BlackScholes formula
Contents:
The BlackScholes formula gives the theoretical “arbitragefree” price of a stock option at any time $t$ before the maturity time $T$.
In financial parlance, an “arbitrage” is a riskless profit. So the prices being “arbitragefree” means that it is impossible to construct a strategy^{1}^{1}Strictly speaking, the strategies described here must be construed as the limiting case, in continuous time, of noarbitrage trading strategies in discrete time, as the time period between trades become small. Otherwise, it is not even obvious what trading in “continuous time” means, and how to actually execute such trades. of trading the stock option and the underlying stock, at the prices given, that starts with no money at time $0$, ends with a balance at time $T$ that is nonnegative almost surely, and is positive^{} with some positive probability.
A stock call option is a contract, in which the vendor gives the buyer a right to purchase some shares of stock in the future, at a fixed strike price agreed upon the writing of the contract. However, the right need not be exercised by the buyer. In fact it would be disadvantageous to do so if the stock price evolves below the strike price since the writing of the contract; if the buyer still wants the stock, it would be cheaper to just buy it from the market.
0.1 BlackScholes pricing formula
Let

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$K$ be the strike price of the call option

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$T$ be the maturity date of the call option

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$r$ be the prevailing interest rate, assumed to be constant

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$\sigma >0$ be the volatility of the stock price (per unit squareroot time)

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$X(t)$ be the stochastic process^{} representing the price of the stock itself at time $t$
Then the fair price $V(t)$, at time $t\le T$, of the call option with the above contract parameters, under the BlackScholes modelling assumptions, is given by:
$$\begin{array}{cc}\hfill V(t)& =x\mathrm{\Phi}\left({d}_{+}(\tau ,x)\right)K{e}^{r\tau}\mathrm{\Phi}\left({d}_{}(\tau ,x)\right),\hfill \\ \hfill {d}_{\pm}(\tau ,x)& =\frac{\mathrm{log}(x/K)+(r\pm \frac{1}{2}{\sigma}^{2})\tau}{\sigma \sqrt{\tau}},\tau =Tt,x=X(t),\hfill \end{array}$$  (1) 
where $\mathrm{\Phi}$ is the cumulative distribution function^{} of a standard normal random variable.
The result (1) is commonly referred to as the BlackScholes pricing formula or just as the BlackScholes formula, after Fischer Black and Myron Scholes, who derived it in the 1970s.
F. Black and M. Scholes first arrived at the solution for $V(t)$ not with our probabilistic calculations, but by describing $V(t)$ in terms of a partial differential equation^{}, and solving the PDE analytically. That PDE, now called, appropriately, the BlackScholes partial differential equation, gives an important alternative formulation of the results here.
0.2 General representation of option price
$V(t)$ in equation (1) is a stochastic process itself, but it happens to depend on $X(t)$ only for its random part.
More generally, for any financial contract, not necessarily a call option, that pays an amount $V(T)$, the BlackScholes model shows that its fair price $V(t)$ at time $t$ must be given by:
$V(t)={e}^{r(Tt)}{\mathbb{E}}^{\mathbb{Q}}[V(T)\mid {\mathcal{F}}_{t}].$  (2) 
for some probability measure^{} $\mathbb{Q}$, and $\{{\mathcal{F}}_{t}\}$ is the filtration describing the information available about the stock price up to time $t$.
If we set $V(T)=\mathrm{max}(X(T)K,0)$, then equation (1) follows from a straightforward calculation starting with equation (2).
Intuitively, (2) states that the option price at the current time is the expected value of the promised payoff amount at the future time, discounted back to the current time at the rate of interest in force.
0.3 The riskneutral measure
The probability measure $\mathbb{Q}$ appearing in (2), is called the riskneutral measure. It depends on the underlying risk factors that drive the contract payoff, but it does not depend on the contract itself. This probability measure is distinct from the realworld or historical probability measure. It does not actually describe “probabilities” in the intuitive sense of the word, but is a mathematical tool to express the solutions to the underlying stochastic differential equations.
Equation (2) is equivalent to saying that $V(t){e}^{rt}$ is a martingale^{} under the probability measure $\mathbb{Q}$.
0.4 Model assumptions
The BlackScholes formula is derived under the following model assumptions.
In the real world, of course, all of these simplifying assumptions are wrong in one way or another; however, the BlackScholes formula is often used to give ballpark figures when quoting option prices.
0.4.1 Stock price
The stock (or any other underlying tradable asset) has a price that follows a geometric Brownian motion process described by the stochastic differential equation:
$dX(t)=\mu X(t)dt+\sigma X(t)dW(t),$  (3) 
where

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$X(t)$ is the price of the stock at time $t\ge 0$; the quantity $X(t)$ is a random variable^{}.
(That is, for each $t$, $X(t)$ is a function of $\omega $, for $\omega $ ranging over the underlying sample space; but the implicit notation is more convenient and widely used.)

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$\mu >0$ is the growth rate of the stock. The quantity $\mu $ has units of reciprocal time.
$\mu $ is assumed to be a constant (independent of time and the state of the system).

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$\sigma >0$ is the volatility of the stock. It can be thought as the standard deviation normalized by time; $\sigma $ has units of the reciprocal square root of time.
$\sigma $ is assumed to be constant.

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$W(t)$ is a standard Brownian motion^{}, also called a Wiener process.
The assumption of geometric Brownian motion means that the relative price movements $X(t+\mathrm{\Delta}t)/X(t)$ after time $t$ are independent of the stock price history before time $t$. In general, relative changes in price are more financially significant and useful to model (e.g. “Google stock has gone up 40% this year”) than absolute changes in price.
We also assume that the stock does not pay dividends, that there are no on the trading of any asset, or borrowing or lending of any amount of cash at the riskfree interest rate, and no transaction costs. Trading is also assumed to be conducted continuously in perfectly divisible amounts, at a single price at each point in time.
0.4.2 Interest on cash
Moreover, cash in the model economy is assumed to grow with time at a riskfree continuouslycompounded interest rate of $r$, and $r$ is constant. Thus the amount $M(t)$ in the “moneymarket account” or “bank account” at time $t$, with an initial infusion of amount $M(0)$ at time $0$, is given by
$M(t)=M(0){e}^{rt}.$ 
Title  BlackScholes formula 

Canonical name  BlackScholesFormula 
Date of creation  20130322 16:29:59 
Last modified on  20130322 16:29:59 
Owner  stevecheng (10074) 
Last modified by  stevecheng (10074) 
Numerical id  15 
Author  stevecheng (10074) 
Entry type  Topic 
Classification  msc 60H10 
Classification  msc 91B28 
Synonym  BlackScholes pricing formula 
Related topic  AnalyticSolutionOfBlackScholesPDE 
Related topic  DerivationOfBlackScholesFormulaInMartingaleForm 