Black-Scholes

From Conservapedia

The Black-Scholes Model

The Black-Scholes model for a stock price assumes that the stock price follows geometric Brownian motion with constant drift and volatility. More precisely, if S(t) the stock price at time t, then

$\frac{dS}{S} = \mu dt + \sigma dW(t)$

where $W (t)$ is a standard Weiner stochastic process.

Loosely speaking, this means that the return $Δ S / S$ of the stock over a very small time interval $Δ t$ can be viewed as a normal random variable with mean $μΔ t$ and variance $σ 2 (Δ t) 2$ . One can make this notion precise by invoking the concepts from the Ito calculus.

The Black-Scholes Formula

The Black-Scholes pricing formula for a European call option can be deduced from the Black-Scholes model for a stock price. A European call option on a stock with strike price $K$ and time to maturity $T$ is a financial contract that gives the holder the option, but not the obligation, to purchase the stock for price $K$ at time $T$ . In other words, a European call on the stock S is a contract that provides a single pay-off of

$max(S (T) - K,0)$

at time $T$ . Let $Φ(t)$ denote the fair value of this contract at time $t < T$ . In deriving a formula for $Φ(t)$ , Black and Scholes' key insight was that by forming a portfolio with the exact right balance of $S$ and the call option, one can completely eliminate risk associated to movements in the stock price $S$ . Moreover, the resulting portfolio, being risk-free, has to grow at the risk free rate. These observations implied that the fair price of the call option had to satisfy the differential equation:

$\frac{\partial\Phi}{\partial t}+r S\frac{\partial \Phi}{\partial S}+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 \Phi}{\partial S^2} = r\Phi$

where $r$ is the continuously compounded risk-free interest rate, and $σ$ is the volatility of the stock. The solution to this differential equation, satisfying the boundary condition

$Φ(T) = max(S (T) - K,0)$

is given by:

$Φ(t) = S (t) N (d 1) - K e - r (T - t) N (d 2)$

Here $N (x)$ is the cumulative normal distribution function,

$d_1 = \frac{\log(S(t)/K)+(r+\frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}$

and $d_2 = d_1-\sigma\sqrt{T}$

This is the famous Black-Scholes formula for the price of a European call. Note that all the variables except for $σ$ can be observed in directly in the market at time $t$ . The volatility, $σ$ of the stock must be estimated using either statistical data, or inferred from the price of options being sold in the market.

Black-Scholes

From Conservapedia

The Black-Scholes Model

The Black-Scholes Formula

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