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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 xj(k) is the payoff a share of security j

purchased an exercise price of k.

- 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.

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

- 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)

- Starting with a two-period economy, we start by

assuming a power utility function for the

representative agent - where ? is the time preference parameter

- The arbitrage condition (selling short a share of

stock and purchasing a call option) then implies

- Next, we assume that x and C are lognormally

distributed - where ? is the correlation coefficient.

- 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

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

- Next, because of the geometric nature of the

distribution function

- The conditional distribution

- Now back to the original integral

- To finish the derivation, we assume
- or that future consumption is discounted at the

risk-free rate of return.

- 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).

(No Transcript)

Itos Lemma formulation

- Stochastic Process the equation of motion
- Defining the Wiener increment (following Kamien

and Schwartzs definitions)

- The expectation and the variance of the equation

of motion for equity can then be defined as

- Thus, we can rewrite the Black and Scholes result

using stochastic process as

Table 1. Black-Scholes Example

Table 1. Black-Scholes Example

Comparing Option Values

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