Black-Scholes Formula

A Black-Scholes Formula is a closed-form option price calculation formula for the prices of European-style options.

• Context:
  • It can (typically) predict an Option's Price at a given time based on factors such as the volatility of the underlying asset, the time to expiration, the risk-free interest rate, and the difference between the option's strike price and the underlying asset's current price.
  • It can (typically) be derived from the Black–Scholes Equation by solving it with specific boundary conditions related to the option type.
  • It can provide the theoretical price of a European call option or European put option by considering the current stock price, strike price, risk-free interest rate, time to expiration, and volatility of the stock.
  • It can (typically) be based on the assumption of a log-normally distributed return on the stock and a constant volatility.
  • It can (typically) assume a frictionless market with no transaction costs or taxes and the ability to borrow and lend money at a risk-free rate.
  • It can be used to calculate the Greeks in options trading, which measure the sensitivity of the option's price to its input parameters.
  • It can (typically) be essential to establish the risk-neutral valuation principle, which implies that the option's price is independent of investors' risk preferences.
  • ...
• Example(s):
  • an extension to price various options and other financial instruments (beyond its initial scope through the Black model).
  • ...
• See: Black–Scholes–Merton Model, Black-Scholes Equation, Option Pricing Theory, Volatility (Finance), Risk-Free Interest Rate, Time to Expiration, European Option, Greeks (Finance).

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Black–Scholes_model#Black–Scholes_formula Retrieved:2024-2-27.
  • The Black–Scholes formula calculates the price of European put and call options. This price is consistent with the Black–Scholes equation. This follows since the formula can be obtained by solving the equation for the corresponding terminal and boundary conditions:

[math]\displaystyle{ \begin{align} & C(0, t) = 0\text{ for all }t \\ & C(S, t) \rightarrow S - K \text{ as }S \rightarrow \infty \\ & C(S, T) = \max\{S - K, 0\} \end{align} }[/math]

The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:

[math]\displaystyle{ \begin{align} C(S_t, t) &= N(d_+)S_t - N(d_-)Ke^{-r(T - t)} \\ d_+ &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\ d_- &= d_+ - \sigma\sqrt{T - t} \\ \end{align} }[/math]

The price of a corresponding put option based on put–call parity with discount factor [math]\displaystyle{ e^{-r(T-t)} }[/math] is:

[math]\displaystyle{ \begin{align} P(S_t, t) &= Ke^{-r(T - t)} - S_t + C(S_t, t) \\ &= N(-d_-) Ke^{-r(T - t)} - N(-d_+) S_t \end{align}\, }[/math]

• • Alternative formulation
    • Introducing auxiliary variables allows for the formula to be simplified and reformulated in a form that can be more convenient (this is a special case of the Black '76 formula):

[math]\displaystyle{ \begin{align} C(F, \tau) &= D \left[ N(d_+) F - N(d_-) K \right] \\ d_+ & = \frac{1}{\sigma\sqrt{\tau}}\left[\ln\left(\frac{F}{K}\right) + \frac{1}{2}\sigma^2\tau\right] \\ d_- &= d_+ - \sigma\sqrt{\tau} \end{align} }[/math]

where: [math]\displaystyle{ D = e^{-r\tau} }[/math] is the discount factor

[math]\displaystyle{ F = e^{r\tau} S = \frac{S}{D} }[/math] is the forward price of the underlying asset, and [math]\displaystyle{ S = DF }[/math]

Given put–call parity, which is expressed in these terms as:

[math]\displaystyle{ C - P = D(F - K) = S - D K }[/math]

the price of a put option is:

[math]\displaystyle{ P(F, \tau) = D \left[ N(-d_-) K - N(-d_+) F \right] }[/math]

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