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Option Pricing using Black-Scholes. Lecture XXIX. The European Call Option. First, we construct the payoff function for a security on which the option is written as: where x j ( k ) is the payoff a share of security j purchased an exercise price of k . .

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the european call option
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.

slide3
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.
slide4
The initial cost of the strategy is:
  • The possible payoffs of this strategy are
slide5
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 can’t make something for nothing)
slide6
Starting with a two-period economy, we start by assuming a power utility function for the representative agent:

where  is the time preference parameter

slide8
Next, we assume that x and C are lognormally distributed:

where  is the correlation coefficient.

slide9
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:
slide10
Some mathematical niceties:
    • Dealing with the lower bound of the integral:
slide14
To finish the derivation, we assume

or that future consumption is discounted at the risk-free rate of return.

slide15
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).

ito s lemma formulation
Ito’s Lemma formulation
  • Stochastic Process: the equation of motion:
  • Defining the Wiener increment (following Kamien and Schwartz’s definitions):