# How to price an Option under the Black-Scholes model?

Deriving Black-Scholes call and put formula

## What is an Option?

An **option** is a contract between a buyer and seller which gives the buyer the *right* to buy or sell a particular security (*underlying asset*) at a later date (*maturity date*) and an agreed price (*strike price*).
There are two option types: *call* and *put*.

For more details, see the article on Options Contract.

## How to price an Option?

The first step to obtain the price of an option is to specify a *model* of the option underlying asset.

In following, one denotes by $F$ the forward price of the underlying asset,
*i.e.*,
the agreed-upon price of the underlying asset in a forward contract delivering the underlying asset at a given maturity date.
Let $T > 0$ be a fixed time to maturity and
$(W_t)_{t\in[0,T]}$ be the standard Brownian motion.

In the **Black-Scholes model** (under the risk-neutral measure), the forward price process $(F^{BS}_t)_{t\in[0,T]}$ is given by
$$
F^{BS}_t = F_0 \exp \Big( \sigma_{BS} W_t - \tfrac{1}{2} \sigma_{BS}^2 t \Big), \quad t\in[0,T],
$$
where the initial forward price is given by $F_0$ and the parameter $\sigma_{BS} > 0$ denotes the *volatility* in the Black-Scholes model.
In a differential form, the dynamics of the forward price $F^{BS}$ are written as:
\begin{align}
d F^{BS}_t &= F^{BS}_t \sigma_{BS} dW_t, \quad t\in[0,T],\\

F^{BS}_0 &= F_0.
\end{align}
Note that $(F^{BS}_t)_{t\in[0,T]}$ is a martingale under the risk-neutral measure $\mathbb{P}$.

## What are the Black-Scholes formula ?

From the stochastic differential equation for the Black-Scholes model, one deduces the derivation of the option price on this framework.

### Black-Scholes call formula

Consider a *European* call with strike price $K$, maturity $T$, whose payoff at $T$ is given by the random variable
$$
C_T = (F^B_T-K)_{+}.
$$

Applying the *No-arbitrage principle*, one obtains the option price at $t=0$
$$
C^{BS}_0 = \mathbb{E}[ (F^{BS}_T-K)_{+} ].
$$

Under the **Black-Scholes model**, the call option price at $t=0$ is given by
$$
C^{BS}_0 = c_{BS}(0, F_0; K, T, \sigma_{BS}),
$$
where $c_{BS}$ is called the **Black-Scholes call formula** and is defined as
$$
c_{BS}(t, x; K, T, \sigma) := x \Phi\big(d_{+}(t, x; K, T, \sigma)\big) - K \Phi\big(d_{-}(t, x; K, T, \sigma)\big).
$$
and
$$
d_{\pm}(t, x; K, T, \sigma) := \frac{\log{x/K}}{\sigma \sqrt{T-t}} \pm \frac{1}{2} \sigma \sqrt{T-t}.
$$

#### Proof

Since the log forward price $\log F^{BS}_T$ follows a normal distribution with mean $\log F_0 - \frac{1}{2} \sigma_{BS}^2 T$ and variance $\sigma_{BS}^2 T$, the option price at $t=0$ writes $$ C^{BS}_0 = \mathbb{E} \bigg[ \Big(F_0 e^{ -\sigma_{BS} \sqrt{T}\ Z - \frac{1}{2} \sigma_{BS}^2 T } - K \Big)_{+} \bigg], $$ where $Z$ is a standard normal random variable.

By developing the payoff with indicator functions, one gets \begin{align} C^{BS}_0 &= \mathbb{E} \bigg[ \Big(F_0 e^{ -\sigma_{BS} \sqrt{T}\ Z - \frac{1}{2} \sigma_{BS}^2 T } - K \Big) \mathbb{1}_{Z \le \frac{\log{F_0/K} - \frac{1}{2} \sigma_{BS}^2 T}{\sigma_{BS} \sqrt{T}} } \bigg] \\ &= F_0 \ \mathbb{E} \bigg[ e^{ -\sigma_{BS} \sqrt{T}\ Z - \frac{1}{2} \sigma_{BS}^2 T }\ \mathbb{1}_{Z \le \frac{\log{F_0/K} - \frac{1}{2} \sigma_{BS}^2 T}{\sigma_{BS} \sqrt{T}} } \bigg] -K \ \Phi \bigg( \frac{\log{F_0/K} - \frac{1}{2} \sigma_{BS}^2 T}{\sigma_{BS} \sqrt{T}} \bigg), \end{align} where $\Phi$ is the cumulative distribution function of the normal distribution.

Let $\phi(x) = \tfrac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ be the probability density function of the normal distribution. So, one obtains the following equality: \begin{align} \mathbb{E}\bigg[ e^{ -w Z - \frac{1}{2} w^2 }\ \mathbb{1}_{Z \le y } \bigg] &= \int_{-\infty}^{y} e^{ - \frac{1}{2} \big( 2 w x + w^2 \big) } \phi(x) dx \\

&= \int_{-\infty}^{y} \phi(x + w ) dx = \int_{-\infty}^{y+w} \phi(x’) dx’ \\

&= \Phi(y+w) = \mathbb{\tilde{E}} [ \mathbb{1}_{\tilde{Z} \le y + w } ]. \end{align} In the previous identity, the assumption of normal distribution of $Z$ is*really*used to obtain the option price. Note that $Z$ is a standard normal random variable under $\mathbb{P}$. By defining $\tilde{Z} = Z+w$, one gets the random variable $\tilde{Z}$ is a normal but not standard under the probability measure $\mathbb{P}$. However, there is another probability measure $\mathbb{\tilde{P}}$, equivalent to $\mathbb{P}$ and defined by $\mathbb{\tilde{P}} := e^{-wZ- \frac{1}{2} w^2} \cdot \mathbb{P}$, under which $\tilde{Z}$ is standard and normal.Therefore, using the previous identity with $y=\frac{\log{F_0/K} - \tfrac{1}{2} \sigma_{BS}^2 T}{\sigma_{BS} \sqrt{T}}$ and $w=\sigma_{BS} \sqrt{T}$, one gets $$ C^{BS}_0 = F_0 \ \Phi \bigg( \frac{\log{F_0/K} + \tfrac{1}{2} \sigma_{BS}^2 T}{\sigma_{BS} \sqrt{T}} \bigg) -K \ \Phi \bigg( \frac{\log{F_0/K} - \tfrac{1}{2} \sigma_{BS}^2 T}{\sigma_{BS} \sqrt{T}} \bigg), $$ which yields the result. $\blacksquare$

### Black-Scholes put formula

Consider a *European* put with strike price $K$ and maturity $T$, whose payoff at $T$ is given by the random variable
$$
P_T = (K-F^{BS}_T)_{+}.
$$

Applying the *No-arbitrage principle*, one obtains the put option price at $t=0$
$$
P^{BS}_0 = \mathbb{E} \big[ \big( K - F^{BS}_T\big)_{+} \big].
$$

Under the **Black-Scholes model**, the put option price at $t=0$ is given by
$$
P^{BS}_0 = p_{BS}(0, F_0; K, T, \sigma_{BS}),
$$
where $p_{BS}$ is called the **Black-Scholes put formula** and is defined as
$$
p_{BS}(t, x; K, T, \sigma) := -x \Phi\big(-d_{+}(t, x; K, T, \sigma)\big) + K \Phi\big(-d_{-}(t, x; K, T, \sigma)\big).
$$

#### Proof

Analogous to the Proof of the Black-Scholes Call Formula. $\blacksquare$

## Do the Black-Scholes formulas satisfy the Call-Put parity?

The **Call-Put parity** can be stated as follows:
$$
C^{BS}_0-P^{BS}_0 \equiv F_0-K,
$$
where the left-hand side corresponds to a portfolio of a long call and a short put, while the right-hand side corresponds to a long forward contract.

Denote by $d_1$ and $d_2$ the following quantities
$$
d_1= \frac{\log{F_0/K}}{\sigma_{BS} \sqrt{T}} + \frac{1}{2} \sigma_{BS} \sqrt{T}, \quad
d_2= \frac{\log{F_0/K}}{\sigma_{BS} \sqrt{T}} - \frac{1}{2} \sigma_{BS} \sqrt{T}.
$$
According to the previously derived formulas, one gets
\begin{align}
C^{BS}_0-P^{BS}_0 &= F_0 \Phi(d_1) - K \Phi(d_2) \\

&-(K \Phi(-d_2) - F_0 \Phi(-d_1)) \\

&= F_0 \big(\Phi(d_1) + \Phi(-d_1)\big) - K \big(\Phi(d_2) + \Phi(-d_2)\big) \\

&\equiv F_0-K,
\end{align}
because the cumulative distribution function $\Phi$ satisfies $\Phi(d)+\Phi(-d)\equiv 1$.
So, the Black-Scholes formulas **do satisfy** the Call-Put parity.