INTRODUCTION TO CALL OPTIONS VALUATION

6.1 INTRODUCTION TO CALL OPTIONS VALUATION

Options are some of the building blocks of modern corporate finance and financial economics. Their mathematical study is in general difficult, however. In this chapter and in the following one, we consider the valuation of options and their use in practice. Terms such as a trading strategy, risk-neutral pricing, rational expectations, etc. will be elucidated in simple mathematical terms. To value an option it is important to define first, and clearly, a number of terms. This is what we do next.

We begin by defining wealth at a given time t, W (t). This is the amount of money an investor has either currently invested or available for investment. Investments can be made in a number of assets, some of which may be risky, providing uncertain returns, while others may provide a risk-free rate of return (as would

be achieved by investing in a riskless bond) which we denote by R f . A risky investment is assumed for simplicity to consist of an investment in securities. Let N 0 be the number of bonds we invest in, say zero coupon of $1 denomination, bearing a risk-free rate of return R f one period hence. Thus, at a given time, our investment in bonds equals N 0 B (t, t + 1) with B(t + 1, t + 1) = 1. This means that one period hence, this investment will be worth B(t, t + 1)N 0 (1 + R f )= N 0 (1 + R f ) for sure. We can also invest in risky assets consisting of m securities each bearing a known price S i (t), i = 1, . . . , m at time t. The investment in securities is defined by the number of shares N 1 , N 2 ,..., N m bought of each security at time t. Thus, a trading strategy at this time is given by the portfolio composition (N 0 , N 1 , N 2 ,..., N m ). The total portfolio investment at time t, is thus given by:

W (t) = N 0 B (t, t + 1) + N 1 S 1 (t) + N 2 S 2 (t) + · · · + N m S m (t)

Risk and Financial Management: Mathematical and Computational Methods. C. Tapiero C 2004 John Wiley & Sons, Ltd ISBN: 0-470-84908-8

OPTIONS AND DERIVATIVES FINANCE MATHEMATICS

For example, for a portfolio consisting of a bond and in a stock, we have: $N

invested in a riskless bond W

(t) = 0

$N 1 S 1 (t)

invested in a risky asset, a stock

W (t) = N 0 +N 1 S 1 (t)

A period later, the bond is cashed while security prices may change in an uncertain manner. That is to say, the price in the next period of a security i is a random variable that we specify by a ‘tilde’, or ˜S i (t + 1). The gain (loss) is thus the random variable:

S i (t) = ˜S i (t + 1) − S i (t), i = 1

Usually, one attempts to predict the gain (loss) by constructing a stochastic process

i (t). The wealth gain (loss) over one period is:

W (t) = ˜ W (t + 1) − W (t)

where, W ˜ (t + 1) = N 0 (1 + R f )+N 1 ˜S 1 (t + 1)

Thus, the net gain (loss) in the time interval (t, t + 1), is:

W (t) = N 0 R f +N 1 S 1 (t)

In general, a portfolio consists of multiple assets such as bonds of various denom- inations and maturities, stocks, options, contracts of various sorts and assets that may be more or less liquid (such as real estate or transaction-cost-prone assets). We restrict ourselves for the moment to an investment in a simple binomial stock and a bond.

Over two periods, future security prices assume two values only, one high S H

(the security price increases), the other low S L (the security price decreases) with 0<S L < S H as well as S L / S≤1+R f ≤S H / S . These conditions will exclude arbitrage opportunities as we shall see later on. Thus stock prices at t and at t + 1 are (see Figure 6.1):

S (t) and

˜S(t + 1) = H

This results in a portfolio that assumes two possible values at time t + 1: N

0 (1 + R f )+N 1 S L In other words, at time t the current time, the price of a stock is known and given

by S = S(t). An instant of time later, at (t + 1), its price is uncertain and assumes the two values (S H , S L ), with S H > S L . As a result, if at t = 0, wealth is invested

in a bond and in a security, we have the investment process given by (see Figure 6.2):

W (0) = N 0 +N 1 S 1 (0) and W ˜ (1) = N 0 (1 + R f )+N 1 ˜S 1 (1)

INTRODUCTION TO CALL OPTIONS VALUATION

Time t

Time t+1

Figure 6.1

where in period 1, wealth can assume two values only since future prices are equal to either of (S H , S L ),S H > S L and the trading strategy is defined by (N 0 , N 1 ). In this specific case, the price process is predictable, assuming two values only. This predictability is an essential assumption to obtain a unique value for the derivative asset, as we shall see subsequently.

For example, say that a stock has a current value of $100 and say that a period hence (say a year), it can assume two possible values of $140 and $70. That is:

H (=140) (t) = S(=100) and ˜S(t + 1) = S L (=70)

The risk-free yearly interest rate is 12%, i.e. R f = 0.12. Thus, if we construct a portfolio of N 0 units of a bond worth each $1 and N 1 shares of the stock, then the portfolio investment and its future value one period hence are:

(0) = N 1

0 (1 + 0.12) + 100N 1 and W ˜ (1) = N 0 (1 + 0.12) + 70N

Now assume that we want to estimate the value of an option derived from such a security. Namely, consider a call option stating that at time t = 1, the strike time,

the buyer of the option has the right to buy the security at a price of K , the exercise or strike price, with, for convenience, S H ≥K≥S L . If the price is high, then the gain for the buyer of the option is S H − K > 0 and the option is exercised

Time t

Time t+1

Figure 6.2

OPTIONS AND DERIVATIVES FINANCE MATHEMATICS

while the short seller of the option has a loss, which is K − S L . If the price is low (below the strike price) then there is no gain and the only loss to the buyer

of the call option is the premium paid for it initially. The problem we are faced with concerns the value/price of such a derived (option) contract. In other words, how much money would the (long) buyer of the option be willing to pay for this right. To find out, we proceed as follows. First, we note the possible payoffs of

the option over one period and denote it by ˜ C (1). Then we construct a portfolio replicating the exact cash flow associated to the option. Let the portfolio worth at the strike time be ˜ W (1):

0 (1 + R f 1 S H (1) = N 0

(1 + R f )+N 1 ˜S(1) =

)+N

0 (1 + R f )+N (1) ≡ ˜ 1 S L To determine this equivalence, the portfolio composition N 0 , N 1 has to be deter-

and W ˜

mined uniquely. If it were not possible to replicate the option cash flow uniquely by a portfolio, then we would not be able to determine a unique price for the option and we would be in a situation we call incomplete. This conclusion is based on the economic hypothesis that two equivalent and identical cash flows have necessarily the same economic value (or cost). If this were not the case, there may be more than one price or no price at all for the derivative asset. Our ability to replicate a risky asset by a portfolio uniquely underlies the notion of the ‘no arbitrage’ assumption, which implies in turn the ‘law of the single price’. Thus, by constructing portfolios that have exactly the same returns with the same risks, their value ought to be the same. If this were not the case, then one of the two assets would be dominated and therefore their value could not be the same. Further, there would be an opportunity for profits that can be made with no in- vestment – or equivalently, an opportunity for infinite rates of returns (assuming perfect liquidity of markets) that cannot be sustained (and therefore not maintain

a state of equilibrium). Thus, to derive the option price, it is sufficient to estimate the replicating portfolio initial value. This is done next. Say that, for a call option, its value one period hence is:

if the security price rises (1) =

H −K

0 if the security price decreases where S L < K<S H . A replicating portfolio investment equivalent to an option

would thus be:

W ˜ (1) = ˜ C (1)

Or, equivalently, W ˜

0 (1 + R f )+N 1 S H =S H −K

(1) = ˜ C (1) ⇔ N 0

(1 + R f )+N 1 S L =0 Note that these are two linear equations in two unknowns and have therefore a

unique solution for the replicating portfolio:

S L (S H −K) N 1 =

S H −K

, N 0 =−

(S H −S L )

(1 + R f )(S H −S L )

135 The procedure followed is summarized below.

INTRODUCTION TO CALL OPTIONS VALUATION

The call option’s payoff is replicated by holding short bonds to invest in a stock (N 0 <

0, N 1 > 0). As the stock price increases, the portfolio is shifted from bonds to stocks. As a result, calling upon the ‘no arbitrage’ assumption, the option price and the replicating portfolio must be the same since they have identical cash flows. That is, as stated above:

W ˜ (1) = ˜ C (1) ⇔ W (0) = C(0) and since: W (0) = N 0 +N 1 S (0) We insert the values for (N 0 , N 1 ) calculated above and obtain the call option

price :

(S(1 + R f )−S L )(S H C −K)

(1 + R f )(S H −S L )

Thus, if we return to our portfolio, and assume that the option has a strike price of $120, then the replicating portfolio is:

1.12 and further, the option price is:

which can be calculated directly from the formula above:

C [S(1 + R )−S

(1 + 0.12)(140 − 70) By the same token, say that the current price of a stock is S = $100 while the price

(1 + R f )(S H −S L )

a period hence (at which time the option may be exercised) is either S H = $120 or S L = $70. The strike price is K = $110 while the discount rate over the relevant period is 0.03. Thus, a call option taken for the period on such a stock has a price, which is given by:

C (0) = (100 − 70)(120 − 110) = $5.825 (1 + 0.03)(120 − 70)

6.1.1 Option valuation and rational expectations

The rational expectations hypothesis claims that an expectation over ‘future prices’ determines current prices (see Figure 6.3). That is to say, assuming that

OPTIONS AND DERIVATIVES FINANCE MATHEMATICS

Future prices based on the current

Current

information Price

Figure 6.3

rational expectations hold, there is a probability measure that values the option in terms of its expected discounted value at the risk-free rate, or

where E ∗ is an expectation taken over the appropriate probability measure as- sumed to exist (in our current case it is given by [ p ∗,1 −p ∗ ]) and therefore:

C (0) = [p ∗ C (1|S ∗ H ) + (1 − p )C(1|S L )]

1+R f

where C(1 |S H )=S H − K , C(1 |S L ) = 0 are the option value at the exercise time and p ∗ denotes a ‘risk-neutral probability’. This probability is not, however,

a historical probability of the stock moving up or down but a ‘risk-neutral prob- ability’, making it possible to value the asset under a risk-neutrality assumption . In this case, the option’s price is the discounted (at a risk-free rate) expected value of the option,

C [p ∗ (S

(0) = ∗ H − K ) + (1 − p )(0)]

1+R f

And, using the value of the option found earlier, we have:

0≤p = S [(1 + R f )S − S L ]≤1

H −S L

In our previous example, we have:

0≤p ∗ = [(1 + 0.03)100 − 70] = 0.66

137 By the same token, we can verify that:

INTRODUCTION TO CALL OPTIONS VALUATION

1.03 with p ∗ = 0.66. This ‘risk-neutral probability’ is determined in fact by traders

in financial markets interacting with others in developing the financial market equilibrium – where profits without risk cannot be realized. For this reason, ‘risk-neutral pricing’ is ‘determined by the market and provides the appropriate discount mechanism to value the asset in the following form (see also Chapter 3 and our discussion on the stochastic discount factor):

C (0) = E{m 1 C ˜ (1)}; m 1 =

1+R f Risk-neutral probabilities, as we have just seen, allow a linear valuation of the

option which hinges on the assumption of no arbitrage. Nonetheless, the existence of risk-neutral probabilities do not mean that we can use linear valuation, for to do so requires markets completeness (expressed by the fact in this section that we were able to replicate by portfolio the option value and derive a unique price of the option). In subsequent chapters, we shall be concerned with market incompleteness and see that this is not always the case. These situations will complicate the valuation of financial assets in general.

6.1.2 Risk-neutral pricing

The importance of risk-neutral pricing justifies our considering it in greater depth. In many instances, security prices can be conveniently measured with respect to a given process – in particular, a growing process called the numeraire, expressing the value of money (money market), a bond or some other asset. That is, allowing us to write (see also Chapter 3):

(S(t)) = ∗ E ∗ (V ( ˜S(t + 1))) = [p V (S H ) + (1 − p )V (S L )]

1+R f 1+R f

p ∗ is said to be a ‘risk-neutral probability’ and R f is a risk-free discount rate.

And for an option (since R f has a fixed value):

In general, for any value (whether it is an option or not) ˜ V i at time i with a risk-free rate R f , we have, over one period: V

0 =E ∗

1+R f V 1

By iterated expectations, we have as well:

V 1 =E ∗

V ˜ 2 and

1+R f

V ˜ 2 =E ∗

OPTIONS AND DERIVATIVES FINANCE MATHEMATICS

and therefore, over n periods:

0 , the information regarding the process initially, then we write

V 0 n (1 + R f ) =E ∗ (˜ V n 0 )

Further, application of iterated expectations has shown that this discounting pro- cess defines a martingale. Namely, we have:

k ; k = 0, 1, 2, . . . and n = 1, 2, 3, . . . or, equivalently,

V (S 0 ) = (1 + R f ) −k V (S k

)=E ∗ (1 + R f ) −(k+n) V (S

k+n

k }; k = 0, 1, 2, . . . and n = 1, 2, 3, . . . This result can be verified next using our binomial model. Set the unit one period

risk-free bond, B(t) = B(t, t + 1) for notational convenience, then discounting

a security price with respect to the risk-free bond yields:

S (t) S (t) =

S (t)

B (t) (t) = (1 + R f ) t and S ∗ (t) is a martingale. Generally, under the risk-neutral measure, P ∗ the dis-

or S ∗

counted process {(1 + R f ) −k S k k }, k = 0, 1, 2, . . . is, as we saw earlier, a martingale. Here again, the proof is simple since:

k = (1 + R f ) k+1 −k S k and

E ∗ (1 + R f ) −(k+1) S

S H +q ∗ S L S k S k

E ∗ (1 + R f ) −(k+1) S k+1 k = (1 + R f ) −(k+1) p ∗

= (1 + R f ) −(k+1) S k [(1 + R f )] = (1 + R f ) −k S k

This procedure remains valid if we consider a portfolio which consists of a bond and m stocks. In this case, dropping for simplicity the tilde over random variables, we have:

W ∗ (t) = N 0 +N 1 S ∗ 1 (t) + N 2 S ∗ 2 (t) + · · · + N m S ∗ m (t) W ∗ (t + 1) = N 0 +N 1 S ∗ 1 (t + 1) + N 2 S ∗ 2 (t + 1) + · · · + N m S ∗ m (t + 1)

and W ∗ (t) = N 1 S ∗ 1 (t) + N 2 S ∗ 2 (t) + · · · + N m S ∗ m (t)

Equating these to the value of some derived asset, a period hence:

(t + 1) = C ∗ (t + 1)

INTRODUCTION TO CALL OPTIONS VALUATION

139 leads to a solution for (N 0 , N 1 , N 2 ,..., N m ) where C ∗ (t + 1) is a vector of assets

we use to construct a riskless hedge and replicate the derivative product we wish to estimate (Pliska (1997) and Shreve et al. (1997) for example).

Example: Options and portfolios holding cost

Consider now the problem of valuing the price of a call option on a stock when the alternative portfolio consists in holding a risky asset (a stock) and a bond, for which there is a ‘holding cost’. This cost is usually the charge a bank may require for maintaining in its books an investor’s portfolio. In this case, the hedging portfolio is given by equating:

0 (1 + R f −c B )+N 1 (S H −c S ) (1) = N 0 (1 + R f −c B )+N 1 (S L −c S )

where c B is the bond holding cost and c S is the stock holding cost. The option’s cash flow is:

˜ if the security price rises

C H (1) = −K

0 if the security price decreases This leads to:

(S H −S L )(1 + R f −c B ) Therefore, the option price is equal instead to:

(S H −S L )(1 + R f −c B ) For example, if we use the data used in the previous option’s example with

H −S L

S = 100, S H = 140, S L = 70, K = 120, R f = 0.12 and the ‘holding costs’ are: c S = 5, c B = 0.02, then

7 − (70)(1.1) = 28.57 − 16.88 = 11.68 which compares to a price of 10.64 without the holding cost. In this sense, holding

C (0) =

costs will increase the price of acquiring the option. A general approach to this problem is treated by Bensoussan and Julien (2000) in continuous-time models. The costs of holding, denoted friction costs, are, however, far more complex, leading to incompleteness.

OPTIONS AND DERIVATIVES FINANCE MATHEMATICS

C = Max , 0 HS HH 2 [ − K ]

C H = Max [ 0 , HS − K ]

C HL = Max , [ 0 HLS − K ]

C LL = Max , [ 0 LS 2 − K

C = Max , 0 LS − K

Figure 6.4

A two-period binomial tree.

6.1.3 Multiple periods with binomial trees

Over two or more periods, the problem remains the same. For one period, we saw that the price of a call option is: C H = Max [0, S H −K];C L = Max [0, S L − K ] and by risk-neutral pricing,

Over two periods, we have:

which we insert in the previous equation, to obtain the option price for two periods (see Figure 6.4). Explicitly, we have the following calculations:

Generally, the price of a call option at time t whose strike price is K at time T can be calculated recursively by:

C (t) = E ∗

C ˜ (t + 1) ; C(T ) = Max [0, S(T ) − K ] 1+R f

Explicitly, if we set, S H = HS, S L = LS, we have :

S−K) +

) 2 (H 2 S−K) +

j =0

p ∗j ∗ ) 2− j {H j L 2− j S−K} (1 − p +

141 We generalize to n periods and obtain by induction:

FORWARD AND FUTURES CONTRACTS

We can write this expression in still another form:

P j ( ˜S j −K) + (1 + R f )

E {(S n −K) n + }=

(1 + R f ) j =0 where

(1 − p ) are the risk-neutral probabilities. This expression is of course valid only under

j = P(S n =H L n− j S )= ∗ p ∗j n− j

the assumption of no arbitrage. This mechanism for pricing options is generally applicable to other types of options, however, such as American, Look-Back, Asiatic, esoteric and other options, as we shall see later on.

The option considered so far is European since exercise of the option is possi- ble only at the option’s maturity. American options, unlike European ones, give the buyer the right to exercise the option before maturity. The buyer must there- fore take into account to optimal timing of his exercise. An option exercised too early may forgo future opportunities, while exercised too late it may lose past opportunities. The optimal exercise time will be that time that balances the live value of the option versus its ‘dead’ or exercise value. The recursive solution of the European call option can be easily modified for the exercise feature of the American option. Proceeding backward from maturity, the option will be exer- cised when its ‘dead’ value is larger than its ‘live’ one. Technically, the exercise time is a stopping time, as we shall see subsequently. Note that early exercise of the option is optimal only if the option value diminishes. For a call option (and in the absence of dividends), it does not diminish over time and therefore it will never pay to exercise an option early. For this reason we note that the price of

a European and an American call are equal. For a put option, the present value of the payoff is a decreasing function of time hence, early exercise is possible irrespective of the existence of dividend payments.