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

1. 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.

2. 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.

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 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.

4. 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.

References

Isaque Pimentel
Quantitative Analyst, Consultant

Ph.D. in Applied Mathematics interested in Quantitative Finance and Data Science.