. For a European call expiring at time T with strike price...

. For a European call expiring at time $T$ with strike price $K$, the Black-Scholes-Merton price at time $t$, if the time- $t$ stock price is $x$, is $$ c(t, x)=x N\left(d_{+}(T-t, x)\right)-K e^{-r(T-t)} N\left(d_{-}(T-t, x)\right), $$ where $$ \begin{aligned} & d_{+}(\tau, x)=\frac{1}{\sigma \sqrt{\tau}}\left[\log \frac{x}{K}+\left(r+\frac{1}{2} \sigma^2\right) \tau\right], \\ & d_{-}(\tau, x)=d_{+}(\tau, x)-\sigma \sqrt{\tau}, \end{aligned} $$

. For a European call expiring at time $T$ with strike price $K$, the Black-Scholes-Merton price at time $t$, if the time- $t$ stock price is $x$, is
$$
c(t, x)=x N\left(d_{+}(T-t, x)\right)-K e^{-r(T-t)} N\left(d_{-}(T-t, x)\right),
$$
where
$$
\begin{aligned}
& d_{+}(\tau, x)=\frac{1}{\sigma \sqrt{\tau}}\left[\log \frac{x}{K}+\left(r+\frac{1}{2} \sigma^2\right) \tau\right], \\
& d_{-}(\tau, x)=d_{+}(\tau, x)-\sigma \sqrt{\tau},
\end{aligned}
$$

Key Concepts

Time to Maturity

Time to maturity is the remaining time until the option expires. It significantly influences the option's price because it determines the duration over which the underlying asset's volatility can affect the probability of the option ending in-the-money.

Volatility

Volatility is a measure of the degree of variation of an asset's price and is a critical input in the Black-Scholes-Merton model. It captures the uncertainty or risk associated with the underlying asset price movements, with higher volatility generally leading to a higher option premium.

Cumulative Normal Distribution

The cumulative normal distribution function, typically denoted as N(), is used in the Black-Scholes formula to represent the probability that a normally distributed variable falls below a specific threshold. In the context of option pricing, it helps quantify the risk-neutral probability that the option will finish in-the-money.

Risk-Free Rate

The risk-free rate is the theoretical rate of return on an investment with zero risk of financial loss, often represented by government bond yields. In the Black-Scholes-Merton model, it is used to discount the expected future payoff of the option to its present value.

Black-Scholes-Merton Pricing Model

The Black-Scholes-Merton model is a mathematical framework for valuing European options. It assumes that the underlying asset follows a geometric Brownian motion with constant volatility and interest rate, leading to a closed-form solution for pricing options under a risk-neutral measure.

European Call Option

A European call option is a financial derivative that gives the holder the right, but not the obligation, to purchase an underlying asset at a predetermined strike price on a specific expiration date. It is distinct from its American counterpart by the fact that it can only be exercised at expiration.

d+ and d- Parameters

The parameters d+ and d- are standardized variables computed within the Black-Scholes-Merton model. They incorporate the logarithm of the ratio of the current asset price to the strike price, adjusted for the risk-free rate, volatility, and time to maturity, effectively serving as metrics for the probability of the option expiring in-the-money under the risk-neutral probability measure.

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