Black-Scholes Market Assumptions and Notation

The Market as a Mathematical Model

The Black–Scholes model begins by fixing an idealized market rather than by starting with an option-pricing formula. It specifies how the safe asset grows, what assumptions are made about the stock, and what a European claim pays at its maturity. These conventions let us state the market precisely before asking how claims can be priced or hedged.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ satisfying the usual conditions: the filtration is complete and right-continuous. The filtration represents information available over time. The space carries a standard Brownian motion $B_t$ adapted to $(\mathcal{F}_t)$. The measure $\mathbb{P}$ is the physical probability measure used to describe the model’s actual randomness. It is not, by definition, a risk-neutral measure.

Time is measured in years, and $t$ denotes the current time. A contract has maturity $T>0$, with $0\le t\le T$. The stock price at time $t$ is $S_t$, with initial value $S_0>0$. The strike of an option is $K\ge0$. The constant $r$ is the risk-free rate, $\mu$ is the stock’s drift coefficient under $\mathbb{P}$, and $\sigma>0$ is its constant volatility.

Definition: The risk-free account $A_t$ starts at $A_0=1$ and grows at the constant continuously compounded rate $r$: $$ A_t=e^{rt}. $$ Its value at time $t$ of one unit of currency invested at time $0$ is $e^{rt}$. The corresponding discount factor from time $t$ to time $0$ is $e^{-rt}$.

Taking $A_0=1$ is a normalization: an initial deposit $a$ grows to $aA_t$. The account is deterministic because its rate is fixed, unlike the stock price. The rate $r$ may be any real constant in the mathematical model; applications often choose $r\ge0$.

Stock-Price Assumption and Its Parameters

In the basic market, the stock pays no dividends and is assumed to have continuous price paths described by the constant-coefficient model

dS_t=\mu S_t\,dt+\sigma S_t\,dB_t.

This is an assumption about the stock, not a formula for an option price. In integral form, for $0\le t\le T$,

S_t=S_0+\int_0^t\mu S_s\,ds+\int_0^t\sigma S_s\,dB_s.

The second integral is an Itô integral. For it, assume that $s\mapsto \sigma S_s$ is progressively measurable and belongs to $\mathcal{L}^2$ on each finite horizon: $\mathbb{E}\int_0^T(\sigma S_s)^2\,ds<\infty$. The first integral is an ordinary time integral. The model takes $S_t>0$; it excludes price jumps and dividend payments. Its explicit solution is not needed to define these assumptions.

Definition: In the stock equation, $\mu$ is the drift coefficient and $\sigma$ is the diffusion coefficient. The drift specifies the coefficient of $S_t\,dt$; the diffusion specifies the coefficient of $S_t\,dB_t$. The safe rate $r$ describes the account $A_t$, not the stock’s drift.

In particular, the stock equation is stated under $\mathbb{P}$. The parameter $\mu$ is not automatically equal to $r$. A risk-neutral measure, conventionally denoted by $\mathbb{Q}$, is a different probability measure used in pricing arguments; its construction and the conditions for changing measure are separate questions. No equality between $\mu$ and $r$, or between $\mathbb{P}$ and $\mathbb{Q}$, is part of the market definition here.

The idealization is frictionless: assets can be bought or sold in arbitrary fractional amounts, trading is allowed continuously, and there are no transaction costs or taxes. Borrowing and short selling are permitted in the idealized model. The same risk-free rate applies to lending and borrowing. The market is taken to have no arbitrage, meaning there is no admissible trading opportunity with zero initial cost, a nonnegative terminal gain with probability one, and a strictly positive terminal gain with positive probability. These assumptions simplify the market description; they do not, by themselves, calculate a claim’s value.

Worked Example: Reading the Market Parameters

Suppose a market is specified by $r=\tfrac1{20}$, $\mu=\tfrac1{10}$, and $\sigma=\tfrac14$. Identify the safe-account growth over one year and the coefficients in the stock equation.

The account has rate $r=\tfrac1{20}$, so over one year its growth factor is $A_1=e^{1/20}$. The stock equation is

dS_t=\frac{1}{10}S_t\,dt+\frac14S_t\,dB_t.

Thus its drift coefficient is $\tfrac1{10}$, and its diffusion coefficient is $\tfrac14$. The square of the diffusion coefficient is $\tfrac1{16}$, but the coefficient multiplying $dB_t$ is $\tfrac14S_t$, not $\tfrac1{16}S_t$.

Check: the account equation uses $r=\tfrac1{20}$, whereas the stock equation uses $\mu=\tfrac1{10}$. These are unequal, so replacing the stock drift by the safe rate would change the stated model. Also, $(\tfrac14)^2=\tfrac1{16}$, confirming the stated squared diffusion coefficient.

European Claims and Their Payoffs

A European claim is a contract whose exercise or cash settlement occurs at its fixed maturity $T$, rather than at a choice of earlier times. A call gives its holder the right to receive the positive part of the excess of the stock price over the strike. A put gives the positive part of the excess of the strike over the stock price. For a real number $x$, write $x^+=\max(x,0)$.

Definition: The terminal payoffs of a European call and a European put with strike $K$ and maturity $T$ are, respectively, $$ (S_T-K)^+\qquad\text{and}\qquad(K-S_T)^+. $$ The call price function is denoted by $C(t,S)$, where $S$ is the current stock-price argument. The maturity is $T$; the strike is $K$. A payoff is the amount specified by the contract at maturity, whereas a price is the amount paid for the contract at an earlier time.

For example, $C(t,S_t)$ denotes the call’s price at time $t$ when the observed stock price is $S_t$. At maturity, the call payoff is $(S_T-K)^+$; it is not generally the same quantity as the price at an earlier time. This tutorial defines the contract and notation only: it does not assert a pricing formula for $C$.

Worked Example: Call and Put Payoffs at Maturity

Take $K=100$. Compute the call and put payoffs when the terminal stock price is $S_T=80$, $100$, and $130$.

At $S_T=80$, the call payoff is $(80-100)^+=(-20)^+=0$, and the put payoff is $(100-80)^+=(20)^+=20$.

At $S_T=100$, the call payoff is $(100-100)^+=0^+=0$, and the put payoff is $(100-100)^+=0^+=0$.

At $S_T=130$, the call payoff is $(130-100)^+=30^+=30$, and the put payoff is $(100-130)^+=(-30)^+=0$.

Check: in each case, call payoff minus put payoff equals $S_T-K$: the three differences are $0-20=-20=80-100$, $0-0=0=100-100$, and $30-0=30=130-100$. This is an identity between terminal payoffs, not a claim about prices at time $t<T$.

Two Elementary Facts About the Setup

Proposition: If the risk-free account starts at $A_0=1$ and satisfies the deterministic rate equation $dA_t=rA_t\,dt$ for constant $r$, then $A_t=e^{rt}$. Consequently, one unit of currency at time $t$ has discounted value $e^{-rt}$ at time $0$.

Proof. The equation is the ordinary differential equation $A'(t)=rA(t)$ with $A(0)=1$. Differentiating $e^{-rt}A_t$ gives $e^{-rt}(A'_t-rA_t)=0$, so this product is constant. Its value at time zero is $1$, hence $e^{-rt}A_t=1$, or $A_t=e^{rt}$. Multiplying the time-$t$ amount $1$ by $e^{-rt}$ expresses it in time-zero units. $\square$

Proposition: For $x,y\ge0$, define the call payoff $c_K(x)=(x-K)^+$ and put payoff $p_K(x)=(K-x)^+$, with $K\ge0$. Both functions are nonnegative and have changes bounded in absolute value by the change in their argument: $$ |c_K(x)-c_K(y)|\le|x-y|,\qquad |p_K(x)-p_K(y)|\le|x-y|. $$ In addition, $c_K(x)-p_K(x)=x-K$.

Proof. The identity follows from the two cases $x\ge K$ and $x<K$. If $x\ge K$, then $c_K(x)=x-K$ and $p_K(x)=0$, so their difference is $x-K$. If $x<K$, then $c_K(x)=0$ and $p_K(x)=K-x$, so their difference is also $x-K$. For the change bound, the map $u\mapsto u^+=\max(u,0)$ satisfies $|u^+-v^+|\le|u-v|$: when $u,v$ have the same sign this follows directly from the formula for the positive part, and when their signs differ the nonzero positive part is no larger than $|u-v|$. Apply this fact first with $u=x-K$, $v=y-K$, and then with $u=K-x$, $v=K-y$. In each case $|u-v|=|x-y|$. Nonnegativity follows from the definition of the positive part. $\square$

Worked Example: Discounting a Fixed Maturity Amount

Suppose $r=\ln(3/2)$, and a contract pays $250$ units of currency at $T=2$. Find the risk-free account growth factor over the two years and the time-zero discounted value of that fixed payment.

The account growth factor is $e^{rT}=e^{2\ln(3/2)}=(3/2)^2=9/4$. The account therefore turns one unit at time zero into $9/4$ units at time two. The discount factor is $e^{-rT}=e^{-2\ln(3/2)}=4/9$, so the discounted value of the payment is $250\cdot\tfrac49=\tfrac{1000}{9}$.

Check: depositing $\tfrac{1000}{9}$ in the account gives $\tfrac{1000}{9}\cdot\tfrac94=250$ at time two. This verifies the discounting calculation by compounding the discounted amount back to maturity.

Common Mistakes and Scope of the Assumptions

Confusing the safe rate with stock drift. A mistaken substitution in the example above would replace $\mu=\tfrac1{10}$ by $r=\tfrac1{20}$, writing $dS_t=\tfrac1{20}S_t\,dt+\tfrac14S_t\,dB_t$. That is not the specified stock model: its drift coefficient is $\tfrac1{10}$. The account and stock have different roles, so their rates must not be silently identified.

Confusing a payoff with a price. The number $(S_T-K)^+$ is determined by the terminal stock price and the contract terms. It does not, on its own, give $C(t,S_t)$ for $t<T$. A price requires a valuation argument and the relevant market assumptions; the payoff definition alone supplies neither.

Remembering what is excluded. Constant volatility means the coefficient $\sigma$ is fixed in this model, not that the stock price itself is constant. The absence of dividends means the stock generates no cash distributions in the model. Continuous stock paths exclude jumps. Relaxing any of these assumptions changes the market specification and may change the mathematical tools needed.

Symbol	Meaning in this model
$A_t$	Risk-free account, $A_t=e^{rt}$
$S_t$	Non-dividend-paying stock price
$r$	Constant risk-free rate
$\mu$	Stock drift coefficient under $\mathbb{P}$
$\sigma$	Constant stock diffusion coefficient, assumed positive
$K,T$	Option strike and maturity
$C(t,S)$	European call price function

Check Your Understanding

Use the definitions and assumptions above to answer the following questions.

Write the stock equation under $\mathbb{P}$ when $r=0$, $\mu=\tfrac{3}{20}$, and $\sigma=\tfrac25$. Which coefficient describes the safe account?
For $K=90$, compute the call and put payoffs when $S_T=75$ and when $S_T=110$.
Explain why the payoff identity between call and put does not by itself give an identity between their prices at time $t<T$.
If $r=\ln 2$, what is the risk-free account growth factor over $T=\tfrac12$ years?
Name two features excluded by the constant-volatility, non-dividend-paying stock assumptions.

Symbol	Meaning in this model
\(A_t\)	Risk-free account, \(A_t=e^{rt}\)
\(S_t\)	Non-dividend-paying stock price
\(r\)	Constant risk-free rate
\(\mu\)	Stock drift coefficient under \(\mathbb{P}\)
\(\sigma\)	Constant stock diffusion coefficient, assumed positive
\(K,T\)	Option strike and maturity
\(C(t,S)\)	European call price function