) [43] is that we calculate our result directly by an ODE method through assumingLaplace transform domain. Mordecki (2002) [43] shows results indirectly by first deriving for generalrepresentations of general Levy processes. Also Mordecki (2002) [43] proof is less detailed and lesseasy to follow. Our computation method, as shown in Kou and Wang (2004)[35] can also be appliedto pricing path-dependent options.

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stable and increasing with positive increments in price volatility. As the likelihood of

a ‘good news’ or ‘good jump’ increases, the option also becomes more valuable. The

opposite also holds for ‘bad-news’. Note that our option value is also an decreasing

function of both η+ and η−, where 1η+ and 1

η− refer to the means of positive jumps

and negative jumps respectively. This is especially interesting for the case of η− as

it sounds counter-intuitive. One reason can be the effect of dividend payments. The

other is that the risk-neutral drift is also dependent on η− 14. Lastly, the trigger value

S∗ shows a positive relationship with volatility; this is consistent with the classical

result.

4 Pricing Real R&D Options With Two Stochastic

Factors

4.1 The investment problem

Real option studies are most commonly written in a continuous framework for a single

underlying stochastic variable. However the existence of good news, bad news, booms

and busts in investments generate discontinuities in returns. A R&D-based investment

often has high instability in outcomes. Some previous notable studies on discontinu-

ity effects of R&D investments under uncertainty are Schwartz and Moon (2000) [49]

and Schwartz and Zozaya-Gorostiza (2003) [50]. Their single-jump characteristics are

modelled with Poisson processes and their two stochastic factors diffusion problems do

not present analytical solutions. Recently Barrieu and Bellamy (2005) [6] applied a

special Levy jump process in a real option valuation context. They considered only

one jump size, following an arbitrary probability density and a benefit/cost ratio but

not separate stochastic factors.

We also consider a moneyness ratio which is driven by two stochastic factors. This

investment ratio acts under the process of two-sided Levy jumps. We first obtain the

first hitting time to invest when the ratio crosses the optimal boundary 15. We later ob-

tain option values based on the paths of the ratio in exponential jumps. We eventually

draw results from the robustness of optimal investment decision under these conditions.

Corollary 1 Consider a filtered probability space (Ω, z, zt, ℘). The investor has to

14This reason is also observed and given by Merton (1976) [40].15See Kou and Wang (2003) [34] and Kou and Wang (2004) [35] for more explanation on the first

passage under the double-exponential concept.

10

decide whether to undertake a given investment project and, if so, when it is optimal

to invest. We assume that the investor has no time limit to make the decision and the

time horizon of the research process is infinite. The investment opportunity value at

time 0 is specified by

V0 = supLE[e−µτL(ΨτL

− 1)+]

where E denotes the expectation with respect to the prior probability measure ℘, (Ψt, t ≥0) is the process of the moneyness ratio, L is the boundary level and τL is the first hit-

ting time of the frontier L.

L∗ is the optimal moneyness conditional when the process Ψ hits the boundary op-

timally. This optimal stopping time is given by

τ ∗L = inft ≥ 0; Ψt ≥ L∗

Hence before the moneyness ratio Ψ reaches the optimal level of L∗, it is optimal for

the investor to wait before undertaking the investment. As soon as Ψ goes beyond this

threshold, it is optimal to invest16.

4.2 The model

We consider an investment opportunity in a R&D project with no restriction on the

time to complete. This problem is similar to the study at time 0 of a perpetual Amer-

ican call option written on the project value. In reality, paths of estimated value and

investment cost in high-risk projects are not deterministic. While products of R&D

may not carry market value until they are launched, we assume that there is an esti-

mated value of the asset upon successful completion of the project.

The stochastic framework of the project is characterised by underlying processes for

asset values and investment cost. Both variables are assumed to follow different but

possibly correlated stochastic variations. We consider below that while investment cost

follows a typical gBm process, the asset values follow a mixed gBm and an exponential

Levy-type jump diffusion processes. Let Vt represent the asset values and Ct the

16For the proof of Corollary 1, see Darling et al (1972) [22] or Mordecki (1999) [42].

11

investment cost sustained by the project. These processes satisfy

dVt = µvVtdt + σvVtdzv + Vtd(N(t)∑

i=1

(Ui − 1))

(19)

dCt = µcCtdt + σcCtdzc (20)

where µv = γv +λ%17 and µc are the expected drift trends; and σv and σc are the volatil-

ities; and dzv and dzc are the increments of Wiener processes of Vt and Ct respectively.

In the variable Vt, we have the additional non-Gaussian jump term where (Nt)t≥0 is

the Poisson process counting the jumps of V , and Ui are jump sizes behaving like i.i.d.

variables. To define the jump process completely, we assign the distribution of jump

sizes, $(J) to follow an asymmetric exponential with a density of the form shown

in Eq.(4)18. Hence the process Vt is a Levy process of a special jump-diffusion type.

Finally, the two stochastic variables in the Gaussian parts may be correlated with a

dependence coefficient ρ which is deterministic 19. The stochastic process of the invest-

ment cost and the Poisson process in the jump term of V , however, are uncorrelated 20.

We refer to the variable ω = VC

as a benefit-cost ratio or ’moneyness’ of the project

and we assume that this variable has a return process given by

m(t) = eω =

(r − 1

2σ2

m − λ%

)t + σmZm(t) +

N(t)∑i=1

Ui (21)

where m(0) = 0. m(t) has a moment-generating function given by E[emM(t)] = eG(m)t,

where the characteristic exponent function, G(m) is defined as

G(m) =1

2σ2m2 + µm + λ

(pη+

η+ −m+

qη−

η− + m− 1

)− α (22)

where there are four roots of βm=1,2,3,4 for G(m) = α as proven for Eq.(13) and Eq.(14).

We let F (V, C) be the value of the investment opportunity. Applying contingent claims

analysis, we consider a portfolio that consists of a long position in one unit of investment

17The drift rate of process increases with jumps intensity and decreases with jump size. Such abehaviour is rather logical. γv is the drift coefficient without the jump term.

18It is particularly important to specify the tail behaviour of $(J) early and correctly depending onthe perception about behaviour of extremal events, because the tail behaviour of the jump measuredetermines to a large extent the tail behaviour of the probability density of the process.

19The time subscripts of Vt and Ct will be suppressed from now on.20Hence there is no risk premium associated with these uncertainties.

12

opportunities, F (V,C) and a short position consisting of ∆1 and ∆2 units in output

and capital respectively. We assume that the investor is risk-neutral 21. Applying

Ito’s lemma, F (V, C) must satisfy the following second-order elliptic partial differential

equation:

Max

[1

2

∂2F

∂V 2σ2

vV2 +

1

2

∂2F

∂C2σ2

cC2 +

∂F

∂V ∂CV Cρσvσc + (r − µv)V

∂F

∂V

+ (r − µc)C∂F

∂C− rF + λ

∫ ∞

−∞

(F (V + J)− F (V )

)$(J)∂J

]= 0 (23)

where r denotes the risk-free rate, J = log(U), Ui is the sequence of i.i.d. variables

and $(J) the probability density function of a double-exponential distribution22.

Wilmott et al. (1995) [55] acknowledges that partial differential equations of elliptic

characteristics often appear in multi-factor models and have remote chances of provid-

ing closed-form solutions. This is because many do not have analytical characteristic

functions. However, the similarity method can simplify some partial differential equa-

tions to second-order ordinary differential equations which may offer less complicated

solutions 23. In our case, we take advantage of the explicit form of the characteristic

Levy exponent to derive our closed-form solutions.

Suppose we imply that F (V,C) = CΨ(m). After appropriate substitutions, Eq.(23)

can be written as:

1

2m2d2Ψ(m)

dm2(σv + σc − 2ρσvσc) + m

dΨ(m)

dm(µc − λ%)− µcΨ(m)

+ λ

∫ ∞

−∞

(Ψ(m + J)−Ψ(m)

)$(J)d(J) = 0 (24)

We show derivation of the above equation in Appendix A.4. For F (V, C) → Ψ(m) and

following the m(t) return process, we re-write Eq.(24) as:

1

2σmΨ

′′(m) + µmΨ

′(m)− µcΨ(m) + λ

∫ ∞

−∞

(Ψ(m + J)−Ψ(m)

)$(J)d(J) = 0 (25)

21Details on equivalent risk-neutral valuation can be found in Dixit and Pindyck (1994) [24]. Theassumption of risk neutrality can be relaxed by adjusting the drifts of V and C to account for a properrisk premium.

22The derivation of Eq.(23) by contingent claim arguments is in Appendix A.3.23Dixit and Pindyck (1994) [24], Quigg (1993) [46] and Paxson and Pinto (2005) [44] are some

examples.

13

where σm = σv + σc − 2ρσvσc and µm = µc − 12σ2

m − λ%− γv.

The solution to this two-factor OIDE Eq.(25) is specified by the following:

Proposition 2 The value of a perpetual investment option under a double-exponential

jump diffusion is

Ψ(υ) =

Pυβ1 + Qυβ2 if υ < υ∗,

υ − C if υ ≥ υ∗.(26)

where υ, υ∗, P and Q are defined as

υ = em

υ∗ =Cβ1β2(η

+ − 1)

η+(β1 − 1)(β2 − 1),

P =(η+ − β1)

[Cβ2(β2 − 1) + η+β2

[υ∗(β2 − 1)− Cβ2

]+ η+2(

υ∗(1− β2) + Cβ2

)]

υ∗β1η+(η+ − 1)(β1 − β2)(1 + η+ − β1 − β2)

Q =(η+ − β2)

[Cβ1(β1 − 1) + η+β1

[υ∗(β1 − 1)− Cβ1

]+ η+2(

υ∗(β1 − 1)− Cβ1

)]

υ∗β2η+(η+ − 1)(β2 − β1)(1 + η+ − β1 − β2)

Z(βn) =1

2β2

n(σv + σc − 2ρσvσc) + βn(µc − 1

2σ2

m − λ%− γv)

+λ

(pη+

η+ − βn

+qη−

η− + βn

− 1

)= µc

or

H(βn) = Z(βn)− µc = 0

where βn=1,2 are the two positive roots of the characteristic exponent function. Note

P ≥ 0 and Q ≥ 0.

The proof for this proposition is given in Appendix A.5. Following this main result,

we can optimise investment in event of positive or negative discontinuities. In the next

section, we evaluate an investment opportunity using the model and illustrate some

relationships between key parameters and their optimal hitting time.

14

5 Value of the investment opportunity and optimal

hitting time

5.1 Main results

In this section, we discuss the value of a project according to variabilities of different

factors and decisions behind the investment problem of the previous section. The in-

vestment problem assumes that the project has a variety of jump sizes to consider.

Therefore, agents do not have to guess whether the jump will be positive or negative

because both probabilities are considered in a double-exponential probability density

function. This may be the case of a loss of patent due to high number of deaths of

patients, or alternatively overwhelming global sales. New competitors can also enter

the market attracted by bright prospects or existing competitors may exit the race for

a patent due to failure in experiments or lack of funds.

[Figure2]

Figure 2 shows the interactions between the project values and the two factors, asset

values and investment cost. The value of the project is equal to zero when the project

is worthless and highest when the cost to completion is close to zero. The value of the

investment project is increasing in the asset value and decreasing with investment cost.

It is easier to appreciate these relations in a two-dimensional graph as shown in Figure 3.

[Figure3]

Note that the investor may not want to invest when NPV turns positive. Instead he

prefers the point of coincidence between option value and NPV. Such a case perfectly

illustrates what MacDonald and Siegel (1986) [36] referred to as the value of waiting

to invest.

5.2 Impact of optimal triggers in bullish and bearish R&D

ventures

The model has made certain assumptions about the probability of jump sizes, i.e.double-

exponential. Figure 4 shows the impact on the investor’s critical level to invest when

he knows exactly the size of the jump. Assuming that p is the positive jump probabil-

ity and q = 1 − p is the probability of no jump or a negative jump, we can see that

15

the critical level to invest in the project increases with higher chances of jumps. This

also means that a rational investor will not mind holding back a while longer before

investing if he knows the project has some positive results ahead. The jump intensity,

λ, also pushes trigger values and option values higher. In R&D under uncertainty,

this confirms the adage of ‘the more discoveries or successful clinical trials, the more

valuable the project’.

When q increases, the trigger level falls and option values decrease. Hence the reverse

holds true such that if the investor feels pessimistic about future results in the process

of R&D, he may consider investing sooner to take advantage of the current neutral

position before any bad news disrupts the progress of the project.

[Figure4]

[Figure5]

Figure 5 illustrates that critical investment values decrease with increasing η+. Recall

that 1η+ is the mean of the exponential distribution representing upside jumps while 1

η−

is the mean of the second exponential distribution representing downside jumps. We

concentrate on η+ because it has a direct effect on our trigger function. A decreasing

trend in η+ hence implies an increasing trend with the mean itself. A R&D project to

develop drugs is worth more if the expected mean of success probability is high. Not

only we find this to be true, the trigger values also suggest deferred investment, like

any option with greater values. Jumps of higher amplitudes will drive option prices

and hence optimal exercise triggers higher as illustrated in Figure 5.

[Figure6]

In most real option models, trigger values tend to increase with volatility. Uncertainty

increases the value of a project as shown in Figure 6. Increasing asset value volatility,

σv results in a higher trigger. Project managers may wait and hold back the decision to

invest because of rising uncertainty. The graph also show positive correlation between

trigger values and drift of investment cost, µc. This implies that a positive upward

drift in cost will drive the threshold to invest higher because of the higher premium

16

paid to enter the project.

The sensitivity of option values to the correlation between asset values, v and invest-

ment cost, C is presented in Figure 7. We can see that the correlation of the two

parameters behave in an opposite way to our project values. This result is hardly

surprising, given that higher correlation causes option values to become lower, partly

due to investment costs increasing as project values increase. Hence option value is

dependent on the direction and the intensity of the relationship between value and

investment cost. Low correlation results in greater option values. We also consider the

correlation effects on option values when favourable jump-size probabilities are differ-

ent. Interestingly in this case, higher probability does not result in significantly larger

option values. Project managers may not have a great incentive to wait to invest if

their R&D programme has a high probability of acquiring good news throughout the

R&D window period.

[Figure7]

6 Applying the model

Our study aims to value an alternative drug-delivery model in multiple specialist dis-

ease areas through personalised medicine, diagnostics and preventions. When the suc-

cess rate of discovering one-size-fits-all drugs diminishes in the future, the healthcare

providers can turn to research of genetically-engineered drugs or nichebusters 24. With

the formation of credible genome databases and the identification of genes and pro-

teins, pharmaceutical R&D is moving away from the old ‘symptom-indication-based’

approach. The growing ability to more accurately define the target population through

DNA screening will help to make medicines more efficient and largely eliminate the

side-effects seen when a drug is applied to a less-suitable patient. It also means that

a previously rejected or strongly restrained drug can be reconsidered for the right pa-

tient group identified on the basis of their genome (Mertens, 2005). Biotechnologies

in pharmagenomics are the most likely beneficiaries of this important market and the

race for new therapies is already on, in particular in the field of oncology where cancer

always involves changes at the level of DNA.

For the purpose of this case study, we select a leading biotechnology firm in gene-

therapy, Human Genome Sciences (HGSI) to calibrate to a R&D-induced jump option

24For more information on personalised medicine, refer to Mertens (2005) [41].

17

model. HGSI operates as a biopharmaceutical company with a pipeline of novel pro-

tein and antibody drugs. The company’s drugs in clinical development include those

targeted for the treatment of rheumatoid arthritis, chronic hepatitis C, hematopoietic

cancers, HIV/AIDS and anthrax infection. It focuses its R&D efforts on novel protein

and antibody drug candidates discovered through genomic-based research and albumin

fusion technology. In the area of protein research, HGSI has isolated the templates of

possibly more than 95% of all human genes. Of these genes, about 75%-80% are fully

functional as they contain the complete compatibility to produce corresponding pro-

teins. HGSI has so far determined the full-length sequence of over 100,000 genes. Our

application of ’jumps’ in clinical discovery may add new dimensions in gauging rare

successful events for this type of biotech research.

6.1 Data and estimation of parameters

Some of the data related to R&D necessary to estimate our model can have propri-

etary rights and, consequently are hard to obtain. Our parameters are estimated using

data collected from stockbroker reports, the company annual report, DataStream, the

Food and Drug Administration (FDA), Human Genome Project (HGP), Tufts Centre

for the Study of Drug Development (CSDD) and PharmaProjects database(PJB). The

stockbrokers’ report provided by Credit Suisse First Boston (CSFB) [4] performed a

weighted valuation on HGSI stock and found that 90% of its value is focused on five

major R&D programs. Table 1 shows the five programs in the pipeline and their re-

spective stages.

[Table1]

The estimation of project values corresponds to the expected present value of the dis-

counted total revenue from sales of the three groups of gene-based drugs. We used

estimated sales from a stockbroker report [4] and consider annual data from 2005 to

2015. For 2016 on, the value of 2015 is considered as a perpetuity. The estimated in-

vestment costs are extracted from the same stockbroker report, CSDD analysis report

[27] and PJB analysis report [45]. We only consider R&D expenditures and exclude

marketing cost at this point. Table 2 shows the gross project values and present values

of R&D cost.

[Table2]

18

We estimate for other key parameters in our two-factor double exponential jump model

(TFDEJ) shown in section 4, by calibrating to market option prices of HGSI. We min-

imize the sum of square root of the difference in both sets of option prices as shown in

Bakshi, Cao and Chen (1997) [5]. To model as realistically as possible the perpetuity

in our TFDEJ model, we choose HGSI call options with maximum time to maturity

of two years. We collected all call option prices of HGSI traded on 15th February

2006 and obtained Table 3 of implied parameters. The same table also includes some

assumed and computed values.

[Table3]

[Table4]

Table 4 presents the results of our computation with the TFDEJ model. HGSI project

value stands at $1.81 billion and the trigger value Y ∗ is 1.98. From this trigger value,

we compute the critical asset value of the project to be $1.4 billion which is below the

expected discounted asset value we found based on analysts’ predictions. As HGSI

projects are all in Phase II of clinical studies, the lower trigger asset value suggests

that HGSI should have entered these trials into Phase III and Phase IV earlier because

$1.4 billion is the optimal level of investment into the next stage. Hence this is an

in-the-money option into stage 3 of R&D that HGSI had yet to cash in. One rea-

son could be because of the poor pre-clinical and Phase I results of these compounds

that slow down progress into later stages of R&D. Hence actual and expected R&D

phase lengths play important roles in optimsation of options within an R&D pipeline

of projects. Evidently, HGSI share price had increased from $10.1 in mid-2005 to a

recent high of $12.5. This shows market optimism in the future development of HGSI

products. Our implied jump rate λ and upside probability p shown in Table 3 also

indicate optimism in successful future clinical tests of these programs.

We also compare our results across different clinical cost estimates. We use CSDD and

PJB expected R&D expenses for clinical Phase II, III and IV, all discounted at 11%.

Differences in project values can be seen in Table 4. Cost estimates from Tufts CSDD

will rank HGSI R&D program as the most valuable. This is not unlikely because the

DiMasi et al. (2003) [27] estimate of $802m for development of an average drug is a

19

lower figure than the $868m cost quoted by Adams and Brantner [1]. The latter found

that different classes of drugs have very different R&D cost and time to completion of

R&D due to individual pharmaceutical company policies. Unfortunately these specific

costs are not always available to the public for further analysis.

Lastly, we compare tractability of our TFDEJ model with some other perpetual Amer-

ican option models. The Dixit and Pindyck (DP) perpetual model is an investment

call option with stochastic variable V . Dixit and Pindyck jump model (DPJ) is a com-

bined gBm and Poisson jump process model that allow V to take downward jumps only.

Quigg (1993) [46] model is a perpetual two-factor model in continuous time. Mordecki

(1999) [42] model presents a single-exponential Levy jump model for a perpetual call

option with one-sided jumps (can only jump up or down). We provide a legend of

computation (analytical) techniques for each of the comparative models.

[Table5]

From Table 5, we can see that our models with jumps i.e. TFDEJ, DPJ and Mordecki

have almost the same project value which is our net present value. We already know

that HGSI has in-the-money projects and according to these results, perpetual invest-

ment option models with no jumps can result in higher project values. This is evident

in both the DP and Quigg figures. Optimal trigger levels are highly sensitive to jumps.

The presence of only downside jumps, i.e. DPJ suggests earlier investment. A mixture

of upside and downside jump risks in TFDEJ also suggest earlier investment, followed

by only upside jumps of the Mordecki model. Both DP and Quigg models are not

exposed to any jump risks and both exemplify higher trigger values, hence suggesting

that it is worthwhile to wait further before investing if there are no expectations of

rare events occurring.

7 Conclusion

Drug product successes and failures are usually prevalent during clinical studies and/or

after they are launched in the market. Investment cost, asset returns, optimal operat-

ing policy and probabilities of success or failure are key issues and primary concerns of

starting a R&D venture. To date, the existence of both positive and negative effects

caused by extreme rare events in a real clinical study environment has either been

rarely considered or not considered at all. The aims of this paper are to investigate

impacts of mixed jumps in a case of healthcare R&D.

20

We study the real options in a R&D project associated with uncertainties in cost, pay-

off from the project and possibility of ’jumps’ which may either end the investment

opportunity or boost its prospects. This specific model evaluates investments made

in risky projects through a twin-exponential jump distribution. The fat-tails of this

statistical process serves as an alternative change to measuring ’value-at-risk’ or ’value-

for-reward’ for R&D discoveries from financial-risk applications.

We highlight the problem of investing under discontinuities and when the characteris-

tics of the jumps are pre-defined and non-arbitrary. This study also provides analysis

of option values and optimal investment rules under various combinations of differ-

ent stochastic processes. We attain closed-form solutions in this class of a Levy jump

process. In terms of investment decisions, we found that jump amplitude, λ, is a more

significant factor than jump size distribution, 1η. Investment under the effects of dis-

continuity also do not appear to be less robust than in a continuous model.

We show the ease of computation in an empirical R&D program that involves condition-

ing human genes into drugs that treat us against killer diseases. Applying our model to

the HGSI pipeline, with parameter values that appear to fit the facts of clinical trials

and realistic jump rates, we have determined that it is optimal to proceed with the

next phase of R&D trial studies. But biotechnological R&D should be pursued with

careful attention to successes and failures of achieving acceptable laboratory results.

If HGSI sales remained as expected and phase II is completed, development into the

next phase should continue as soon as possible; however if ’disasters’ or ’good news’ in

trials occurs or cost estimates suddenly escalate, then jump factors and probabilities in

this R&D should be re-evaluated and investment decisions may be altered. This case

approach exemplifies the tractability of modelling high-risk clinical trials.

Evaluating R&D with abrupt disturbance of events is a realistic but complex problem.

In order to engage in this issue, double exponential Levy jumps are considered to be

useful approaches. However, one needs a robust method for estimating jump processes’

project values and risk parameters. We also regard investigating time-to-build models

in R&D with Levy jumps as an important area for future research. Of course, inside

expert knowledge would be preferred to the heroic assumption that traded option prices

across moneyness might embody this expert knowledge.

21

Appendix

A.1. Proof of Lemma 1

We show that Eq.(13) has four roots such that the relationship Eq.(14) holds. We con-

sider f(x) as a convex function on the interval (−η−, η+) with f(0) = λ(p+ q− 1) = 0.

Since limx→η+− f(x) = +∞ and limx→−η−− f(x) = +∞, there is at least one root, β1

in the interval (0, η+) and another one, β2 in the interval (η−, 0).

Since limx→η++ f(x) = −∞ and limx→+∞ f(x) = +∞, there is at least one root, β3 in

the interval (η+,∞). Lastly the final root, β4 can be found in the interval (−∞,− 1η− )

implied by limx→−∞ f(x) = −∞ and limx→−η−+ f(x) = ∞. Hence this completes our

proof on the existence of four real and distinct roots in Eq.(13).

A.2. Proof of Proposition 1

Here, we prove equation (18) for pricing vanilla call options under double-exponential

jumps. We will make use of Eq.(17) to solve our differential equation problem. An

initial guess to the solution of Eq.(17) appears to have the exponential form v(x) =

exβi , i = 1, 2, 3, 4 which leads to the corresponding characteristic Eq.(13).

Let v = ex and define

ψ(x) =

Aexβ1 + Bexβ2 if x < x∗,

ex −K if x ≥ x∗.

First we need to compute∫ +∞−∞ [ψ(x + J)]$(J)(dJ), which is necessary for the purpose

of evaluating the generator shown in Eq.(11). For x < x∗,

∫ +∞

−∞ψ(x + J)$(J)(dJ) =

∫ 0

−∞

(Aeβ1(x+J) + Beβ2(x+J)

)qη−eJη−dJ+

∫ x∗−x

0

(Aeβ1(x+J) + Beβ2(x+J)

)pη+e−Jη+

dJ +

∫ ∞

x∗−x

(ex+J −K)pη+e−Jη+

dJ

=

[Aqη−eJ(β1+η−)+β1x

β1 + η−+

Bqη−eJ(β2+η−)+β2x

β2 + η−

]0

−∞

+

[Apη+eJ(β1−η+)+β1x

β1 − η++

Bpη+eJ(β2−η+)+β2x

β2 − η+

]x∗−x

0

+

[pη+eJ(1−η+)+x

1− η++ Kpe−Jη+

]∞

x∗−x

22

=Aqη−eβ1x

β1 + η−+

Bqη−eβ2x

β2 + η−+

Apη+eβ1x+(x∗−x)(β1−η+)

β1 − η++

Bpη+eβ2x+(x∗−x)(β2−η+)

β2 − η+

−Apη+eβ1x

β1 − η+− Bpη+eβ2x

β2 − η+− pη+ex+(x∗−x)(1−η+)

1− η+−Kpe−(x∗−x)η+

=

(Aqη−eβ1x

β1 + η−+

Bqη−eβ2x

β2 + η−

)+ pe−(x∗−x)η+

(η+ex∗

η+ − 1−K

)

+Apη+

β1 − η+

(eβ1x+(x∗−x)(β1−η+) − eβ1x

)+

Bpη+

β2 − η+

(eβ2x+(x∗−x)(β2−η+) − eβ2x

)

Following Eq.(11), we use Eq.(17) and the above result for∫ +∞−∞ [ψ(x + J)]$(J)(dJ) to

obtain:

Lψ(x)− αψ(x)

=1

2σ2Aβ2

1eβ1x +

1

2σ2Bβ2

2eβ2x + µAβ1e

β1x + µBβ2eβ2x − (α + λ)Aeβ1x − (α + λ)Beβ2x

+λpe(x−x∗)η+

(η+ex∗

η+ − 1−K +

Aη+

β1 − η+eβ1x∗ +

Bη+

β2 − η+eβ2x∗

)+

λAqη−eβ1x

β1 + η−+

λBqη−eβ2x

β2 + η−− λApη+eβ1x

β1 − η+− λBpη+eβ2x

β2 − η+

Straightforward algebraic simplification and collection of terms Aeβ1x and Beβ2x yield

Aeβ1x

(1

2σ2β2

1 + µβ1 − α + λ

(pη+

η+ − β1

+qη−

η− + β1

− 1

))

+Beβ2x

(1

2σ2β2

2 + µβ2 − α + λ

(pη+

η+ − β2

+qη−

η− + β2

− 1

))

+λpe(x−x∗)η+

(Aη+eβ1x∗

β1 − η++

Bη+eβ2x∗

β2 − η++

η+ex∗

η+ − 1−K

)

= Aeβ1xf(β1) + Beβ2xf(β2) + λpe(x−x∗)η+

(Aη+eβ1x∗

β1 − η++

Bη+eβ2x∗

β2 − η++

η+ex∗

η+ − 1−K

)

We know that f(β1) = 0 and f(β2) = 0 from Eq.(13). As for the last term with

coefficient λpe(x−x∗)η+, we refer to Eq.(13). Assuming λ > 0, the term

(Aη+eβ1x∗

β1−η+ +

23

Bη+eβ2x∗

β2−η+ + η+ex∗

η+−1−K

)is equal to zero25. Hence this completes our proof for Eq.(17).

Our boundary conditions are imposed as follows:

Aeβ1x∗ + Beβ1x∗ = ex −K

β1Aeβ1x + β2Aeβ2x = ex

(Aη+eβ1x∗

β1 − η++

Bη+eβ2x∗

β2 − η++

η+ex∗

η+ − 1−K

)= 0

With three unknown parameters, A, B and x∗, we have the above three equations. Us-

ing algebraic manipulation, we can attain theclosed-form expression shown in Eq.(18)

A.3. Proof of Eq.(23)

We buy an option of F (V,C). At the same time, we short m units of output and n

units of capital. By Ito’s Lemma, we have

∂(F −mV − nC) =

(∂F

∂V−m

)∂V +

(∂F

∂C− n

)∂C +

1

2

(∂2F

∂V 2σ2

vV2 +

∂2F

∂C2σ2

cC2+

2∂F

∂V ∂CV Cρσvσc + 2µvV

∂F

∂V+ 2λ

∫ ∞

−∞

(F (V + J)− F (V )

)$(J)∂J)

))

∂t

Substitute m = Fv and n = Fc into the equation to get rid of these terms and make

the portfolio riskless. Then the owner of the portfolio will have capital gain of

π =1

2

(∂2F

∂V 2σ2

vV2 +

∂2F

∂C2σ2

cC2 + 2

∂F

∂V ∂CV Cρσvσc

+2λ

∫ ∞

−∞

(F (V + J)− F (V )

)$(J)∂J

)∂t

Since the portfolio does not pay further dividends, the capital gain is the portfolio

25Refer to Kou and Wang (2003) [34], p.510 for an extended proof.

24

return. The owner will pay dividends for shorting m and n units. They are

(mµvV )∂t

and

(nµcC)∂t

or (µvV

∂F

∂V

)∂t

and (µcC

∂F

∂C

)∂t

Hence equating these two components and our capital gain to our total riskless return,

r

(F − ∂F

∂VV − ∂F

∂CC

)∂t

gives us

π −(

µvV∂F

∂V

)∂t−

(µcC

∂F

∂C

)∂t− r

(F − ∂F

∂VV − ∂F

∂CC

)∂t = 0

=1

2

(∂2F

∂V 2σ2

vV2 +

∂2F

∂C2σ2

cC2 + 2

∂F

∂V ∂CV Cρσvσc

+2λ

∫ ∞

−∞

(F (V + J)− F (V )

)$(J)∂J

)∂t−

(µvV

∂F

∂V

)∂t−

(µcC

∂F

∂C

)∂t

−r

(F − ∂F

∂VV − ∂F

∂CC

)∂t = 0

Cancel all ∂t and collect terms, we have

1

2

∂2F

∂V 2σ2

vV2 +

1

2

∂2F

∂C2σ2

cC2 +

∂F

∂V ∂CV Cρσvσc + (r − µv)V

∂F

∂V+ (r − µc)C

∂F

∂C− rF

+λ

∫ ∞

−∞

(F (V + J)− F (V )

)$(J)∂J = 0

This is our Eq.(23).

25

A.4. Derivation of Eq.(24)

We define m = e

(VC

)and

F (m) = F (V,C) = CΨ(e

VC

)= CΨ(m)

Successive differentiations give:

∂F (V, C)

∂V=

∂Ψ(m)

∂m

∂F (V, C)

∂C= Ψ(m)−m

∂Ψ(m)

∂m

∂2F (V, C)

∂2V=

1

C

∂2Ψ(m)

∂2m

∂2F (V, C)

∂2C=

m2

C

∂2Ψ(m)

∂2m

∂2F (V,C)

∂V ∂C= −m

C

Ψ2(m)

∂m2

Substitute these differentiations into Eq.(23) and through algebraic simplifying gives

us Eq.(24).

1

2m2d2Ψ(m)

dm2(σv + σc − 2ρσvσc) + m

dΨ(m)

dm(µc − λ%)− µcΨ(m)

+λ

∫ ∞

−∞

(Ψ(m + J)−Ψ(m)

)$(J)d(J)

A.5. Proof of Proposition 2

Here we prove the formula (26) for pricing an investment option model with two sto-

chastic factors and a double-exponential jump diffusion.

We let υ = em and define

Ψ(m) =

Pemβ1 + Qemβ2 if m < m∗,

em − C if m ≥ m∗.

26

We already show in Proposition 1 the computation of

∫ +∞

−∞[Ψ(m + J)]$(J)(dJ) =

(Pqη−eβ1m

β1 + η−+

Qqη−eβ2m

β2 + η−

)+ pe−(m∗−m)η+

(η+em∗

η+ − 1−C

)

+Ppη+

β1 − η+

(eβ1m+(m∗−m)(β1−η+) − eβ1m

)+

Qpη+

β2 − η+

(eβ2m+(m∗−m)(β2−η+) − eβ2m

)

Substitute the expressions of Ψ(m) and∫ +∞−∞ [Ψ(m + J)]$(J)(dJ) into Eq.(25) yield

1

2σ2

mPβ21e

β1m +1

2σ2

mQβ22e

β2m + µmPβ1eβ1m + µmQβ2e

β2m − (µc + λ)Peβ1m

−(µc + λ)Qeβ2m + λpe(m−m∗)η+

(η+em∗

η+ − 1− C +

Pη+

β1 − η+eβ1m∗

+Qη+

β2 − η+eβ2m∗

)+

λPqη−eβ1m

β1 + η−+

λQqη−eβ2m

β2 + η−− λPpη+eβ1m

β1 − η+− λQpη+eβ2m

β2 − η+

Further simplification and we end up with expression similar to that in Proposition 2:

Peβ1m

(1

2σ2

mβ21 + µmβ1 − µc + λ

(pη+

η+ − β1

+qη−

η− + β1

− 1

))

+Qeβ2m

(1

2σ2

mβ22 + µmβ2 − µc + λ

(pη+

η+ − β2

+qη−

η− + β2

− 1

))

+λpe(m−m∗)η+

(Pη+eβ1m∗

β1 − η++

Qη+eβ2m∗

β2 − η++

η+em∗

η+ − 1− C

)

= Peβ1mH(β1) + Beβ2mH(β2) + λpe(m−m∗)η+

(Pη+eβ1m∗

β1 − η++

Qη+eβ2m∗

β2 − η++

η+em∗

η+ − 1− C

)

= 0

To solve for our unknown parameters P , Q and m∗, we need to impose some boundary

conditions.

Value-matching condition

Pem∗β1 + Qem∗β2 = em∗ − C

27

Smooth-pasting condition

β1Pem∗β1 + β2Qem∗β2 = em∗

Additional equation from computation of Eq.(25)

Pη+eβ1m∗

β1 − η++

Qη+eβ2m∗

β2 − η++

η+em∗

η+ − 1− C = 0

We solve for P , Q and em∗algebraically and represent them in Eq.(26).

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32

33

Figures and Tables

Figure 1: Sensitivity of Perpetual American Call Options ‘DEVC’ to Changes in Various Parameters.

PERPETUAL AMERICAN DOUBLE-EXPO JUMP CALL OPTIONS

0

50

100

1500 40 80 120

160

200

S

Opt

ion

valu

e

V(u)

NPV

DEVC OPTION VALUE SENSITIVITY TO λ λ λ λ

19.319.419.519.619.719.819.9

1 2 3 4 5 6 7 8 9 1

λλλλ

Opt

ion

valu

e

V(S)

DEVC OPTION VALUE SENSITIVITY TO σ σ σ σ

0

10

20

30

40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

σσσσ

Opt

ion

valu

e

V(S)

DEVC OPTION VALUE SENSITIVITY TO q

19.519.519.519.519.619.619.619.6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

q

Opt

ion

valu

e

V(s)

DEVC OPTION VALUE SENSITIVITY TO η η η η+

17.2

17.4

17.6

17.8

18

18.2

18.4

18.6

10 20 30 40 50 60 70 80

ηηηη+

Opt

ion

valu

e

V(S)

DEVC OPTION VALUE SENSITIVITY To ηηηη-

17.2

17.4

17.6

17.8

18

18.2

18.4

18.6

18.8

10 20 30 40 50 60 70 80

ηηηη-

Opt

ion

Val

ue

V(S)

DEVC TRIGGER S* SENSITIVITY TO σ σ σ σ

0

50

100

150

200

250

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

σσσσ

Opt

ion

Val

ue

S*

DEVC OPTION VALUE SENSITIVITY TO p

19.5

19.5

19.6

19.6

19.7

0.1 0.3 0.5 0.7 0.9

p

Op

tio

n v

alu

e

V(S)

34

Figure 2: Project Values As Function of Asset Values and Investment Cost. 0.3 0.3 0 0.06 0.3 66.67 83.33 0.1 0.1 , , , , , , , , v c c vr pσ σ ρ η η µ γ+ −= = = = = = = = =

Figure 3: Project Values As Function of Asset Values.

1000.3 0.3 0 0.06 0.3 66.67 83.33 0.1 0.1 , , , , , , , , , v c c vr p cσ σ ρ η η µ γ+ −= = = = = = = = = =

20 60 100

140

18020

1600

50

100

150

200

PROJECT VALUES

INVESTMENT COST

ASSET VALUES

TWO-FACTOR DOUBLE-EXPONENTIAL JUMP CALL VALUE FUNCTION

TWO-FACTOR DOUBLE-EXPONENTIAL CALL OPTION VALUE

020406080

100120

0 20 40 60 80 100 120 140 160 180 200

ASSET VALUES, V

PR

OJE

CT

VA

LUE

S

V(v)NPV

35

Figure 4: Trigger Values Sensitivity to Probability of Upward Jumps 200 200 0.3 0.3 0 3,6,9 0.06 66.67

83.33 0.1 0.1

, , , , , , , ,

, , v c

c v

V C rσ σ ρ λ ηη µ γ

+

−

= = = = = = = =

= = =

Figure 5: Trigger Values Robustness to Mean of Upside Exponential Distribution. 200 200 0.3 0.3 0 0.3 3,6,9 0.06

83.33 0.1 0.1

, , , , , , , ,

, , v c

c v

V C p rσ σ ρ λη µ γ−

= = = = = = = =

= = =

Sensitivity of Investment Trigger Values to Upward Jump Sizes

347.8348

348.2348.4348.6348.8

349

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

p , probability of up jumps

Trig

ger

valu

esλ=3λ=6λ=9

Sensitivity of trigger values to mean of exponential distribution 1/ηηηη+

345

347

349

351

353

355

357

10 20 30 40 50 60 70 80 90

ηηηη+

Trig

ger

valu

es

λ=3λ=6λ=9

36

Figure 6: Triggers Value Robustness to Value Volatility

200 200 0.3 0.3 0 3 0.06 66.67

83.33 0.1

, , , , , , , ,

, c

v

V C p rσ ρ λ ηη γ

+

−

= = = = = = = =

= =

Figure 7: Trigger Values Robustness to Correlation Coefficient

200 200 0.3 0.3 0.6 3 0.06 66.67

83.33 0.1 0.1

, , = , , , , , ,

, , v c

c v

V C p rσ σ λ ηη µ γ

+

−

= = = = = = =

= = =

Trigger Sensitivity To Value Volatilities For Different Levels of µµµµc

300

320

340

360

380

400

420

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Value volatility, σσσσv

Trig

ger

valu

es µχ = 0.05µχ = 0.1µχ = 0.3

Option values sensitivity to correlation coefficient, ρρρρ

0

10

20

30

40

50

60

-1 -0.7 -0.5 -0.3 0 0.3 0.5 0.7 1

Correlation coefficent, ρρρρ

Op

tio

n v

alu

es

37

Table 1: Human Genome Sciences (HGSI) R&D Pipeline Source: HGSI website [32]

Table 2: Forecasts of HGSI Project Values and R&D Costs.

Data Source: Credit Suisse First Boston Equity Research Report (2005) [21]

!""#"$%&'

()*%+ %, -

()*% )"# - +

()%. )"# - +

! "

/"! #)0)1

#%&2$3' +45 64 674 -64 -6-45-546-4

%8/*9$3' + - 5 -+ - - +

/"! #0%8/*9$3'++4 ++4- +4+ 4 545 74 4 -4--+476+4

$3' 1814

38

%": 4 "

/";!< ="; γϖ +4 #

/"%8/"<µ! -4 "

<""σ< 4 #

&<<""σ! -4 "

"!"!"< # "

"<<""ρ

"" ; 554 #

""#=; 4 #

2""<; ">#"" "η+ 4 #

2?"<; ">#"" "η− 4+ #

. λ -4 #

< $@)3""' +46 #

%8/"<$@)3""' 546 #

Table 3: Estimated Base Case Parameters Risk-free rate – 3 months U.S Treasury bills downloaded from Datastream. Drift of R&D investment – assumed. Correlation coefficient – assumed. Investment volatility – assumed. Rest of the parameters (excluding V and C) – Back out by minimising sum of squared errors (SSE) between theoretical option price and market option price.

%"< "<! -45

"??< 9;! +46

;!< $@)3""' ++4-

;!< $=)//"'$@)3""' +6+74

;!< $=.A"'$@)3""' ++4+

Table 4: Results of Two-Factor Double Exponential Jump ‘TFDEJ’ model with Different R&D Cost Estimates.

Tufts CSDD – Tufts Center of Study of Drug Development PJB - Pharmaprojects

39

Table 5: Comparative Table On Perpetual Investment Options

TFDEJ – Two-Factor Double Exponential Jump Model DP- Dixit and Pindyck Model:

( )( )

( ) ( )*

* 2

*

1( ) 1 0

1 2, , , v

V IF V AV V I A r r

V

ββ

β σ β β δ ββ

−= = = − + − − =

−

All notations are similar to base case parameters in Table 3. DPJ – Dixit and Pindyck [24] Jump Model:

( )( )

( ) ( )*

* 2

*

1( ) 1 ( ) (1 ) 0

1 2, , ,

V IF V AV V I A r r

V

β ββ

β σ β β δ β λ λ φβ

−= = = − + − − + + − =

−

All notations are similar to base case parameters in Table 3. Additionally, dividend yield, δ = 0, jump size, φ = 0.1. Quigg – Quigg 1993 [46] Model:

( ) ( )

* * *

1222 2 2 2

2 2 2

(1 )( ) ( ) ( 1 )( )

1

1 12 ,

2 4

2

c c c , , = , , = - ,

, ,

j jc

c

c v v x c v v v v

c v c v

j k zF V C Az k A z k z z k v

j r v

Vj v v v v r v v z v

S ϖ

β µ θ σ

ω ω ω ω γ θ σ

ω σ ρσ σ σ

−

−

+= + = − − =− −

= + − + − − + + − = = −

= − + All notations are similar to base case parameters in Table 3. Additionally, β = 0,

0.1 0.1 , c vθ θ= =

# $% & %$%'$

()*%%+ ()*%%+

,-. ++4+ +-6746

++74+ 76-4-

. ++ 674-

/ +4 -74

01 ++4+ +64+

40

Mordecki – Mordecki 1994 [42] Model:

( ) ( )* * * *1 2 *

2 2 1 1

*2 1 1 2

2 22 2 *

1,2 1,2 2

(1 ) (1 )( )

10 ln

2 2 22 1

, ,

, ,

V V V V V V

V

v vv

v

e C e CAe Be if V VF V A Be C if V V

aa a V C

a

β β β β β ββ β β β

λασ σ λασ β β α λ

σ λα

− − + − + − + <= = = − −− >

−

+ + + − = = + − + −

11

, >a αα

<−

All notations are similar to base case parameters in Table 3. β = 0, 0.1 0.1 , c vθ θ= = . Additionally, 2 5 , aα = = .