The Black and Scholes formula

Fischer Black and Myron Scholes offered a break-through option valuation model in a publication in 1973. It was the first ever closed-form formula that allowed to compute precisely the right price for a generic European stock option. By "closed-form" understand that the formula is sufficient in itself, and doesn't require iterations or additional processing, simply input the right variables in the formula and you get the result.

Here I will simply state the Black and Scholes formula and the main assumptions that accompany it.

I won't explain here the meaning and interpretation of each chunk of the formula, but I explain a bit more here on the intuitions and explanations behind the Black and Scholes formula.

The Black & Scholes option pricing formula

Without further introduction (and I know most of you have no time to lose in useless digressions), behold the so called Black and Scholes option call and put pricing formula:

Call option price

$$C(S_0,t)=N(d1)S_0 - N(d2)K e^{-r(T-t)} $$

Put - Call parity


Put option price

$$P(S_0,t)=N(-d2)K e^{-r(T-t)}-N(-d1)S_0 $$


$$d1=  1/(σ√(T-t)) [ln(S_0/K)+(r+σ^2/2)(T-t)]$$

$$d2=  1/(σ√(T-t)) [ln(S_0/K)+(r-σ^2/2)(T-t)]$$

$$d2= d1- σ√{T-t}$$

Here are the main parameters:

  • σ is the (flat) volatility of the underlying. It represent the risk and uncertainty of the return distribution, it is used in d1 and d2.
  • K is the strike price of the option, $S_0$ is the underlying price when the option is traded,
  • t is the time at which we compute the option price, T is the time to maturity (t < T obviously),
  • r is the (flat) risk-free interest rate,
  • N() is the standard normal cumulative probability distribution, defined as $N(x)= 1/{√{2π}} ∫_{-∞}^x e^{-{x^2}/2} dz$
  • N(d1) is the probability of expiring in the money under the equivalent exponential martingale measures using the stock as numéraire and
  • N(d2) is the probability of expiring in the money using the risk-free asset as numéraire.

The assumptions behind the Black and Scholes formula

Like many advanced financial models, the Black and Scholes approach to option pricing is crippled by its assumptions, that one could dub unrealistic and naïve. The Black and Scholes formula has even been described as the model that requires "the wrong number in the wrong formula to get the right price".

Here are the main assumptions of the Black and Scholes model:

Assumptions on the market

The Black and Scholes model assumes that:

  • The market offers at least two assets: a risk free asset and a risky asset (that is the underlying of the option), and they are available in any amount. Regarding the risk free asset, it is possible to borrow at the risk free rate.
  • Short selling is allowed;
  • The market has no arbitrage opportunity;
  • There are no transaction costs in the market. The market can be said then to be “frictionless”.
  • It is possible to trade an option on the risky asset in this market.

Assumptions on the assets

The Black and Scholes model assumes that:

  • On the “risk free asset”: the return on the risk free asset is constant.
  • On the “risky asset”:
    • In this version of the BS model, the risky asset doesn’t pay a dividend (it is however easy to modify the formula so this assumption isn’t necessary anymore);
    • The instantaneous log return on the risky asset is a geometric Brownian motion, with a constant drift and volatility. In other words the log return on the risky asset is normally distributed.



Matthieu Liatard
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