Option Pricing using BlackScholes

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Option Pricing using Black-Scholes

• Lecture XXIX

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The European Call Option

• First, we construct the payoff function for a
  security on which the option is written as
• where xj(k) is the payoff a share of security j
  purchased an exercise price of k.

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• The price of the European call option is then
  defined as
• Consider the strategy of selling one share of the
  security to buy one European call option written
  on the security.

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• The initial cost of the strategy is
• The possible payoffs of this strategy are

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• In the first case, you exercise the option buying
  back the stock at the original price while in the
  second case the investor makes money because the
  stock price decreased (you make profit equal to
  the decrease in the stock price).
• Therefore, to avoid a risk-less profit (you cant
  make something for nothing)

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• Starting with a two-period economy, we start by
  assuming a power utility function for the
  representative agent
• where ? is the time preference parameter

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• The arbitrage condition (selling short a share of
  stock and purchasing a call option) then implies

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• Next, we assume that x and C are lognormally
  distributed
• where ? is the correlation coefficient.

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• This assumption implies that ln(S/x) (the return
  on the short sale) and ?ln(C/C0) are normally
  distributed
• The value of the call option can then be written
  as

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• Some mathematical niceties
• Dealing with the lower bound of the integral

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• Next, because of the geometric nature of the
  distribution function

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• The conditional distribution

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• Now back to the original integral

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• To finish the derivation, we assume
• or that future consumption is discounted at the
  risk-free rate of return.

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• In addition, we assume
• which is implicitly the pricing condition of
  stock in period 0 given its utility distribution
  in period 1 (enforces a zero arbitrage condition
  on the stock price).

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Itos Lemma formulation

• Stochastic Process the equation of motion
• Defining the Wiener increment (following Kamien
  and Schwartzs definitions)

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• The expectation and the variance of the equation
  of motion for equity can then be defined as

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• Thus, we can rewrite the Black and Scholes result
  using stochastic process as

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Table 1. Black-Scholes Example
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Table 1. Black-Scholes Example
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Comparing Option Values