Black-Scholes Formula: Derivation, Intuition & Python Calculator 2026

The Black-Scholes Formula

The Black-Scholes formula gives the theoretical price of a European call or put option using five observable inputs: the current stock price, the strike price, time to expiry, the risk-free interest rate, and the volatility of the underlying asset. It's the single most important equation in options pricing, and understanding it thoroughly is essential for anyone working in quantitative finance in 2026.

The formula for a European call option is:

C = S * N(d1) - K * exp(-rT) * N(d2)

The formula for a European put option is:

P = K * exp(-rT) * N(-d2) - S * N(-d1)

where:

d1 = (ln(S / K) + (r + sigma^2 / 2) * T) / (sigma * sqrt(T))

d2 = d1 - sigma * sqrt(T)

These two formulas, published by Fischer Black and Myron Scholes in 1973 with key contributions from Robert Merton, transformed derivatives pricing from guesswork into a mathematically rigorous discipline. For a broader treatment of the model's history and context, see our Black-Scholes model overview. This article focuses specifically on the formula itself - what each piece means, how it's derived, and how to compute it in practice.

---

Understanding Each Component

Every symbol in the Black-Scholes formula has a concrete financial meaning. Let's break them down one at a time.

S - the current stock price. This is the spot price of the underlying asset at the time of valuation. For equity options, it's simply the current share price. As S increases relative to K, a call option becomes more valuable and a put option less valuable.

K - the strike price. The price at which the option holder can buy (call) or sell (put) the underlying asset at expiry. This is fixed at the time the option contract is written and doesn't change over the option's life.

T - time to expiry. Measured in years. A 90-day option has T = 90/365 = 0.2466. More time generally means more value for both calls and puts, because there's more opportunity for the stock to move favourably. As T approaches zero, the option price converges to its intrinsic value.

r - the risk-free interest rate. The continuously compounded annual rate on a riskless investment, typically proxied by government bond yields. It enters the formula in two places: in the discounting term exp(-rT) and inside d1. A higher risk-free rate increases call values (because the present value of paying K later is lower) and decreases put values.

sigma - the volatility of the underlying. Annualised standard deviation of the stock's log returns. This is the only unobservable input - you can't look it up on a screen in the same way as the stock price or interest rate. In practice, traders often reverse-engineer sigma from observed option prices, producing the implied volatility. Higher volatility increases the value of both calls and puts because it raises the probability of large price movements.

N(x) - the cumulative standard normal distribution function. This gives the probability that a standard normal random variable is less than or equal to x. N(0) = 0.5, N(1.96) ≈ 0.975, and N(-infinity) = 0. It maps any real number to a value between 0 and 1, which is why the formula always produces sensible option prices.

exp(-rT) - the discount factor. This converts a future cash flow to its present value using continuous compounding. The term K * exp(-rT) is the present value of paying the strike price at expiry.

---

The d1 and d2 Terms Explained

The d1 and d2 terms are the heart of the Black-Scholes formula. They look technical, but each has a clear probabilistic interpretation that connects directly to how options work.

d2 is the risk-neutral probability of exercise.

More precisely, N(d2) is the probability that the option finishes in the money under the risk-neutral measure. This is the probability that S_T > K for a call (or S_T < K for a put), assuming the stock drifts at the risk-free rate r rather than its actual expected return.

Breaking d2 apart:

d2 = (ln(S / K) + (r - sigma^2 / 2) * T) / (sigma * sqrt(T))

The numerator contains two terms. The first, ln(S / K), measures how far the stock currently is from the strike on a logarithmic scale - the "moneyness" of the option. If S = K (at the money), this term is zero. The second, (r - sigma^2 / 2) * T, is the expected drift of the log stock price under the risk-neutral measure over time T. The denominator, sigma * sqrt(T), normalises everything by the total standard deviation of the log price over the option's life. The result is a z-score: how many standard deviations the expected log stock price is above or below the log strike.

d1 is d2 shifted by one standard deviation.

d1 = d2 + sigma * sqrt(T)

N(d1) is the option's delta - the hedge ratio, or equivalently, the probability of exercise under the stock-price measure (sometimes called the share measure). The difference between d1 and d2 reflects a subtle but important concept: when you evaluate the expected payoff weighted by the stock price itself (as you must for the S * N(d1) term), you're working under a different probability measure where the stock's drift is r + sigma^2 rather than r. This extra sigma^2 / 2 term arises from the convexity of the exponential function and is a consequence of Jensen's inequality.

Practical intuition:

• When the option is deep in the money, both d1 and d2 are large positive numbers, so N(d1) and N(d2) are both close to 1. The call price approaches S - K * exp(-rT), which is the intrinsic value discounted.
• When the option is deep out of the money, both are large negative numbers, so N(d1) and N(d2) approach 0 and the call price approaches zero.
• When the option is at the money (S ≈ K * exp(-rT)), d1 and d2 are close to zero, and the formula gives a price driven primarily by volatility and time.

---

Intuitive Derivation of the Black-Scholes Formula

The full rigorous derivation of Black-Scholes involves stochastic calculus, Ito's lemma, and partial differential equations. Here we'll walk through the economic logic in an accessible way, focusing on the risk-neutral pricing approach that most practitioners use.

Step 1: Stock prices follow a log-normal distribution.

Under the Black-Scholes model, the stock price follows geometric Brownian motion:

dS = mu * S * dt + sigma * S * dW

where dW is a Brownian motion increment. Applying Ito's lemma to ln(S), we find that the log of the stock price at time T is:

ln(S_T) = ln(S) + (mu - sigma^2 / 2) * T + sigma * sqrt(T) * Z

where Z is a standard normal random variable. This means S_T is log-normally distributed - it can take any positive value, with the distribution skewed to the right.

Step 2: Switch to the risk-neutral measure.

The fundamental theorem of asset pricing tells us that in a complete, arbitrage-free market, the price of any derivative is the expected value of its discounted payoff under a special probability measure - the risk-neutral measure Q. Under Q, the stock's drift mu is replaced by the risk-free rate r:

ln(S_T) = ln(S) + (r - sigma^2 / 2) * T + sigma * sqrt(T) * Z

This isn't saying the stock actually grows at rate r. It's a mathematical change of measure that correctly prices all derivatives without needing to know the stock's true expected return. The volatility sigma stays the same under both measures.

Step 3: Write the expected payoff of a call.

Under Q, the call price is:

C = exp(-rT) * E_Q[max(S_T - K, 0)]

This is the present value of the expected payoff. To evaluate this expectation, we need to integrate over all possible values of S_T where S_T > K.

Step 4: Evaluate the integral.

Splitting the expectation:

C = exp(-rT) * E_Q[S_T * 1(S_T > K)] - exp(-rT) * K * P_Q(S_T > K)

The second term is straightforward: P_Q(S_T > K) is the risk-neutral probability that the stock ends above the strike, which works out to N(d2).

The first term requires computing E_Q[S_T * 1(S_T > K)]. Because S_T is log-normal, this expectation involves a weighted integral of the normal distribution. Working through the algebra (completing the square inside the exponent), we get:

E_Q[S_T * 1(S_T > K)] = S * exp(rT) * N(d1)

Step 5: Combine.

Putting it together:

C = exp(-rT) * [S * exp(rT) * N(d1) - K * N(d2)]

C = S * N(d1) - K * exp(-rT) * N(d2)

This is the Black-Scholes formula. The exp(rT) from the first term cancels with the exp(-rT) discount factor, leaving just S * N(d1). The result is elegant: the call price is the delta-weighted stock price minus the probability-weighted present value of the strike.

---

Step-by-Step Worked Example

Let's price a European call option by hand, showing every calculation. Suppose:

• Stock price S = 100
• Strike price K = 105
• Time to expiry T = 0.5 years (6 months)
• Risk-free rate r = 5% (0.05)
• Volatility sigma = 20% (0.20)

Step 1: Compute d1.

d1 = (ln(100 / 105) + (0.05 + 0.20^2 / 2) * 0.5) / (0.20 * sqrt(0.5))

First, ln(100 / 105) = ln(0.9524) = -0.04879

Then, (0.05 + 0.02) * 0.5 = 0.07 * 0.5 = 0.035

Numerator: -0.04879 + 0.035 = -0.01379

Denominator: 0.20 * 0.7071 = 0.14142

d1 = -0.01379 / 0.14142 = -0.09751

Step 2: Compute d2.

d2 = d1 - sigma * sqrt(T) = -0.09751 - 0.14142 = -0.23893

Step 3: Look up N(d1) and N(d2).

N(-0.09751) = 0.46116

N(-0.23893) = 0.40557

Step 4: Calculate the call price.

C = 100 * 0.46116 - 105 * exp(-0.05 * 0.5) * 0.40557

The discount factor: exp(-0.025) = 0.97531

K * exp(-rT) = 105 * 0.97531 = 102.408

C = 46.116 - 102.408 * 0.40557

C = 46.116 - 41.535

C = 4.581

The European call is worth approximately 4.58. Notice that the option is out of the money (S < K), so its entire value is time value - there's no intrinsic value. The 4.58 represents the market's assessment of the probability-weighted gain from the stock potentially rising above 105 over the next six months.

Sanity checks:

• The price is positive (good - options always have non-negative value)
• It's less than S = 100 (good - a call can never be worth more than the underlying)
• It's less than S - K * exp(-rT) = 100 - 102.41 = -2.41... which is negative, confirming the option is out of the money and its value comes entirely from the chance of finishing in the money

---

Black-Scholes Calculator in Python

Here's a complete Python implementation that prices European calls and puts using the Black-Scholes formula and includes sensitivity analysis across different input parameters.

import numpy as np
from scipy.stats import norm


def black_scholes(S, K, T, r, sigma, option_type="call"):
    """
    Price a European option using the Black-Scholes formula.

    Parameters
    ----------
    S     : Current stock price
    K     : Strike price
    T     : Time to expiry in years
    r     : Risk-free rate (annualised, continuous)
    sigma : Volatility (annualised)
    option_type : "call" or "put"

    Returns
    -------
    Option price (float)
    """
    d1 = (np.log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)

    if option_type == "call":
        return S * norm.cdf(d1) - K * np.exp(-r * T) * norm.cdf(d2)
    else:
        return K * np.exp(-r * T) * norm.cdf(-d2) - S * norm.cdf(-d1)


# --- Price the worked example from above ---
S, K, T, r, sigma = 100, 105, 0.5, 0.05, 0.20

call_price = black_scholes(S, K, T, r, sigma, "call")
put_price = black_scholes(S, K, T, r, sigma, "put")

print(f"Call price: {call_price:.4f}")
print(f"Put price:  {put_price:.4f}")
print(f"Put-call parity check: C - P = {call_price - put_price:.4f}")
print(f"S - K*exp(-rT)               = {S - K * np.exp(-r * T):.4f}")


# --- Sensitivity: price across a range of stock prices ---
print("\nCall and Put Prices Across Stock Prices")
print("-" * 50)
print(f"{'Stock':>8}  {'Call':>8}  {'Put':>8}  {'Intrinsic (C)':>14}")

for stock in range(80, 121, 5):
    c = black_scholes(stock, K, T, r, sigma, "call")
    p = black_scholes(stock, K, T, r, sigma, "put")
    intrinsic = max(stock - K, 0)
    print(f"{stock:>8}  {c:>8.3f}  {p:>8.3f}  {intrinsic:>14.3f}")


# --- Sensitivity: price across volatilities ---
print("\nCall Price Across Volatilities (S=100, K=105, T=0.5)")
print("-" * 40)

for vol in [0.10, 0.15, 0.20, 0.25, 0.30, 0.40, 0.50]:
    c = black_scholes(S, K, T, r, vol, "call")
    print(f"  sigma = {vol:.0%}  ->  Call = {c:.4f}")

The put-call parity check in the code verifies our implementation: C - P should equal S - K * exp(-rT). If those two values match (they will, to machine precision), the call and put formulas are consistent.

---

Computing the Greeks from Black-Scholes

The Greeks are partial derivatives of the Black-Scholes formula with respect to its inputs. They tell you exactly how an option's price will respond to small changes in the underlying variables. Because the Black-Scholes formula is in closed form, every Greek has an analytical expression - no numerical approximation needed.

Here are the formulas for a European call. In each, N() is the cumulative normal CDF and N'() is the standard normal PDF: N'(x) = exp(-x^2 / 2) / sqrt(2 * pi).

Delta (sensitivity to stock price):

Delta_call = N(d1)

Delta_put = N(d1) - 1

Delta tells you how many shares to hold to hedge the option. A call with delta 0.55 moves roughly 0.55 for every 1.00 move in the stock. Delta also approximates the probability of finishing in the money (under the stock-price measure).

Gamma (rate of change of delta):

Gamma = N'(d1) / (S * sigma * sqrt(T))

Gamma is the same for calls and puts. It's highest for at-the-money options near expiry, where delta can swing rapidly between 0 and 1. High gamma means the hedge needs frequent rebalancing.

Theta (time decay):

Theta_call = -(S * N'(d1) * sigma) / (2 * sqrt(T)) - r * K * exp(-rT) * N(d2)

Theta_put = -(S * N'(d1) * sigma) / (2 * sqrt(T)) + r * K * exp(-rT) * N(-d2)

Theta is almost always negative - options lose value as time passes. The first term captures the erosion of optionality (less time for the stock to move), and the second captures the interest rate effect on the strike's present value.

Vega (sensitivity to volatility):

Vega = S * sqrt(T) * N'(d1)

Vega is the same for calls and puts and is always positive - higher volatility means higher option prices. Vega is largest for at-the-money, long-dated options.

Rho (sensitivity to interest rate):

Rho_call = K * T * exp(-rT) * N(d2)

Rho_put = -K * T * exp(-rT) * N(-d2)

Rho is typically the smallest Greek for short-dated equity options. It becomes more significant for long-dated options or in interest-rate-sensitive environments.

Here's a Python implementation of all five Greeks with a worked example:

import numpy as np
from scipy.stats import norm


def bs_greeks(S, K, T, r, sigma, option_type="call"):
    """
    Compute Black-Scholes price and all first-order Greeks.

    Returns a dict with keys: price, delta, gamma, theta, vega, rho.
    Theta is per calendar day. Vega is per 1% vol move.
    """
    d1 = (np.log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)

    n_d1 = norm.cdf(d1)
    n_d2 = norm.cdf(d2)
    npdf_d1 = norm.pdf(d1)
    disc = np.exp(-r * T)

    gamma = npdf_d1 / (S * sigma * np.sqrt(T))
    vega = S * np.sqrt(T) * npdf_d1 / 100  # per 1% move

    if option_type == "call":
        price = S * n_d1 - K * disc * n_d2
        delta = n_d1
        theta = (-(S * npdf_d1 * sigma) / (2 * np.sqrt(T))
                 - r * K * disc * n_d2) / 365
        rho = K * T * disc * n_d2 / 100
    else:
        price = K * disc * norm.cdf(-d2) - S * norm.cdf(-d1)
        delta = n_d1 - 1
        theta = (-(S * npdf_d1 * sigma) / (2 * np.sqrt(T))
                 + r * K * disc * norm.cdf(-d2)) / 365
        rho = -K * T * disc * norm.cdf(-d2) / 100

    return {
        "price": round(price, 4),
        "delta": round(delta, 4),
        "gamma": round(gamma, 4),
        "theta": round(theta, 4),
        "vega": round(vega, 4),
        "rho": round(rho, 4),
    }


# --- Example: ATM call, 6 months, 20% vol ---
greeks = bs_greeks(S=100, K=105, T=0.5, r=0.05, sigma=0.20, option_type="call")

print("Black-Scholes Greeks (Call, S=100, K=105, T=0.5yr)")
print("-" * 48)
for name, value in greeks.items():
    print(f"  {name:<8}: {value:>10.4f}")

print()

# --- Compare call and put Greeks ---
call_g = bs_greeks(100, 100, 0.25, 0.05, 0.25, "call")
put_g = bs_greeks(100, 100, 0.25, 0.05, 0.25, "put")

print("ATM Call vs Put Greeks (S=K=100, T=0.25, sigma=25%)")
print("-" * 52)
print(f"  {'Greek':<8}  {'Call':>10}  {'Put':>10}")
for key in call_g:
    print(f"  {key:<8}  {call_g[key]:>10.4f}  {put_g[key]:>10.4f}")

Notice that gamma and vega are identical for calls and puts at the same strike and expiry. This follows directly from put-call parity: since the difference between a call and put is a forward (which has zero gamma and zero vega), the second-order and volatility sensitivities must match.

---

The Put Formula and Put-Call Parity

The Black-Scholes put formula can be derived independently using the same risk-neutral pricing approach, but there's a simpler route: put-call parity.

Put-call parity is a model-free relationship that holds for European options:

C - P = S - K * exp(-rT)

This says the difference between a call and a put at the same strike and expiry equals the forward price of the underlying minus the discounted strike. It's an arbitrage relationship - if it were violated, you could construct a risk-free profit.

Rearranging to solve for the put price:

P = C - S + K * exp(-rT)

Substituting the Black-Scholes call formula:

P = [S * N(d1) - K * exp(-rT) * N(d2)] - S + K * exp(-rT)

P = S * [N(d1) - 1] - K * exp(-rT) * [N(d2) - 1]

Using the symmetry property of the normal distribution, N(x) - 1 = -N(-x):

P = -S * N(-d1) + K * exp(-rT) * N(-d2)

P = K * exp(-rT) * N(-d2) - S * N(-d1)

This is exactly the Black-Scholes put formula. The derivation through put-call parity confirms that the call and put formulas are internally consistent. In practice, this relationship also serves as a critical check - if your implementation gives call and put prices that violate put-call parity, there's a bug.

Quick numerical verification using our earlier example:

C = 4.581, S = 100, K = 105, T = 0.5, r = 0.05

P = C - S + K * exp(-rT) = 4.581 - 100 + 105 * exp(-0.025) = 4.581 - 100 + 102.408 = 6.989

The put is worth roughly 6.99 - more than the call, which makes sense because the option is out of the money for the call (S < K) but in the money for the put.

---

Assumptions and When They Break Down

The Black-Scholes formula is exact given its assumptions. The question is whether those assumptions hold in real markets. Understanding where each one fails - and how badly - is what separates textbook knowledge from practical trading competence.

Constant volatility. The formula assumes sigma doesn't change over the option's life. In reality, volatility moves constantly. The existence of the volatility smile - different implied volatilities at different strikes - directly contradicts this assumption. For at-the-money options, the impact is small. For deep out-of-the-money puts, where the smile effect is strongest, Black-Scholes can significantly underprice protection. This is the most consequential assumption failure, and it's why practitioners developed stochastic volatility models like Heston.

Log-normal stock prices (no jumps). The model assumes the stock price moves continuously without sudden gaps. In reality, stocks gap at earnings, on geopolitical events, and during flash crashes. These jumps create fat tails in the return distribution that the log-normal model misses. The practical effect: Black-Scholes underestimates the probability of extreme moves. This matters most for short-dated, far out-of-the-money options - precisely where tail risk pricing is critical.

No dividends. The basic formula assumes the stock pays no dividends. For stocks with significant dividend yields, ignoring dividends overstates call prices and understates put prices. The standard fix is to replace S with S * exp(-qT), where q is the continuous dividend yield. This adjustment is simple and works well for index options. For individual stocks with discrete dividends, more care is needed around ex-dividend dates.

Constant risk-free rate. The formula treats r as fixed. For short-dated equity options (less than a year), this is usually fine - interest rate changes over a few months have minimal impact on option prices. For long-dated options (LEAPS) or in environments with rapidly shifting rate expectations, the assumption is more problematic. Most practitioners simply use the yield on the zero-coupon bond matching the option's maturity.

Continuous trading with no transaction costs. The derivation assumes you can rebalance your delta hedge continuously at zero cost. In practice, every trade costs something - spreads, commissions, market impact. The result is that perfect hedging is impossible. Traders typically rebalance at discrete intervals, accepting some hedging error as the price of keeping transaction costs manageable. The Leland (1985) adjustment adds a correction to the volatility to account for discrete hedging.

European exercise only. The formula prices options that can only be exercised at expiry. American options - exercisable at any time - require different methods (binomial trees, finite differences, or the Barone-Adesi and Whaley approximation). For calls on non-dividend-paying stocks, early exercise is never optimal, so the Black-Scholes call formula is correct. For puts, and for calls on dividend-paying stocks, the early exercise premium can be significant.

---

Beyond Black-Scholes: How Practitioners Adjust

The Black-Scholes formula is the starting point, not the final answer. Every serious options pricing desk in 2026 uses models that extend or correct Black-Scholes to handle the failures listed above.

Local volatility (Dupire, 1994). Instead of a single sigma, define a volatility function sigma(S, t) that varies with both the stock price and time. The local volatility surface can be uniquely extracted from observed option prices, giving a model that perfectly matches the current market smile. The downside: its predictions for how the smile will evolve over time are unrealistic. The model implies that the smile flattens too quickly as you look further into the future.

Stochastic volatility (Heston, 1993). The Heston model treats volatility as a separate random process correlated with the stock price. It produces realistic smiles and more believable dynamics for how volatility evolves. Five parameters - initial variance, long-run variance, mean-reversion speed, vol of vol, and correlation - control the shape and behaviour of the implied volatility surface.

Jump-diffusion (Merton, 1976). Adds sudden jumps to the stock price process on top of the continuous diffusion. This generates the steep short-dated skew that pure stochastic volatility struggles to produce. Jumps matter most for options with less than a month to expiry, where there isn't enough time for diffusive volatility to generate extreme tail probabilities.

Local-stochastic volatility hybrids. These combine local volatility for exact calibration to current market prices with a stochastic volatility component for realistic forward dynamics. They're the current standard on many exotic derivatives desks, though computationally expensive. The Black-Scholes formula remains embedded at the core - even these sophisticated models express their outputs in Black-Scholes implied volatility terms.

In practice, Black-Scholes serves as the universal translator. Traders and quants run complex models internally but communicate prices and risks in Black-Scholes implied vol. When someone quotes "25-delta put vol at 22%", they mean the Black-Scholes implied volatility for a put with 25-delta is 22%. The formula is the Rosetta Stone of options markets.

---

Frequently Asked Questions

What is the Black-Scholes formula used for?

The Black-Scholes formula is used to calculate the theoretical price of European call and put options. Given the current stock price, strike price, time to expiry, risk-free rate, and volatility, it produces a fair value for the option under a set of idealised assumptions. Traders use it for initial pricing, for computing the Greeks (sensitivities to input changes), and as the common language for quoting option prices in terms of implied volatility. It's also the basis for computing implied volatility - feeding a market price back into the formula and solving for sigma. Every options desk in 2026 uses Black-Scholes or an extension of it.

How do you derive the Black-Scholes formula?

The formula can be derived through several equivalent routes. The most common is risk-neutral pricing: assume the stock follows geometric Brownian motion, switch to the risk-neutral measure (where the stock drifts at the risk-free rate), compute the expected discounted payoff of the option, and evaluate the resulting integral over the log-normal distribution of the terminal stock price. This involves completing the square in a Gaussian integral and produces the N(d1) and N(d2) terms. An alternative approach uses the Black-Scholes PDE, derived by constructing a riskless portfolio of the option and the underlying, then applying the no-arbitrage condition. Both routes yield the same formula.

What is the difference between the Black-Scholes formula and the Black-Scholes equation?

The Black-Scholes equation (or PDE) is a partial differential equation that any derivative's price must satisfy under the model's assumptions. It is: dV/dt + (1/2) * sigma^2 * S^2 * d^2V/dS^2 + r * S * dV/dS - r * V = 0. The Black-Scholes formula is the specific solution to this PDE for the boundary condition of a European call or put. The equation is general and applies to any European payoff; the formula is the closed-form answer for the particular case of a vanilla option. Other payoffs (digital options, power options) satisfy the same PDE but have different solutions.

How accurate is the Black-Scholes formula?

For at-the-money options on liquid underlyings with moderate time to expiry, the formula produces prices close to market values when you use the correct implied volatility. Its accuracy degrades for deep out-of-the-money options (where the volatility smile means a single sigma isn't sufficient), very short-dated options (where jump risk dominates), and during market stress (where volatility is far from constant). The formula is most useful as a framework for thinking about option pricing rather than as a source of highly precise prices. In practice, traders use the formula's structure but adjust the volatility input to account for smile and skew effects.

Can I use the Black-Scholes formula for American options?

Not directly. The formula assumes European exercise (only at expiry). However, for American calls on non-dividend-paying stocks, early exercise is never optimal, so the European call price equals the American call price and the formula is correct. For American puts, and for calls on stocks that pay dividends before expiry, early exercise can be optimal and the formula will undervalue the option. In those cases, use numerical methods - binomial trees, finite difference solutions of the Black-Scholes PDE, or analytical approximations like Barone-Adesi and Whaley.

What inputs do I need for the Black-Scholes formula?

You need five inputs: the current stock price (S), the strike price (K), time to expiry in years (T), the continuously compounded risk-free interest rate (r), and the annualised volatility of the underlying (sigma). The first four are directly observable. Volatility is the tricky one - you can estimate it from historical returns (historical volatility) or extract it from current option prices (implied volatility). The choice of volatility input has the largest impact on the output price, which is why volatility estimation and implied volatility analysis are central skills in options trading.