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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: p Strike Price Price of the call 45 o 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: Distribution Payoff x S 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. Px p 1 p x 0,1 x 1 x • Assume a very simple payoff structure y 100 10 x 1 2 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? f 100;1, p 5 p 01 p If p=.5, then the call option is worth $2.5. How much is the put option worth? g 100;1, p 0 p 51 p Again, if p=.5, the put option is worth $2.5. 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: z 0 x1 0 x2 0 z 1 x1 1 x2 0 or x1 0 x2 1 z 2 x1 1 x2 1 • Extending the payoff formulation y $100 10z 1 • 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. • Now the call option is worth f 100;2, p 10 p 0 2 p1 p 01 p 2 2 which again equals 2.5 if p=.5. The call option for a strike price of $95 is now f 95;2, p 15 p 5 2 p1 p 01 p 2 2 which equals 6.25 if p=.5. Binomial Distribution 3z 2z p3 z p2 2z p z p2(1-p) 0 p(1-p) z 0 1-p p(1-p)2 0 (1-p)2 (1-p)3 • Mathematically, the probability of r “heads” out of n draws becomes n r Pr n, p p 1 p p 1 p nr n! r nr r r!n r ! 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 c S N d1 Xe N d 2 rf T d1 X r T 1 ln S f T T 2 d 2 d1 T 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 z 2 N d 1 2p exp 2 dz 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: d1 ln 50 45 .06 14 1 .2 1 .65 2 4 .2 1 4 d 2 d1 .2 1 .4264 4 • 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: d V 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 d= to the the dividend associated with owning the asset. is the capital gain while “operating” return. – If d is less than or equal to zero, the option will never be exercised. Thus, d >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 F (V ) FV (V ) V 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. • 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 -d 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 FV dV d V FV dt r F FVV dt • Combining this expression with the original geometric process and applying Ito’s Lemma we derive the combined zero-profit and zero-risk condition 1 2 2 V FVV r d V FV rF 0 2 In addition to this differential equation we have three boundary conditions F (0) 0, F (V ) V I , FV (V ) 1 * * * The solution of the differential equation with the stated boundary conditions is: F (V ) V V * I * V V * 1 I 1 1 r d r d 1 2 r 2 2 2 2 2 2 2 then simplifies to 1 1 8r 2 1 1 2 2 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 N t CFi Vt i 1 (1 r ) i • Converting this value to an infinite streamed investment then involves: Vt APVt 1 1 (1 r ) N r APVt Vt * r • The parameters of the stochastic process can then be estimated by dV d lnV V lnVt 1 lnVt 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.

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posted: | 12/1/2011 |

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