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Black Scholes Theory

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The proposal by Black and Scholes in 1973 was the first explicit solution to obtain the price of a simple option and gave rise to the modern theory of option pricing. To obtain the valuation formula for an option based on the share price, Black and Scholes, and like the binomial method, assume a series of ideal conditions or market starting hypotheses, both for the stock and the option. These basic hypotheses are as follows:

1. Perfect and competitive capital markets are assumed, in which there are no taxes, transaction costs, or limitations on overdraft operations.

2. There are no restrictions on short-term purchases or sales of securities and options, so the volume of transactions will not affect the market price of the securities.

3. There is a risk-free interest rate, known and constant over time. It is possible to lend and borrow any amount of money at the interest rate without risk.

4. Trading in the markets is continuous.

5. The basic stock or title does not pay dividends or any other type of distribution of profits, reserves, or capital.

6. The price of the call option only depends on the price of the basic security, the time, and variables that are assumed to be known constants.

7. The instantaneous rate of return on the price of the safety follows a diffusion procedure with a constant mean, a, and constant instantaneous variance 2 (Wiener process).

Black-Scholes Price Valuation Formula for a European Call Choice:

C = S N (d 1) - E e -r, t N (d 2) Being:

Black Scholes Theory

In which:

c = call option premium.

S = current price of the share.

E = option strike price.

rf = interest rate without risk.

t = time remaining for option expiration.

2 = instantaneous variance of the stocks return.

N (d) = normal distribution function.

In the case of a put preference:

P = E e -r, t N (d2) - S N (d1)

Where p is the put selection premium.

The Black and Scholes option pricing model is very simple, since once d 1 and d 2 have been calculated, to obtain N (d) you only have to consult the standardized tables of the normal distribution.

Subsequently, Black-Scholes developed in 1976 a formula for valuing options on futures contracts based on the following hypotheses:

1. The price distribution of the futures contract follows a lognormal.

2. The prices of the contracts vary continuously, without jumps.

3. There is a risk-free interest rate, rf, known and constant over time. It is possible to lend and borrow any amount of money at the interest rate without risk.

4. The standard deviation of contract return (volatility) is constant.

5. There are no transaction costs or taxes.

6. There are no restrictions on buying or selling short or fractional.

The resulting formula is as follows:

Formula Black-76 for European CALL options on futures:

C = e -rt (F N (d1) - E N (d2))

Black Scholes Theory Academic Assistance through Online Tutoring

In which:

E = option strike price.

F = price of the future.

t = time remaining for option expiration.

= volatility of the future.

N (d) = normal distribution function.

In the case of a PUT option:

P = (E - F) e -rt + c

Black Scholes Theory Academic Assistance through Online Tutoring Black Scholes Theory Academic Assistance through Online Tutoring

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Black Scholes Theory Academic Assistance through Online Tutoring

Black Scholes Theory Academic Assistance through Online Tutoring

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