Option Valuation

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Option Valuation

• Lecture XXI

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• 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.

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• 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.

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• 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.

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• 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

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• 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.

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• Given these differences let F(S,tT,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,tT,x) is the price of a European call
  option, G(S,tT,x) is the price of an American
  put option, and g(S,tT,x) is the price of the
  European put option.

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• 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,TT,x)f(S,TT,x)Max(S-x,0),
  G(S,TT,x)g(S,TT,x)Max(S-x,0).
• Proposition 3 F(S,tT,x) ? S-x, G(S,tT,x) ?
  x-S.
• Proposition 4 For T2gtT1 F(.T2,x) ? F(.T1,x),
  G(.T2,x) ? G(.T1,x).

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• Proposition 5 F(.) ? f(.) and G(.) ? g(.).
• Proposition 6 For x1 gt 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,tT,x) ?
  f(S,tT,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.

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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).

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• 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

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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.

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• Assume a very simple payoff structure
• Under the Bernoulli structure, the value of the
  payoff is y95 with probability (1-p) and y105
  with probability p.

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• Assume a strike price for a call option of 100.
  If the event is x1, implying that y105, the
  value of the call option is 5. However, if the
  event is x0 implying that y95, the value of
  the call option is 0.
• The question is then How much is the call option
  worth?

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• 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.

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• 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

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• Extending the payoff formulation
• In this case, y110 if z2 which occurs with
  probability p2, y100 if z1 which occurs with
  probability 2p(1-p), and y90 if z0 which
  occurs with probability (1-p)2.

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• 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.

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Binomial Distribution
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• Mathematically, the probability of r heads out
  of n draws becomes

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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

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• 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

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• 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.

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• 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 s consistent
  with this increment in time. Assume it to be .2.
• The two constants can then be computed as

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• 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.

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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.

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• 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.

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• 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.

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• 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

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• 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

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• Comparing the two stochastic processes leads to a
  comparison of a and m.
• The relationship between these two values gives
  rise to the execution of the option.
• Defining dm-a to the the dividend associated
  with owning the asset. a is the capital gain
  while m operating return.
• If d is less than or equal to zero, the option
  will never be exercised. Thus, d gt0 implies that
  the operating return is greater than the capital
  gain on a similar asset.

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• 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.

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• 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).

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• Combining this expression with the original
  geometric process and applying Itos Lemma we
  derive the combined zero-profit and zero-risk
  condition
• In addition to this differential equation we
  have three boundary conditions

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

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• b then simplifies to

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• Estimating b
• In order to incorporate risk into an investment
  decision using the Dixit and Pindyck approach we
  must estimate s.
• This one approach to estimating s is through
  simulation. Specifically, simulating the
  stochastic Net Present Value of an investment as

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• Converting this value to an infinite streamed
  investment then involves

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

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• 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.

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