Option Valuation

Lecture XXI

What is an option?• In a general sense, an option is exactly

what its name implies - An option is the opportunity to buy or sell one share of stock or lot of commodity at some point in the future at some state price.

– For example, a call option entitles the purchaser to purchase a stock or commodity in the future at some state price.

– Assume that a call contract stated that the holder of the contract was entitled to purchase one share of IBM at $120 at some point in the future. Suppose further that the right cost $2 per share. At the time the contract matured, suppose that the price of IBM was $130. The gain to the holder of the instrument would be $8. In other words, the investor would exercise the option to purchase one share of IBM at $120 and sell the share for $130. Hence, the investor would gross $10. Of this $10, the call cost $2.

– On the flip side, the instrument that gives the bearer the right to sell a stock at a fixed price in the future is called a put.

• Both of these contracts are called contingent claims. Specifically, the claim only has value contingent on certain outcomes of the economy.

– In the call security, suppose that the market price for IBM was $110. The rational investor would choose not to exercise their right.

– Exercising their right would mean purchasing a stock for $120 and selling it for $110 for a gross loss of $10 and a total loss of $12.

– Graphically (ignoring the time value of money) the payoff on buying a call is:

45 oPrice of the call

Strike Price

Stock Price

Factors affecting the price of options• Technically, there are two types of options:

a European option and an American Option.

– A European option can only be exercised on the expiration date.

– The American option can be exercised on any date up until the expiration date.

Given these differences let: F(S,t;T,x) denote the value of an American call option with stock price S on date t and an expiration data T for an exercise price of X. Given this notation f(S,t;T,x) is the price of a European call option, G(S,t;T,x) is the price of an American put option, and g(S,t;T,x) is the price of the European put option.

• Risk Neutral Propositions - Simply assume that investors prefer more to less.

– Proposition 1: F(.) 0, G(.) 0, f(.) 0, g(.) 0.– Proposition 2: F(S,T;T,x)=f(S,T;T,x)=Max(S-

x,0), G(S,T;T,x)=g(S,T;T,x)=Max(S-x,0).– Proposition 3: F(S,t;T,x) S-x, G(S,t;T,x) x-

S.

– Proposition 4: For T2>T1 F(.;T2,x) F(.;T1,x), G(.;T2,x) G(.;T1,x).

– Proposition 5: F(.) f(.) and G(.) g(.).

– Proposition 6: For x1 > x2 F(.,x1) F(.,x2) and f(.,x1) f(.,x2), and G(.,x1) G(.,x2) and g(.,x1) g(.,x2)

– Proposition 7: S = F(S,t;,0) F(S,t;T,x) f(S,t;T,x). The first equality involves the definition of a stock in a limited liability economy. If you purchase a stock, you purchase the right to sell the stock between now and infinity. Further, given limited liability, you will not sell the stock for less than zero.

– Proposition 8: f(0,.)=F(0,.)=0.

Valuing Options

Intuitive Determinants of European Option Prices.• Three of the previous results bear restating

– The value of a call option is an increasing function of the spot stock price (S).

– The value of a call option is a decreasing function of the strike price (x).

– The value of a call option is an increasing function of the time to maturity (T).

• The value of an option is an increasing function of the variability of the underlying asset. To see this, think about imposing the probability density function over a “zero price” option:

xS

Payoff

Distribution

Binomial Pricing Model

The simplest form of option pricing model is referred to as a binomial pricing model. It is based on a series of Bernoulli gambles.• A Bernoulli event is the probability

distribution function used for a coin toss.

1,01 1 xppxP xx

• Assume a very simple payoff structure

Under the Bernoulli structure, the value of the payoff is y=95 with probability (1-p) and y=105 with probability p.

2110100 xy

• Assume a strike price for a call option of $100. If the event is x=1, implying that y=$105, the value of the call option is $5. However, if the event is x=0 implying that y=$95, the value of the call option is $0.

• The question is then: How much is the call option worth?

pppf 105,1;100

If p=.5, then the call option is worth $2.5. How much is the put option worth?

Again, if p=.5, the put option is worth $2.5.

pppg 150,1;100

The binomial probability function is the sum of a sequence of Bernoulli events.• For example, if we link to coin tosses

together we have three possible outcomes: 2 heads, 2 tails or one head and one tail.

• Let z be the sum of two Bernoulli events. z could take on the value of zero, one or two:

112

10011

000

21

2121

21

xxz

xxorxxz

xxz

• Extending the payoff formulation

• In this case, y=$110 if z=2 which occurs with probability p2, y=$100 if z=1 which occurs with probability 2p(1-p), and y=$90 if z=0 which occurs with probability (1-p)2.

110100$ zy

• Now the call option is worth

which again equals 2.5 if p=.5. The call option for a strike price of $95 is now

which equals 6.25 if p=.5.

22 1012010,2;100 pppppf

22 1012515,2;95 pppppf

Binomial Distribution

p

1-p

z

0

(1-p)2

p(1-p)

p2

2z

z

0

0

(1-p)3

p(1-p)2

z

p2(1-p)

2z

p3

3z

• Mathematically, the probability of r “heads” out of n draws becomes

rnrrnr pprnr

npp

r

npnrP

1

!!

!1,

Black-Scholes

The Black-Scholes pricing model extends the binomial distribution to continuos time.

The derivation of the Black-Scholes model is beyond this course. However, the formula for pricing a call option is

Tdd

TT

TrXS

d

dNXedNSc

f

Tr f

12

1

21

2

1ln

where S is the price of the asset (stock price), X is the exercise price, rf is the riskless interest rate, and T is the time to expiration. N(.) is the integral of the normal density function:

d

dzz

dN2

exp2

1 2

Example: • Assume that the current stock price is $50,• the exercise price of the American call

option is $45, • the riskless interest rate is 6 percent,• and the option matures in 3 months.

• Given that the interest rate is specified as an annual interest rate, T is implicitly in years. 3 months is then ¼ of a year.

• In addition, we need an estimate of consistent with this increment in time. Assume it to be .2.

• The two constants can then be computed as:

4264.412.

65.412.

2

1

412.

4106.45

50ln

12

1

dd

d

• The two N(.) can be derived from a standard normal table as N(d1)=.742 and N(d2)=.6651. Plugging these values back into the option formula yields a call price of $7.62.

Option Value of Investments

Moss, Pagano, and Boggess. “Ex Ante Modeling of the Effect of Irreversibility and Uncertainty on Citrus Investments.”• Traditional courses in financial management

state that an investment should be undertaken if the Net Present Value of the investment is positive.

• However, firms routinely fail to make investments that appear profitable considering the time value of money.

• Several alternative explanation for this phenomenon have been proposed. However, the most fruitful involves risk.

– Integrating risk into the decision model may take several forms from the Capital Asset Pricing Model to stochastic net present value.

– However, one avenue which has gained increased attention during the past decade is the notion of an investment as an option.

• Several characteristics of investments make the use of option pricing models attractive.

– In most investments, investors can be construed to have limited liability with the distribution being truncated at the loss the the entire investment.

– Alternatively, Dixit and Pindyck have pointed out that the investment decision is very seldomly a now or never decision. The decision maker may simply postpone exercising the option to invest.

Derivation of the value of waiting• As a first step in the derivation of the value

of waiting, we consider an asset whose value changes over time according to a geometric Brownian motion stochastic process:

dV V dt V dt

• Given the stochastic process depicting the evolution of asset values over time, we assume that there exists a perfectly correlated asset that obeys a similar process

dx x dt x dz

r vm

• Comparing the two stochastic processes leads to a comparison of and .

– The relationship between these two values gives rise to the execution of the option.

– Defining = to the the dividend associated with owning the asset. is the capital gain while “operating” return.

– If is less than or equal to zero, the option will never be exercised. Thus, >0 implies that the operating return is greater than the capital gain on a similar asset.

• Next, we construct a riskless portfolio containing one unit of the option to some level of short sale of the original asset

P is the value of the riskless portfolio, F(V) is the value of the option, and FV(V) is the derivative of the option price with respect to value of the original asset.

P F V F V VV ( ) ( )

• Dropping the Vs and differentiating the riskfree portfolio we obtain the rate of return on the portfolio. To this differentiation, we append two assumption:

– The rate of return on the short sale over time must be - V (the short sale must pay at least the expected dividend on holding the asset).

– The rate of return on the riskfree portfolio must be equal to the riskfree return on capital r(F-FVV).

dF F dV V F dt r F F V dtV V V

• Combining this expression with the original geometric process and applying Ito’s Lemma we derive the combined zero-profit and zero-risk condition

In addition to this differential equation we have three boundary conditions

1

202 2 V F r V F rFVV V

F F V V I F VV( ) , ( ) , ( )* * *0 0 1

The solution of the differential equation with the stated boundary conditions is:

F V V( )

V I

V

*

*

V I*

1

1

2

1

222 2

2

2

1

2r r r

then simplifies to

1

21 1

82

1

2r

Estimating • In order to incorporate risk into an

investment decision using the Dixit and Pindyck approach we must estimate .

• This one approach to estimating is through simulation. Specifically, simulating the stochastic Net Present Value of an investment as

VCF

rti

ii

N t

( )11

• Converting this value to an infinite streamed investment then involves:

APVV

rr

tt

N

11

1( )

VAPV

rtt*

• The parameters of the stochastic process can then be estimated by

dVd V

V Vt t

V

ln

ln ln1

Application to Citrus• The simulated results indicate that the present

value of orange production was $852.99/acre with a standard deviation of $179.88/acre.

• Clearly, this investment is not profitable given an initial investment of $3,950/acre.

• The average log change based on 7500 draws was .0084693 with a standard deviation of .0099294.

• Assuming a mean of the log change of zero, the computed value of is 25.17 implying a /(-1) of 1.0414.

• Hence, the risk adjustment raises the hurdle rate to $4113.40. Alternatively, the value of the option to invest given the current scenario is $163.40.