Option Valuation Lecture XXI

n 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 o Price of the call  Strike Price Stock Price

n 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.

n 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 T 2 >T 1 F(.;T 2,x)  F(.;T 1,x), G(.;T 2,x)  G(.;T 1,x).

–Proposition 5: F(.)  f(.) and G(.)  g(.). –Proposition 6: For x 1 > x 2 F(.,x 1 )  F(.,x 2 ) and f(.,x 1 )  f(.,x 2 ), and G(.,x 1 )  G(.,x 2 ) and g(.,x 1 )  g(.,x 2 ) –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 n 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:

Binomial Pricing Model n 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.

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.

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?

n 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.

n 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:

Extending the payoff formulation In this case, y=$110 if z=2 which occurs with probability p 2, 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.

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.

Binomial Distribution

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

Black-Scholes n The Black-Scholes pricing model extends the binomial distribution to continuos time. n The derivation of the Black-Scholes model is beyond this course. However, the formula for pricing a call option is

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

n 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:

The two N(.) can be derived from a standard normal table as N(d 1 )=.742 and N(d 2 )= Plugging these values back into the option formula yields a call price of $7.62.

Option Value of Investments n 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.

n 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:

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

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 F V (V) is the derivative of the option price with respect to value of the original asset.

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- F 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

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

 then simplifies to

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

Converting this value to an infinite streamed investment then involves:

The parameters of the stochastic process can then be estimated by

n 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 with a standard deviation of

Assuming a mean of the log change of zero, the computed value of  is implying a  /(  -1) of Hence, the risk adjustment raises the hurdle rate to $ Alternatively, the value of the option to invest given the current scenario is $