# How to Price an Option under the Bachelier Model?

Deriving Bachelier call and put formulas

## 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-upon 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 evaluate the price of an option is to specify a *model* for 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 maturity and
$(W_t)_{t\in[0,T]}$ be the standard Brownian motion.

In the **Bachelier model**, one considers a properly scaled Brownian motion as a model for the forward price.
In other words, the forward price process $(F^{B}_t)_{t\in[0,T]}$ writes
$$
F^{B}_t = F_0 + \sigma_B W_t, \quad t\in[0,T],
$$
where the initial forward price is given by $F_0$ and the parameter $\sigma_B > 0$ denotes the *volatility* in the Bachelier model.
In a differential form, the dynamics of the forward price $F^{B}$
is written as:
\begin{align}
d F^{B}_t &= \sigma_{B} dW_t, \quad t\in[0,T],\\

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

## What is the Bachelier formula?

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

### Bachelier call formula

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

Applying the *Bachelier fundamental principle*, one obtains the call option price at $t=0$
$$
C^B_0 = \mathbb{E}[ (F^B_T-K)_{+} ].
$$

Under the **Bachelier model**, the call option price at $t=0$ is given by
$$
C^B_0 = c_B(0, F_0; K, T, \sigma_B),
$$
where $c_B$ is called the **Bachelier call formula** and is defined as
$$
c_B(t, x; K, T, \sigma) := (x - K) \Phi\Big(\frac{x-K}{\sigma \sqrt{T-t}}\Big) + \sigma \sqrt{T-t}\ \phi\Big(\frac{x-K}{\sigma \sqrt{T-t}}\Big).
$$

#### Proof

Since the forward price $F^B_T$ follows a normal distribution with mean $F_0$ and variance $\sigma_B^2 T$, the option price at $t=0$ writes $$ C^B_0 = \mathbb{E} \big[ \big(F_0 - K - \sigma_B \sqrt{T} \ Z\big)_{+} \big], $$ where $Z$ is a standard normal random variable.

By developing the payoff with indicator functions, one gets \begin{align} C^B_0 &= \mathbb{E}\Big[ \big(F_0 - K - \sigma_B \sqrt{T} \ Z\big) \mathbb{1}_{Z \le \frac{F_0-K}{\sigma^B \sqrt{T}} } \Big] \\ &= (F_0 - K) \Phi\Big(\frac{F_0-K}{\sigma^B \sqrt{T}}\Big) - \sigma_B \sqrt{T}\

\mathbb{E}\Big[ Z \mathbb{1}_{Z \le \frac{F_0-K}{\sigma^B \sqrt{T}} } \Big], \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 applies the relation $\phi’(x) = - x \phi(x)$ to obtain the following equality: $$ \mathbb{E}\Big[ Z \mathbb{1}_{Z \le y } \Big] = \int_{-\infty}^{y} x \phi(x) dx = -\phi(y). $$ In the previous identity, the assumption of normal distribution of $Z$ is

*really*used to obtain the option price.Therefore, using the previous identity with $y=\frac{F_0-K}{\sigma^B \sqrt{T}}$, one gets $$ C^B_0 = (F_0 - K) \Phi\Big(\frac{F_0-K}{\sigma^B \sqrt{T}}\Big) + \sigma_B \sqrt{T} \phi\Big(\frac{F_0-K}{\sigma^B \sqrt{T}}\Big), $$ which yields the result. $\blacksquare$

### Bachelier put formula

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

Applying the *Bachelier fundamental principle*, one obtains the put option price at $t=0$
$$
P^B_0 = \mathbb{E} \big[ \big( K - F^{B}_T\big)_{+} \big].
$$

Under the **Bachelier model**, the put option price at $t=0$ is given by
$$
P^B_0 = p_B(0, F_0; K, T, \sigma_B),
$$
where $p_B$ is called the **Bachelier put formula** and is defined as
$$
p_B(t, x; K, T, \sigma) := (K - x) \Phi\Big(\frac{K-x}{\sigma \sqrt{T-t}}\Big) + \sigma \sqrt{T-t}\ \phi\Big(\frac{K-x}{\sigma \sqrt{T-t}}\Big).
$$

#### Proof

Analogous to the Proof of the Bachelier call formula. $\blacksquare$

## Do the Bachelier formulas satisfy the Call-Put parity?

The **Call-Put parity** can be stated as follows:
$$
C^B_0-P^B_0 \equiv F_0-K,
$$
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 $d$ the following quantity
$$
d=\frac{F_0-K}{\sigma_B \sqrt{T}}.
$$
According to the previously derived formulas, one gets
\begin{align}
C^B_0-P^B_0 &= (F_0-K) \Phi(d) + \sigma_B \sqrt{T}\ \phi (d) \\

&-(K - F_0) \Phi(-d) - \sigma_B \sqrt{T}\ \phi(-d) \\

&= (F_0-K) \big(\Phi(d) + \Phi(-d)\big) + \sigma_B \sqrt{T} \big(\phi(d)-\phi(d)\big) \\

&\equiv F_0-K,
\end{align}
because the probability density function $\phi$ is even, i.e., $\phi(d)\equiv\phi(-d)$ and
the cumulative distribution function $\Phi$ satisfies $\Phi(d)+\Phi(-d)\equiv 1$.
So, the Bachelier formulas **do satisfy** the Call-Put parity.