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Black-Scholes Models

What is the Black Scholes Model?

The Black Scholes model was the first widely used model for option pricing. It is used to calculate the theoretical value of European-style options by using current stock prices, expected dividends, expected interest rates, the option’s strike price, time to expiration and expected volatility.

The model makes certain assumptions:

  • The option is European and can only be exercised at expiration.

  • No dividend is paid out during the life of the option.

  • Markets are efficient and cannot be predicted.

  • No transaction costs are incurred when buying an option.

  • The risk-free rate and volatility of the underlying are known and constant.

  • The returns on the underlying are normally distributed.

The Black-Scholes Formula

The Black Scholes formula is calculated by multiplying the stock price by the cumulative standard normal probability distribution function. Then, the net present value (NPV) of the strike price multiplied by the cumulative standard normal distribution is subtracted from the resulting value of the previous calculation. C = SN(d1) - Ke^(-rT)*N(d2).

Conversely, the value of a put option could be calculated using the formula: P = Ke^(-rT)N(-d2) - S*N(-d1). In both of these formulas, S is the stock price, K is the strike price, r is the risk-free interest rate and T is the time to maturity. The formula for d1 is: (ln(S/K) + (r + (annualized volatility)^2 / 2)*T) / (annualized volatility * (T^(0.5))). The formula for d2 is: d1 - (annualized volatility)*(T^(0.5)).

Criticisms of the Model

  • Tail-risk fragility or other extreme randomness: Generally, returns do not follow a normal distribution. The p-value on the Normality Test is 0.0 when applied to S&P returns meaning market returns are leptokurtic.

  • The structure does not reflect present realities: it assumes a market is using European options. It also does not allow for dividends.

  • Assumes the interest rate is risk-free.

  • Assumption of cost-less trading: there are usually trading fees such as the cost to buy or sell stocks and options, cost of time etc. These costs are not included in the model.

  • Gap-Risk: the model assumes that trading continues continuously, unlike reality, where markets close overnight and then reopen with different prices to reflect new information.

For more on the Black Scholes Formula:

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