Perspectives in Science (2016) 7, 10—18

Available online at www.sciencedirect.com

ScienceDirect

j ournal homepage: www.elsevier.com/pisc

Real options with changing volatilityଝ

Miroslav Culíkˇ

VSB — Technical University Ostrava, Faculty of Economics, Department, Sokolska trida 33,

701

21 Ostrava, Czech Republic

Received 25 September 2015; accepted 11 November 2015

Available online 10 December 2015

JEL Summary This paper aims at the valuation of real options with changing . Volatility

CLASSIFICATION change is a typical feature of real projects, where the riskiness of cash flow gener-

G11; ated by the project can change significantly during the project life span. In this paper, there is

G12; explained how the problem of changing volatility can be considered if binomial lattice and repli-

G13; cation strategy is used for real valuation. There are recombining and non-recombining

G31 lattice used and constant and increasing volatility are analysed and results compared. In situ-

KEYWORDS ation when volatility is changing, two approaches overcoming this problem are employed and compared.

Real options;

© 2015 Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license Valuation; (http://creativecommons.org/licenses/by-nc-nd/4.0/). Risk-neutral; Transition probabilities;

Binomial lattice; Recombining; Non-recombining;

Volatility

Introduction Compared to the traditional passive valuation approaches

(NPV, IRR, etc.), real option approach takes into consid-

eration two important aspects: (a) riskiness of cash flow

Real options methodology is a relatively new approach for

generated by the assets and (b) flexibility, i.e. capability

solution of a wide range of valuation and decision-making

of management to change past decision or to make new

issues. Here, traditional methods and models used for finan-

ones in already undertaken projects. These future possi-

cial option valuation are used for real assets valuation.

ble decisions (depends on the future state of the world) are

modelled as a formal call and put options, which have their

ଝ value and can be exercised by company’s management. Real

This article is part of a special issue entitled ‘‘Proceedings of

asset value provided by the real option methodology appli-

the 1st Czech-China Scientific Conference 2015’’.

E-mail

address: [email protected] cation is given as a sum of two components: of

http://dx.doi.org/10.1016/j.pisc.2015.11.004

2213-0209/© 2015 Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Real options valuation with changing volatility 11

directly measurable cash flows and flexibility value, which III. Management has flexibility.

captures managerial possibilities (real options). IV. Flexibility strategies (real options) are creditable and

As mentioned, assets value and future managerial oppor- executable.

tunities captured in real options are quantified by financial V. Management is rational in executing real options.

option valuation models. These models are based on some

assumptions, which are sometimes difficult to keep due to Future managerial investment opportunities captured in

the specific feature of real and real options real options and quantified by financial option valuation

captured in them (constant volatility, risk free rate, etc.). models represent the flexibility component (active part) of

That is why it is necessary to adjust these models to spe- the project value. The total project’s NPV then consists

cific conditions of a given project otherwise these are not of two components: the traditional static (passive) NPV of

applicable. directly measurable expected cash flows, and the flexibil-

ity value capturing the value of real options under active

management, i.e.,

Expanded NPV = standard (static, passive) NPV of directly measurable cash flows

+ flexibility value (value of real options from active management).

Valuation procedure by applying real options method-

The goal of this paper is the application of financial option

ology when discrete valuation model is applied can be

valuation models on the real asset under specific condition

described by the following steps:

typical for most real investments — changing risk (cash flow

volatility) during the expected life span.

1. Estimation of the type and parameters of the underlying

The paper is structured as follows. First, real option

asset random evolution (return mean, standard deviation

methodology is described and classification of financial

of returns);

option valuation models and application possibilities is

2. Simulation of the future random underlying asset evolu-

stated. In the subsequent part, mathematical background

tion for each discrete node of the tree;

for lattice valuation models is provided including the situa-

3. The option’s intrinsic value calculation (for given type of

tion of variable parameters. In the end, illustrative example

real option or portfolio of options) for each discrete node

is stated.

of the tree;

4. Flexibility and asset’s value quantification; and

Real options methodology and flexibility 5. Recommendation of the optimal decision.

valuation

For flexibility quantification, traditional models for finan-

cial options pricing are employed. These models can be

Real options methodology represents an approach where

classified as follows: (a) analytical (Black—Scholes model),

financial options pricing theory and models are applied on

discrete (binomial, trinomial, multinomial), and simulation real assets valuation.

(Monte Carlo).

There are many applications in corporate finance where

Because the real options are mostly the American options

the financial option valuation models are applied on the real

(decisions can be made at any time until the investment

assets valuation (real options approach). Black and Scholes

opportunity disappears), more possible decisions can exist

(1973) were the first authors to state that it is possible

at a given point of time, so the discrete valuation model on

to take the equity in a levered company as a

the basis of replication strategy is frequently applied.

on the company value. Since then, new pioneering work

appeared developing real option analysis; see for example

Merton (1973), Cox et al., 1979 or Brennan and Schwartz

Replication strategy for option valuation

(1985). During the last two decades, significant increase in

publishing activities on this topic is obvious. This area has

This approach relies on the fact that it is possible to set

been studied and developed by many authors, and new pos-

up a portfolio of the underlying asset and risk free borrow-

sible applications appear for solutions for a wide array of

ing, whose value replicates the payoff of the option for any

financial-decision and valuation problems. The key papers

state of the underlying asset value. Because there are two

and books that focus on the real options methodology appli-

assets (portfolio, option) providing identical payoffs, in the

cation are those of Dixit and Pindyck (1994), Smith and

absence of opportunities, their current price must

Nau (1995), Trigeorgis (1999), Brennan and Trigeorgis (2000),

be the same. This makes it possible to work out the cost of

Copeland and Antikarov (2003), Grenadier (2000), Brach

setting up the portfolio and, thus, the option’s price.

(2002), Trigeorgis and Schwartz (2001), Trigeorgis and Smith

The following symbols are used for the option’s price

(2004) or Damodaran (2006).

derivation: h is the quantity (number of units) of the under-

Real options methodology application on real projects is

lying asset, St is the price of the underlying asset at t, Bt is

justifiable only if:

the monetary amount of the risk free borrowings, Ct is the

option’s price at t, t is the value of the replication port-

I. There is risk.

folio, Rf is the risk free rate, and u (d) is the proportional

II. Risk drives project value.

increase (or decrease) in the price of the underlying asset.

ˇ 12 M. Culík

t t + dt t t + dt t t + dt

The cost of setting up the replication portfolio consisting

of h units of the underlying asset and risk free borrowings at time t is:



= h · St + Bt, (2.1)

t

and in the absence of arbitrage opportunities it must hold n

that the value of the replication portfolio and the option’s

Figure 1 Discrete stochastic lattices (left-binomial, middle-

price must be identical:

 trinomial, right-multinomial).

=

Ct = h · St + Bt. (2.2)

t

Real option valuation under changing volatility

If the underlying asset moves up at the time t + dt, it

holds for the portfolio value:

There are a lot methods and approaches in finance theory, u

u u dt

which are applicable for option pricing. These methods and

= = · + · + Ct+dt h St+dt Bt (1 Rf ) , (2.3)

t+dt

approaches range from analytical equations (Black—Scholes

model), lattice models (binomial, trinomial, multinomial),

and if it moves down:

simulation (Monte Carlo) to using partial-differential equa- d

d d dt

=

= · + · + tions (finite difference method). Ct+dt h St+dt Bt (1 Rf ) . (2.4)

t+dt

Generally, real options can be quantified by applying any

Under the assumption that the payoff of the European of these approaches. Due to specific features of real options

call option at T is equal to its intrinsic value, discrete lattices are mostly employed. The reason is as fol-

u u u d d d

= = − = = − lows:

that is CT IVT max(ST X; 0) and CT IVT max(ST

X; 0); then, by solving the set of Eqs. (2.3) and (2.4) for

h

B

and and substituting this number into (2.2), we get for — easy calculation, interpretation,

the option’s price: — easily accommodate most types of real options problems,

 

— valuation of both plain vanilla (call, put) and exotic

dt d

· + −

dt u St (1 Rf ) St+dt (Bermudian, Asian, etc.) options,

· + = ·

Ct (1 Rf ) Ct+dt u d

− — managerial strategic decisions are made rather at dis-

St+dt St+dt

  crete time than continuously,

dt

u — valuation of multinomial real options (more possible deci-

− · +

d St+dt St (1 Rf )

+ ·

Ct+dt u d , (2.5) sions are available at given time moment), −

St+dt St+dt

— valuation of real options with multiple sources of risk,

— valuation of real options with variable parameters

where (.) on the right-side of (2.5) represents the risk-

(changing volatility, price, risk free rate, etc.).

neutral probabilities of up and down movements.

Formula (2.5) can be reduced to

u u d u −dt 1

Simulation via discrete lattice

= · + · − · +

Ct [Ct+dt p Ct+dt (1 p )] (1 Rf ) . (2.6)

It is apparent from (2.6) that the European option’s price

Discrete lattice is a , where the stochas-

at t is equal to its expected payoff at the subsequent period

tic variable can change only after passing certain time-step

t

+ dt discounted at the risk-free rate.

(stepping time) and can take on given number of new val-

The procedure for the American options is similar to that

ues. A given time period (T − t), during which the stochastic

of the European options, i.e., we work back through the tree

process is simulated, is divided into finite number of time

from the end to the beginning, and, moreover, we are testing

steps where dt is the length of one discrete time step (time

at each node whether the early exercise is optimal. The

interval). For any discrete moment at time t has the stochas-

value of the American call option at maturity (end nodes of

tic process at t + dt (i.e. after passing stepping time) finite

the tree) is the same as for the European options; at earlier

possible number of values which can take on. According to

nodes, the option’s price is greater than its expected payoff

the number of values at the end of stepping time we work

at the subsequent period t + dt, discounted at the risk-free

with the following processes: binomial (at t + dt takes on

rate or the payoff (intrinsic value) IVt from early exercise,

two values), trinomial (at t + dt takes on three values) or

i.e., multinomial

(at t + dt takes on n values), see Fig. 1.      

dt d u dt

· + − − · +

u St (1 Rf ) St+dt d St+dt St (1 Rf ) −dt

= + +

Ct max Ct+dt u d Ct+dt u d (1 Rf ) ; IVt ,

− −

St+dt St+dt St+dt St+dt and after simplification,

u u d u −dt

= · + · − · + Ct max[(Ct+dt p Ct+dt (1 p )) (1 Rf ) ; IVt].

1

(2.7) More details are available in Hoek and Elliot (2006).

Real options valuation with changing volatility 13

Simulation via binomial lattice is the simplest discrete t t + dt

.

process. Here, the value of underlying (random) asset takes u St

u

on at time t the value St; at the end of discrete interval (i.e. p

u

after passing stepping time) it can either jump up to St+dt or

d St

down to St+dt with some transition (risk-neutral) probability.

If the process can take on only positive values (typical for d u p = 1 - p .

d St

most financial variables), we work with geometric version,

where it holds for upward jump,

Figure 2 One period binomial lattice (geometric process).

u

S = u · S (3.1)

t+dt t t t + dt and downward jump, S + u pu t

d

= ·

St+dt d St (3.2)

and where u(d) are up and down factors. For these two fac-

rfdt

tors it holds following: u ≥ 1; 0 < d ≤ 1 and d < e < u. The d u

p = 1 - p

upward and downward factors u and d and transition (risk- St - d

neutral) probabilities are set uniquely in order to determine

Figure 3 One period binomial lattice (arithmetic process).

the evolution of underlying asset. Due to the fact that

expected return of any asset is assumed to be risk-free, the t t + dt t + 2dt

rf·dt

at the end of discrete step equals St · e . 2 .

u St

It follows that the expected value of the underlying asset .

u St

can be written as,

·

rf dt u u . .

= · = · · + − · · E(St+dt) St e p u St (1 p ) d St (3.3) St d u St .

d St

and after some rearrangements, · 2 .

rf dt u u d St

e = p · u + (1 − p ) · d (3.4)

u Figure 4 Two-period recombining binomial lattice (geometric

From (3.4) the risk-neutral probability of upward jump p is process). given as,

rf ·dt

e − d

pu = (3.5) In situation, that the simulated process can take on both

−

u d positive and negative values, we work with arithmetic ver-

u d 2

= + = −

sion, where St+dt St u and St+dt St d, see Fig. 3.

and for downward jump must follows,

d = u

−

p (1 p ). (3.6) Simulation via binomial lattice with volatility

change

The of the underlying asset between two sub-

2

sequent discrete nodes at time t and t + dt is dt. And

If we assume the constant volatility over the period (T − t),

because the variance of random variable is generally given

2 2 2 the up and downward factors given according to (3.10) and

as (S) = E(S ) − [E(S)] , it is possible to write

(3.11) are constant throughout the whole and

2

2 u 2 u 2 u u due to (3.9) it holds u · d = d · u = 1, which follows in the result

dt = p · u + (1 − p ) · d − [p · u + (1 − p ) · d] . (3.7)

the lattice recombines. This means that the nodes recon-

u

S u · d · S S d · u · S

Substituting for p from (3.5)—(3.7) and after some rear- nect, i.e. t+2dt = t or t+2dt = t. Furthermore, the

rangements we get, value of the underlying asset in any node of the binomial

lattice can be expressed as,

2 rf ·dt rf ·dt

dt = e · (u + d) − u · d − 2 · e . (3.8)

j i−j

St+i·dt = u · d · St (3.12)

Solving (3.4) and (3.8) and under the condition,

If (3.10) and (3.11) are unchanged and under the assump-

1 tion of constant risk-free rate, the transition probabilities

u = , (3.9)

d between any two subsequent nodes according to (3.5) and

(3.6) are constant, as well. Fig. 4 illustrates two-period

we get for upward and downward factors u and d following

recombining binomial lattice can be depicted as it is shown

formulas,

in Fig. 4.

√

· dt When an up move followed by a down move does not

u = e , (3.10)

√ reconnect in the same node as a down move followed by an

−· dt = d e . (3.11)

2

Fig. 2 shows one period binomial lattice with geometric More details on derivation can be found in Tich´y (2008), Hull

process. (2014), etc. ˇ 14 M. Culík

t t + dt t + 2dt dt1 dt2 dt3 2 . . u S t u S t

. . u d S t S t . .

. d u S t

d S t 2 .

d S t

Figure 5 Two-period non-recombining binomial lattice (geo-

metric process).

Figure 6 Three-period recombining binomial lattice with

S =/ u ·

up move, i.e. t+2dt d · St or St+2dt =/ d · u · St, the lattice

unequal length of time steps.

is called non-recombining. Non-recombining lattice is used

for analysis when there are multiple sources of uncertainty

or when volatility changes over time. The main difference

risk-free rate is rf = 8% p.a. and the annual volatility = 25%.

between recombining and non-recombining lattice is, that

The illustration example is structured as follows:

n the lattice with periods the recombining have (n + 1) final

n

unique values, whereas non-recombining 2 values, which

I. first, it is assumed that the volatility is constant,

are not unique. Fig. 5 shows two-period non-recombining

II. next, volatility of cash flow increases as the project

binomial lattice.

expected end-life is approaching,

If the assumption of constant volatility is relaxed, it fol-

III. in the end, volatility of cash flow increases is assumed

lows that upward and downward factors according to (3.10)

again, for problem solution, an approach suggested by

and (3.11) are not constant throughout the lattice any more.

Guthrie (2011) is employed.

The same is true about the transition risk-neutral proba-

bilities. In such situations, there are a few ways how to

overcome this issue: (a) for each period and volatility cal-

Problem solution I

culate corresponding upward and downward factors; the

same is true about the transition probabilities (b) set the

u d Problem solution is decomposed into the following steps:

transition probabilities p = p = 0.5 throughout the binomial

lattice and calculate for each period upward and downward

(a) Calculation of upward and downward factor. Substitut-

factors according to given volatility,

√ ing into (3.10) and (3.11) we get u = 1.284 and d = 0.7788. 2

(rf − /2)·dt+ dt

=

u e (3.13) (b) Calculation the risk-neutral probabilities. Applying (3.5)

√ u d

2 p p

(rf − /2)·dt− dt and (3.6) we get following: = 60.27% and = 39.73%.

d = e (3.14)

(c) Simulation of the underlying asset via the recombining

or (c) the size of up and down movements and their cor- and non-recombining lattice.

responding transition probabilities are constant throughout (d) The intrinsic value calculation. For option to expand it

4

IV V − I

the lattice, but the time periods are of unequal length. is defined as t = max( t E ; 0).

When volatility is high, the time periods are short, so that (e) Option value calculation according to (3.7) by applying

the state variable changes frequently by the standardized backward-induction approach with risk-neutral proba-

amount. When volatility is lower, the periods are longer so bilities.

3

that the changes in the state variable are less frequent.

This binomial lattice is presented in Fig. 6.

Following Fig. 7 shows numerical results; Fig. 8 shows

the histogram of end node values for recombining and non-

Application — valuation American real option recombining lattice. The results obtained are identical no

matter which approach is used (option value equals 45 c.u.).

valuation with changing volatility

This part of the paper is focused on the application of dis-

Problem solution II

crete binomial lattice on the real option valuation (option to

expand a project). The option is an American-type option,

Real option valuation with changing volatility includes fol-

i.e. the project can be expanded at any time during the

lowing steps:

expected life span. It is assumed that the underlying asset

is the cash flow generated by the project; the initial value

FCF (a) Calculation of upward and downward factors for each

0 = 100 c.u., and evolves according to the GBM. Company

period and volatility according to (3.10) and (3.11).

has the option to expand the project at any time during

Recall, that the volatility 1 applies for the first two

the life span with the costs on expansion IE = 80 c.u. The

3 4

For more details and mathematical background see Guthrie For more details see for example Trigeorgis (1999), Trigeorgis

(2011) or Haahtela (2010). (2000) Mun (2003) or Mun (2005).

Real options valuation with changing volatility 15

Non-recombining und erlying asset lacce Non-recombining intrinsic value lace Non-recombining valuaon lace

272 192 192

212 132 138

165 85 85

165 85 97

165 85 85

128 48 55

100 20 20

128 48 66

165 85 85

128 48 55

100 20 20

100 20 34

100 20 20

78 0 11

61 0 0

100 20 45

165 85 85

128 48 55

100 20 20

100 20 34

100 20 20

78 0 11

61 0 0

78 0 21

100 20 20

78 0 11

61 0 0

61 0 6

61 0 0

47 0 0

37 0 0

Recombining und erlying asset latce Recombining intrinsic value lace Recombining valuaon lace

272 192 192

212 132 138

165 165 85 85 97 85

128 128 48 48 66 55

100 100 100 20 20 20 45 34 20

78 78 0 0 21 11

61 61 0 0 6 0

47 0 0

37 0 0

Figure 7 Real option valuation lattice (recombining and non-recombining lattice, constant volatility).

6 3

5

4 2 ency ency 3 qu qu Fre Fre 2 1

1

0 0

37 62 100 165 272 37 61 100 165 272

e nd node values end node values

Figure 8 End node values of underlying asset value for non-recombining lattice (left) and recombining lattice (right).

Table 1 Up (down) ward factors and transition probabilities for given volatility level.

u

Volatility (%) Period Upward factor (u) Downward factor (d) p (%)

1 40 0—1 and 1—2 1.4918 0.6703 50.3

2 45 3 1.5683 0.6376 47.9

2 50 4 1.6487 0.6065 45.7

ˇ 16 M. Culík

Non-recombining und erlying asset lace Non-recombining intrinsic value lace Non-recombining valuaon lace

575 495 495

349 269 275

212 132 132

223 143 154

234 154 154

142 62 68

86 6 6

149 69 91

259 179 179

157 77 83

95 15 15

100 20 42

105 25 25

64 0 11

39 0 0

100 20 52

259 179 179

157 77 83

95 15 15

100 20 42

105 25 25

64 0 11

39 0 0

67 0 22

116 36 36

70 0 15

43 0 0

45 0 7

47 0 0

29 0 0

17 0 0

Recombining und erlying asset lace Recombining intrinsic value lace Recombining valuaon lace

575 495 495

349 269 299

223 259 143 179 177 179

149 157 69 77 104 94

100 100 116 20 20 36 60 49 36

67 70 0 0 26 15

45 47 0 0 7 0

29 0 0

17 0 0

Figure 9 Real option valuation lattice (recombining and non-recombining lattice, increasing volatility).

Table 2 Up (down) ward factors and transition probabilities for given volatility level.

u

Volatility (%) Period Upward factor (u) Downward factor (d) p (%)

1 40 0—2 1.4918 0.6703 50

2 45 3 1.5353 0.6242 50

2 50 4 1.5762 0.5798 50

periods, 2 for period three and 3 for period four. of the recombining and non-recombining underlying asset

Results are summarized in the following Table 1. lattice.

(b) Simulation of the underlying asset via the recombining

and non-recombining lattice.

(c) The intrinsic values calculation.

Problem solution III

(d) Option value calculation according to (3.7) by applying

backward-induction approach with risk-neutral proba-

Real option valuation with changing volatility includes fol- bilities.

lowing steps:

Fig. 9 shows resulting lattices; Fig. 10 shows the

(a) Calculation of the upward and downward factors for

histogram of end node values for recombining and non-

each period and volatility according to (3.13) and (3.14);

recombining lattice. The results obtained are no more

the transition probability is set to equals 50% for all

identical; the difference is caused primarily by the larger

periods, see Table 2.

differences in the end nodes values (and their frequencies)

Real options valuation with changing volatility 17

3 3

2 2 ency ency qu qu Fre Fre 1 1

0 0

17 39 43 47 86 95 105 116 212 234 259 575 37 61 100 165 272

End node values end node values

Figure 10 End node values for non-recombining lattice (left) and recombining lattice (right).

Non-recombining und erlying asset lace Non-recombining intrinsic value lace Non-recombining valuaon lace

539 459 459

342 262 266

198 118 118

223 143 153

219 139 139

139 59 64

81 1 1

149 69 89

242 162 162

154 74 79

89 9 9

100 20 40

98 18 18

62 0 8

36 0 0

100 20 51

242 162 162

154 74 79

89 9 9

100 20 40

98 18 18

62 0 8

36 0 0

67 0 21

109 29 29

69 0 13

40 0 0

45 0 6

44 0 0

28 0 0

16 0 0

Figure 11 Real option valuation lattice (non-recombining lattice, increasing volatility).

2

(b) Simulation of the underlying asset via the non-

recombining lattice. Due to the fact that the u · d =/ 1,

it is not possible to construct the recombining lattice.

(c) The intrinsic value calculation.

(d) Option valuation according to (3.7) by backward- ency 1

qu

induction approach and applying risk-neutral probabili-

ties. Fre

0

16 36 40 44 81 89 98 109 198 219 242 539

Results are again depicted in the following Fig. 11; Fig. 12

End node values

shows frequency of underlying asset end node values. It is

apparent that this approach provides the same results as the

5 Figure 12 End node values of underlying asset for non-

one applied in Problem solution II.

recombining lattice.

Conclusion

5

Small difference in the results is caused by the number of the

steps in valuation lattice. For five and more steps the results world This paper focuses at the real option valuation under chang-

be identical. ing volatility. Change in the volatility structure (i.e. change ˇ

18 M. Culík

in project cash flows riskiness) is a typical feature for most Brach, M.A., 2002. Real Options in Practice. Wiley, New Jersey.

of the long-term project, where the prediction of the cash Brennan, M.J., Schwartz, E.S., 1985. Evaluating natural resource

investments. J. Bus. 58 (2), 135—157.

flows in the far future is difficult and associated with the

Brennan, M.J., Trigeorgis, L., 2000. Project Flexibility Agency

higher risk.

and Competition: New Developments in the Theory and Appli-

Due to the fact that most real options are American-style

cation of Real Options, first ed. Oxford University Press,

options, there are discrete binomial models applied. More-

London.

over, other option valuation models (like B—S model) rely on

Copeland, T.E., Antikarov, V. , 2003. Real Options Revised Edition: A

some assumptions (valuation European-style options, con-

Practitioner’s Guide, second ed. Texere, New York.

stant volatility, risk free rate) and their application may be

Cox, J.C., Ross, S.A., Rubinstein, M., 1979. Option pricing: a sim-

limited for real options issues solutions. plified approach. J. Financ. Econ. 7 (3), 229—263.

It has been shown that, under changing volatility Damodaran, A., 2006. Damodaran on Valuation: Analysis

application of non-recombining lattice with risk-neutral for Investment and , second ed. Wiley, New

Jersey.

probabilities is a useful tool helping the analysts to over-

Dixit, A.K., Pindyck, R.S., 1994. Investment under Uncertainty, first

come the option-valuation problems associated with the

ed. Princeton University Press, New York.

variable parameters.

Grenadier, S., 2000. Game Choices: The Intersection of Real Options

There were two approaches employed to evaluate real

and Game Theory. Riskbooks, London.

option with the volatility change. The former one is based on

Guthrie, G., 2011. Learning options and binomial trees. Wilmott J.

the assumption that the entire lattice is divided into stages

3 (1), 1—23.

with constant volatility; at the end of the constant volatility

Haahtela, T. , 2010. Recombining for real option

period, each resulting point becomes starting point of a new valuation with changing volatility. In: Annual Real Option

lattice. For each period, corresponding up and down factors Conference.

and transition probabilities must be calculated. The latter Hoek, J., Elliot, R.J., 2006. Binomial Models in Finance. Springer,

New York.

approach sets the transition probabilities to 50% throughout

Hull, J.C., 2014. Options, Futures, and Other Derivatives, ninth ed.

the entire lattice and adjusts the up and downward factors

Prentice Hall.

with respect to the volatility during given period. Due to

Merton, R., 1973. The theory of rational option pricing. Bell J. Econ.

the fact that the centrality is broken, only non-recombining

Manag. Sci. 4 (1), 141—183.

lattice can be used. Anyway, for sufficient number of steps

Mun, J., 2005. Real Option Analysis. Tool and Techniques for Valu-

both approaches provide the same results.

ing Strategic Investments and Decisions, second ed. Wiley, New

Jersey.

Conflict of interest Mun, J., 2003. Real Options Analysis Course: Business Cases and

Software Applications. Wiley, New Jersey.

Smith, J.E., Nau, R.F., 1995. Valuing risky projects: option

The author declares that there is no conflict of interest.

pricing theory and decision analysis. Manag. Sci. 14, 795—816.

Acknowledgements

Tich´y, T. , 2008. Lattice Models. Pricing and Hedging at (In)omplete

ˇ

Markets. VSB-TU Ostrava.

This paper was supported within Operational Pro- Trigeorgis, L., 1999. Real Options and Business Strategy: Applica-

gramme Education for Competitiveness — Project No. tions to Decision Making. Riskbooks, London.

Trigeorgis, L., Schwartz, E.S., 2001. Real Options and Investments

CZ.1.07/2.3.00/20.0296 and the Project No. GP14-15175P.

under Uncertainty. MIT Press, Cambridge.

References Trigeorgis, L., Smith, H.T.J., 2004. Strategic Investment: Real

Options and Games. Princeton University Press, New York.

Black, F. , Scholes, M., 1973. The pricing of options and corporate

liabilities. J. Polit. Econ. 81 (3), 637—654.