How to Price an Option under the Bachelier Model?

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

1. 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.

2. 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.

3. 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.

4. 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.

References

• Option (Wikipedia)
• Put-Call Parity (Wikipedia)