Black-Scholes formula

The Black-Scholes formulagives the theoretical “arbitrage-free” price of a stock optionat any time $t$ before the maturity time $T$.

In financial parlance, an “arbitrage” is a risklessprofit. So the prices being “arbitrage-free”means that it is impossible to constructa strategyof 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 $T$ thatis non-negative almost surely,and is positivewith some positive probability.

A stock call option is a contract, in which the vendor givesthe 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 priceevolves below the strike price since the writing of the contract;if the buyer still wants the stock, it would be cheaper to justbuy it from the market.

0.1 Black-Scholes pricing formula

Let

• $K$ be the strike price of the call option

• $T$ be the maturity date of the call option

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

• $\sigma >0$ be the volatility of the stock price (per unit square-root time)

• $X(t)$ 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 thecall option with the above contract parameters,under the Black-Scholes modelling assumptions,is given by:

(1)

where $\mathrm{\Phi }$ is the cumulative distribution function of a standard normal random variable.

The result (1)is commonly referred to as the Black-Scholes pricing formulaor just as the Black-Scholes 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 apartial differential equation, and solving the PDE analytically.That PDE, now called, appropriately,the Black-Scholes 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 $X(t)$ onlyfor its random part.

More generally, for any financial contract, not necessarilya call option, that pays an amount $V(T)$,the Black-Scholes model shows that its fair price $V(t)$at time $t$must be given by:

$V(t)=e^{-r(T-t)}{𝔼}^{\mathbb{Q}}[V(T)\mid {ℱ}_t].$(2)

for some probability measure $\mathbb{Q}$,and $\{{ℱ}_t\}$ is the filtrationdescribing the information available about the stock price up to time $t$.

If we set $V(T)=\max (X(T)-K,0)$, thenequation (1)follows from a straightforwardcalculation starting with equation (2).

Intuitively, (2) states thatthe option price at the current time is the expected valueof the promised pay-off amount at the future time,discounted back to the current time at the rate of interest in force.

0.3 The risk-neutral measure

The probability measure $\mathbb{Q}$ appearing in (2),is called the risk-neutral measure.It depends on the underlying risk factors that drive the contract pay-off,but it does not depend on the contract itself.This probability measure is distinct from the real-world or historicalprobability measure.It does not actually describe “probabilities” in the intuitive senseof the word, but is a mathematical tool to express the solutionsto the underlying stochastic differential equations.

Equation (2) is equivalent tosaying that $V(t)e^{-rt}$ is a martingale under the probabilitymeasure $\mathbb{Q}$.

0.4 Model assumptions

The Black-Scholes formula is derived under the followingmodel assumptions.

In the real world, of course, all of these simplifying assumptionsare wrong in one way or another; however,the Black-Scholes formula is often usedto 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 motionprocess described by the stochastic differential equation:

$dX(t)=\mu X(t)dt+\sigma X(t)dW(t),$(3)

where

• $X(t)$ is the price of the stock at time $t\ge 0$; the quantity $X(t)$ isa random variable.

  (That is, for each $t$, $X(t)$ is a function of $\omega$,for $\omega$ ranging over the underlying sample space; but the implicitnotation is more convenient and widely used.)

• $\mu >0$ is the growth rate of the stock. The quantity $\mu$ has unitsof reciprocal time.

  $\mu$ is assumed to be a constant (independent of time and the stateof the system).

• $\sigma >0$ 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.

• $W(t)$ is a standard Brownian motion, also called a Wiener process.

The assumption of geometric Brownian motion meansthat 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 priceare 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 risk-free interest rate,and no transaction costs.Trading is also assumed to be conducted continuouslyin 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 growwith time at a risk-free continuously-compounded interest rate of $r$,and $r$ is constant. Thus the amount $M(t)$ in the “money-market account”or “bank account”at time $t$,with an initial infusion of amount $M(0)$ at time $0$,is given by

$M(t)=M(0)e^{rt}.$