Investment Modeling Using Real Options Approach

Július Bems,

Ing., Dept. of Economics, Management and Humanities, Faculty of Electrical Engineering, Czech Technical University (инженер, отдел экономики, менеджмента и гуманитарных наук, факультет электротехники, Чешский технический университет) (Technická 2, 166 27 Prague, Czech Republic; e-mail: bemsjuli@fel. cvut. cz)

Oldrich Stary,

CSc., prof. Ing., Dept. of Economics, Management and Humanities, Faculty of Electrical Engineering, Czech Technical University (кандидат наук, профессор, отдел экономики, менеджмента и гуманитарных наук, факультет электротехники, Чешский технический университет) (Technická 2, 166 27 Prague, Czech Republic; e-mail: bemsjuli@fel.cvut.cz)

Аннотация. В статье сравниваются стандартные оценочные методы с методом, основанным на теории реальных опционов.Стандартные методы расширяются теорией реальных опционов для достижения сложной проектной оценки в очень изменчивой экономической обстановке. Стандартные методы оценки, такие как чистая стоимость или внутренняя норма прибыли ограничены, потому что эти методы не учитывают гибкость будущих решений. Чистая стоимость основана на учетной ставке, которая является постоянной в течение всего периода вычисления и не может эффективно отразить будущую неопределенность. Теория реальных опционов расширяет стандартные методы оценки учетом гибкости в управленческих решениях, которые могут быть предприняты в будущем. Это может включать величину неуверенности в проектной оценке. Итоговая проектная ценность может быть вычислена комбинацией статических методов, моделирований и оценкой реального опциона.

Abstract. This paper compares standard appraisal methods to method based on real options theory. It also extends standard methods by real option theory to achieve complex project valuation in highly volatile economic environment. Standard valuation methods such as Net Present Value or Internal Rate of Return are limiting because these methods do not involve flexibility in future decisions. Net Present Value is based on discount rate, which is constant during all calculation period and cannot effectively reflect the future uncertainty. Real Option theory extends standard valuation methods by valuating of flexibility in managerial decisions which can be undertaken in the future. This can include value of the high uncertainty in project valuation. The final project value can be calculated by combination of static methods, simulations and real option valuation.

Ключевые слова: реальные опционы, экономика, оценки. Keywords: real options, economics, valuations.

1. Introduction to Options

1.1. Options History

One of the first mentions about options was described by Aristotle in 4th century coram nobis. Thales, pre-Socratic Greek philosopher, predicted a great olive harvest, so he bought the "right" for the use of olive presses. When the harvest time came, Thales rented the presses and achieved high prof-it.[1]

Buy options have been often used to assemble large parcels of land from smaller plots owned by separate owners. A potential buyer pays for the right to buy several plots, but he does not have to buy these plots unless he can buy all of it.

In London, put options and "refusals" (call options) at first became well-known trading instruments in the 1690s during the reign of William and Mary.[2]

The modern options market was established in the year 1973. The Chicago Board of Trade (CBOT) opened the Chicago Board Options Exchange (CBOE). The CBOE instituted a new "exchange traded options contract", which was standardized in its terms and conditions. Contract parties no longer had to negotiate the contract conditions every time they wanted to make an option deal.[3]

1.2. Basic Characteristics

Option is a financial derivative contract that provides the right to buy or sell underlying by a certain time in the future at a fixed price. There are two

types of options, a call option and a put option. A call option grants the right to buy underlying. A put option grants the right to sell underlying. The option buyer has the right but not obligation to sell/buy an underlying. The option seller is obliged to sell/buy an underlying. The option buyer also called option holder is in the long position and option seller also called option writer is in the short position.[4]

To obtain the right, the option buyer pays the seller option price when the option contract is initiated. The option price is often called the option premium.

The price at which the option holder can buy or sell the underlying is called strike or exercise price. The exercising the option means to use the right to buy or sell the underlying. The option holder cannot exercise the option after expiration date.

If the option holder is exercising a call, he pays the exercise price and receives underlying or an equivalent cash settlement. The option writer receives the exercise price and delivers the underlying or pays equivalent cash settlement. If the option holder is exercising a put, he delivers the stock and receives the exercise price or a cash equivalent from option writer.

1.3. Option Styles

Options can be divided into a several categories according to exercise style and pay-off value calculation. Vanilla and exotic options are the two major categories. Vanilla options are the regular, most traded options with standard features like expi-

Экономика и предпринимательство, № 3, 2013 г.

ration date and exercise price. European and American options belong to the vanilla option category. Vanilla options are usually traded through the stock exchange. Exotic options mostly differ from the vanilla options in the calculation of their pay-off value. The trading is usually realized in over-the-counter manner.

European option can be exercised only in option expiration time.

American option can be exercised anytime between the time of contract initiation and option expiration.

1.4. Moneyness of an Option

Moneyness of an option refers to the relationship between the exercise price and the price of the underlying.

In-the-money are options in which exercising would produce more cash inflow than cash outflow. The call options are in-the-money when the value of underlying is higher than exercise price. The put options are in-the-money when the value of underlying is lower than exercise price.

Out-of-money are options with a negative intrinsic value. These are the opposite of the in-the-money options.

At-the-money are options with exercise price equal or nearly equal to the current underlying price.

1.5. Option Types

Financial options are options in which the underlying is a financial asset, interest rate or a currency.

There are several types of financial options. These are stock options, index options, bond options, interest rate options and currency options. The stock options, also called equity options, are the most popular.

There are also other types of options such as option on futures, commodity options, real options and others. In options on futures the underlying is futures contract. Options in which the asset underlying the futures is a commodity, such as oil, gold, rice are called commodity options. The real options will be discussed in the later part of this paper.

2. Option Valuation

Call option value = Max(0, ST - E) (1)

Put option value = Max(0, E - ST) (2)

where St is price of the underlying at the expiration time and E is exercise price.

Tab. 1 Option values

ST —

Option Minimum value Maximum value

European call 0 S0

American call 0 S0

European put 0 E/(1+rf)'

American put 0 E

The table above explains boundary values for various types of options. So is price of the underlying at the time 0, rf is risk-free rate and T is time to expiration.

2.1. Asian option

Asian option exercise price is based on average value during its life. Exercise style can be both American and European. The determining underlying value of the Asian option at the time T is equal to arithmetic average.

1 t

- J S(t)dt

1 N

St — N ? S

(3)

(4)

This option is widely used in energy sector because the price of the oil, gas, electricity or other commodities are highly volatile and these are used as the underlying or the underlying depends on the value of these commodities. The advantage of using the Asian option is that determining underlying value is not as highly volatile as the value of underlying.

Also the other kinds of averages can be used for the calculation of determining underlying value. Harmonic, geometric or weighted averages are often used.

2.2. Option Pricing Models 2.2.1. Black-Scholes Model In the year 1973, Fischer Black and Myron Scholes published the article about option pricing which included also the option pricing formula for European put and call options on a stock that does not pay dividend. [1]

Call price c = N(dl)S0 - N(d2 )Ee~rfT (4) Put price p = N(-d2)Ee~rfT - N(-dl)S0 (5)

d1 —

-f EI+

f

'f +

T

crJT d2 — d1 - (jyfT

(6) (7)

N(di) is rate of change of the option price with respect to the price of underlying asset and N(d2) is probability of exercise in risk-neutral world.

Black-Scholes model assumptions

• No dividends paid during the option's life

• European exercise terms are used

• The stock price follows a geometric Brownian motion with constant volatility

• Possibility to borrow and lend money at a known constant risk-free interest rate

• There are no transaction costs

• There are no restriction on short selling

2.3. Binomial model

The binomial option pricing model is very popular due to its simplicity. It can be applied in more situations than Black-Scholes model. It can be also applied for American options and also takes into account dividends paid.

The idea of the binomial model is in expanding tree. The root node is the current value of underlying. This tree expands and each node represents possible price of the underlying at a particular time. At each step, the underlying will move up o

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