Black-Scholes Formula: European Option Pricing & Greeks

In the previous chapter, we derived the Black-Scholes-Merton partial differential equation (PDE), which describes how the price of a derivative must evolve to prevent arbitrage opportunities. That PDE, while elegant, doesn't immediately tell us what an option is worth. To price an option, we need to solve the PDE subject to the appropriate boundary conditions.

This chapter presents the solution: the famous Black-Scholes formula for European options. Published by Fischer Black and Myron Scholes in 1973, with an alternative derivation and extensions published by Robert Merton that same year, this formula transformed finance by providing a closed-form European option price grounded in dynamic replication and no-arbitrage reasoning. The formula gave market participants a systematic way to connect option prices with the cost of dynamically hedging their payoffs. Cboe reports growth from 5.6 million contracts in 1974 to roughly 1.3 billion across its exchanges in 2014, including futures. The 1997 Nobel committee credited the pricing methodology with advances in valuation and risk management.

The formula expresses an option's price in terms of five observable or estimable quantities: the current stock price, the strike price, time to expiration, the risk-free interest rate, and the volatility of the underlying asset. The expected return of the stock does not appear in the formula. As we learned when studying risk-neutral valuation, this non-dividend-paying stock has expected growth rate $r$ under the pricing measure. More generally, the martingale condition applies to appropriately discounted total gains, including dividends, rather than to every ex-dividend asset price.

We'll work through the solution to the Black-Scholes PDE, interpret each component of the resulting formula, implement it in Python, and verify that it satisfies fundamental relationships like put-call parity. We'll close with an introduction to the Greeks, the sensitivity measures that quantify how option prices respond to changes in the underlying parameters.

Recall from the previous chapter that the Black-Scholes PDE for a derivative $V(S,t)$ on a non-dividend-paying stock is:

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$

where:

• $V$: price of the derivative
• $t$: time
• $S$: price of the underlying asset
• $\sigma$: volatility of the underlying asset
• $r$: risk-free interest rate

This equation captures a fundamental requirement: the price of any derivative must evolve in time so no riskless arbitrage profit can be extracted. Each term in the equation plays a specific role. The first term represents how the option price changes with the passage of time alone. The second term, involving the second derivative with respect to the stock price, captures the effect of the stock's random fluctuations, which is precisely why the volatility parameter appears squared and multiplied by $S^2$. The third term accounts for the drift in the option price due to the expected growth of the stock under risk-neutral dynamics. The last term discounts the derivative value at rate $r$. Together, the PDE terms make the self-financing delta-hedged position with value $V-SV_S$ earn the risk-free rate; the unhedged derivative need not have that expected return under the physical measure.

To find the price of a European call option, we must solve this PDE subject to the boundary condition at expiration:

$$C(S,T) = \max(S - K, 0)$$

where:

• $C(S,T)$: value of the call option at expiration
• $S$: stock price at expiration
• $K$: strike price
• $T$: expiration time

This boundary condition states something intuitive: at the moment of expiration, an option is worth exactly its intrinsic value. If the stock price exceeds the strike, the call holder can buy the stock at $K$ and immediately sell it at $S$, pocketing the difference. If the stock price falls below the strike, the call expires worthless. The mathematical challenge lies in determining what the option should be worth at any time before expiration, given that it will satisfy this terminal condition.

We seek the financially admissible solution, with $0\leq C(S,t)\leq S$ and at most linear growth as $S$ increases. These restrictions exclude nonfinancial solutions on the unbounded stock-price domain.

Assume $S,K>0$, constant $\sigma>0$ and constant $r$. Introduce remaining time $\tau=T-t$, measured in years when the rate and volatility use annual units. As $\tau$ increases, we move backward from the known expiry payoff. Set $x=\ln(S/K)$, the dimensionless log-moneyness. Multiplying $S$ by a factor adds the logarithm of that factor to $x$. For example, when K=\100$, stock prices of$$50$and$$200$map to$-\ln 2$and$+\ln 2$.

Normalize and discount the option value by writing

$$V(S,t)=K e^{-r\tau}v(x,\tau).$$

Here $v$ is dimensionless; $\tau$ still has units of time. The chain rule gives

$$V_t=K e^{-r\tau}(rv-v_\tau),\qquad V_S=\frac{K e^{-r\tau}}{S}v_x,\qquad V_{SS}=\frac{K e^{-r\tau}}{S^2}(v_{xx}-v_x).$$

Substituting these derivatives into the pricing PDE and cancelling $K e^{-r\tau}$ leaves a constant-coefficient equation:

$$v_\tau=\frac{\sigma^2}{2}v_{xx}+ \left(r-\frac{\sigma^2}{2}\right)v_x.$$

Write $a=\sigma^2/2$ and $b=r-\sigma^2/2$. To remove both the drift term and the zero-order term that the next substitution creates, set

$$v=e^{\alpha x+\beta\tau}u,\qquad \alpha=-\frac{b}{2a}=\frac12-\frac{r}{\sigma^2},\qquad \beta=-\frac{b^2}{4a}=-\frac{\sigma^2\alpha^2}{2}.$$

The constant $\alpha$ is dimensionless, while $\beta$ has inverse-time units. Direct differentiation gives $u_\tau=a u_{xx}+(2a\alpha+b)u_x+(a\alpha^2+b\alpha-\beta)u$. Our choices make both extra coefficients zero, so

$$u_\tau=\frac{\sigma^2}{2}u_{xx}.$$

This is the heat equation with diffusivity $\sigma^2/2$. The terminal call payoff becomes its initial condition at $\tau=0$:

$$u(y,0)=e^{-\alpha y}(e^y-1)^+.$$

The notation $z^+=\max(z,0)$ clips negative values to zero. Thus this initial function is zero for $y\leq0$. For $\tau>0$, the whole-line heat kernel is the Gaussian density

$$G(x-y,\tau)= \frac{1}{\sigma\sqrt{2\pi\tau}} \exp\!\left(-\frac{(x-y)^2}{2\sigma^2\tau}\right).$$

It weights transformed payoff values at log-price endpoints $y$. The kernel integrates to one; $u(y,0)$ is a transformed payoff, not a probability density. Convolving the two gives

$$u(x,\tau)= \int_0^\infty \left[e^{(1-\alpha)y}-e^{-\alpha y}\right]G(x-y,\tau)\,dy.$$

To evaluate this integral, let $z=\sigma\sqrt{\tau}$ and define $I_c=\int_0^\infty e^{cy}G(x-y,\tau)\,dy$. Completing the square in its exponent gives

$$cy-\frac{(x-y)^2}{2z^2} =-\frac{(y-x-cz^2)^2}{2z^2}+cx+\frac{c^2z^2}{2}, \qquad I_c=e^{cx+c^2z^2/2}N\!\left(\frac{x+cz^2}{z}\right).$$

Here $N$ is the standard normal cumulative distribution function. Hence $u=I_{1-\alpha}-I_{-\alpha}$. Multiplying by the two scaling factors recovers $C=K e^{-r\tau}e^{\alpha x+\beta\tau}u$. Because $\beta=-\sigma^2\alpha^2/2$ and $\alpha\sigma^2=\sigma^2/2-r$, the first term simplifies to $SN(d_1)$ and the second to $Ke^{-r\tau}N(d_2)$, where

$$d_1=\frac{x+(r+\sigma^2/2)\tau}{\sigma\sqrt{\tau}}, \qquad d_2=\frac{x+(r-\sigma^2/2)\tau}{\sigma\sqrt{\tau}}.$$

The next section states the resulting prices and interprets these standardized terms.

Working backward through the transformations, reversing each change of variables in sequence, the solution for a European call option emerges as:

$$C(S,t) = S \cdot N(d_1) - K e^{-r(T-t)} N(d_2)$$

where:

• $C(S,t)$: call option price
• $S$: current stock price
• $N(\cdot)$: cumulative standard normal distribution function
• $d_1, d_2$: standardized moneyness terms
• $K$: strike price
• $r$: risk-free interest rate
• $T-t$: time to expiration

This formula expresses the option price as the difference between two terms, each involving the cumulative normal distribution function $N(\cdot)$. The function $N(x)$ gives the probability that a standard normal random variable is less than $x$, and it appears here because the underlying stock price follows a lognormal distribution under geometric Brownian motion.

The terms $d_1$ and $d_2$ are defined as:

$$d_1 = \frac{\ln(S/K) + (r + \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$$

where:

• $S$: current stock price
• $K$: strike price
• $r$: risk-free rate
• $\sigma$: volatility
• $T-t$: time to expiration

The numerator of $d_1$ contains two parts. The first, $\ln(S/K)$, measures the current moneyness of the option in logarithmic terms. A positive value indicates the option is in-the-money, while a negative value indicates it is out-of-the-money. The second part, $(r + \sigma^2/2)(T-t)$, equals the risk-neutral expected log-price drift $(r - \sigma^2/2)(T-t)$ plus the variance adjustment $\sigma^2(T-t)$. It shifts $d_2$ to $d_1$; it is not the expected log-price drift itself. The denominator, $\sigma\sqrt{T-t}$, normalizes this quantity by the standard deviation of the log-price at expiration, converting it into a standardized z-score.

The second standardized term is:

$$d_2 = d_1 - \sigma\sqrt{T-t} = \frac{\ln(S/K) + (r - \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$$

where:

• $d_1$: first standardized term
• $\sigma$: volatility
• $T-t$: time to expiration
• $S$: current stock price
• $K$: strike price
• $r$: risk-free rate

Notice that $d_2$ differs from $d_1$ only in the sign of the $\sigma^2/2$ term in the numerator. This sign change determines the probabilistic interpretation developed below. The relationship $d_2 = d_1 - \sigma\sqrt{T-t}$ shows that the two quantities differ by exactly one standard deviation of the log-return.

For a European put option, we can either solve the PDE with the put boundary condition $P(S,T) = \max(K-S, 0)$, or use put-call parity. The result is:

$$P(S,t) = K e^{-r(T-t)} N(-d_2) - S \cdot N(-d_1)$$

where:

• $P(S,t)$: put option price
• $K$: strike price
• $r$: risk-free rate
• $T-t$: time to expiration
• $S$: current stock price
• $N(\cdot)$: cumulative standard normal distribution function
• $d_1, d_2$: standardized moneyness terms

The put formula has a parallel structure to the call formula but with the terms reversed and the signs of the arguments to $N(\cdot)$ flipped. This symmetry reflects the complementary nature of calls and puts: a call benefits when the stock rises above the strike, while a put benefits when the stock falls below the strike. The relationship between the two formulas is made precise by put-call parity.

These are the celebrated Black-Scholes formulas. Let's examine what each term means.

The call option formula $C = SN(d_1) - Ke^{-r\tau}N(d_2)$ (where $\tau = T-t$) has a clear interpretation when viewed through the lens of risk-neutral valuation. To understand the formula, we must connect it to risk-neutral pricing, where a derivative's price is its discounted expected payoff:

$$C = e^{-r\tau} \mathbb{E}^{\mathbb{Q}}[\max(S_T - K, 0)]$$

where:

• $C$: call option price
• $r$: risk-free rate
• $\tau$: time to expiration
• $\mathbb{E}^{\mathbb{Q}}$: expectation under the risk-neutral measure $\mathbb{Q}$
• $S_T$: stock price at expiration
• $K$: strike price

This expectation says that we imagine a world where all investors are indifferent to risk, so the stock grows at the risk-free rate on average. In this risk-neutral world, we calculate the average payoff of the option and then discount it back to today. This procedure gives the correct arbitrage-free price even though the real world is not risk-neutral.

This expectation can be decomposed into two pieces:

$$C = e^{-r\tau} \mathbb{E}^{\mathbb{Q}}[S_T \cdot \mathbf{1}_{S_T > K}] - e^{-r\tau} K \cdot \mathbb{P}^{\mathbb{Q}}(S_T > K)$$

where:

• $C$: call option price
• $r$: risk-free rate
• $\tau$: time to expiration
• $S_T$: stock price at expiration
• $K$: strike price
• $\mathbf{1}_{S_T > K}$: indicator function (1 if $S_T > K$, 0 otherwise)
• $\mathbb{P}^{\mathbb{Q}}$: probability under the risk-neutral measure

This decomposition separates the option's payoff into its two components: receiving the stock and paying the strike. The indicator function $\mathbf{1}_{S_T > K}$ equals 1 when the option finishes in-the-money and 0 otherwise. The first term captures the expected value of receiving the stock in those scenarios where exercise occurs. The second term captures the expected cost of paying the strike price, again only in scenarios where exercise occurs.

The second term is straightforward: $K e^{-r\tau} N(d_2)$ is the present value of paying the strike, weighted by the risk-neutral probability $N(d_2)$ that the option finishes in-the-money. This term has a clean economic interpretation. You can think of it as the cost of a contingent payment: you will pay $K$ at expiration, but only if $S_T > K$. The probability of this event under risk-neutral dynamics is $N(d_2)$, and we discount the expected payment back to today at the risk-free rate.

The first term requires more care. $S \cdot N(d_1)$ is the discounted expected stock receipt restricted to exercise outcomes: $e^{-r\tau}\mathbb{E}^{\mathbb{Q}}[S_T\mathbf{1}_{S_T>K}]$. This is an unconditional expectation with an indicator, not an expectation conditional on exercise. If $A=\{S_T>K\}$, then $e^{-r\tau}\mathbb{E}^{\mathbb{Q}}[S_T\mid A]=S N(d_1)/N(d_2)$ when the exercise probability is positive. High-stock-price outcomes receive more weight in the expected stock receipt than in the exercise probability. The next paragraph gives $N(d_1)$ its precise share-measure interpretation.

Technically, $N(d_1)$ is the probability that $S_T > K$ under a different probability measure, sometimes called the stock measure or share measure. This alternative measure reweights outcomes according to the stock price, giving more weight to high-stock-price scenarios. Under this measure, the probability of exercise is higher than $N(d_2)$ because we are overweighting exactly those outcomes where exercise occurs. The difference between $N(d_1)$ and $N(d_2)$ captures this reweighting effect.

The five inputs to the Black-Scholes formula each affect option prices in intuitive ways. Understanding these relationships builds intuition for how options behave and why traders care about each parameter.

• Stock price ($S$): Higher stock prices increase call values and decrease put values. This is the most direct relationship: a call gives you the right to buy at $K$, so the higher $S$ is relative to $K$, the more valuable that right. If the stock is already trading at $120 and your strike is $100, the option has $20 of intrinsic value plus whatever time value remains. Mathematically, the stock price enters the formula directly in the first term and indirectly through $d_1$ and $d_2$.

• Strike price ($K$): Higher strikes decrease call values and increase put values when the other inputs are fixed. A call has a positive expiry payoff only when the stock finishes above its strike. For example, calls struck at \100$and$$110$have different exercise thresholds. A purchased call's profit also depends on its premium and financing; ignoring financing, expiry break-even is the strike plus the premium. The strike enters both the discounted payment and the log-moneyness in$d_1$and$d_2$.

• Time to expiration ($\tau = T-t$): More remaining time often raises an option's value, but the effect depends on the option type and inputs. A European put can fall in value as remaining maturity increases, even when interest rates are positive. Near expiration, at-the-money options can experience especially rapid time decay. Theta captures this local calendar-time effect with other inputs fixed.

• Risk-free rate ($r$): Higher rates increase call values and decrease put values. The intuition is that buying a call is like a leveraged position in the stock. You get stock exposure while keeping your capital earning interest elsewhere. Higher rates make this implicit financing benefit more useful. Additionally, higher rates reduce the present value of the strike payment, making it cheaper in today's dollars to acquire the stock at expiration.

• Volatility ($\sigma$): Higher volatility increases both call and put values with the other inputs fixed. Under the model's pricing measure, changing volatility leaves the terminal stock-price mean $Se^{r\tau}$ unchanged and changes the dispersion of outcomes. The convex vanilla payoffs benefit from that extra dispersion: a call struck at \100$pays$$20$if the stock finishes at$$120$and zero if it finishes at$$80$. These are payoff examples, not probabilities assigned to exact endpoints. Greater volatility does not make every upper-tail event more likely. At very low volatility the call approaches$\max(S-Ke^{-r\tau},0)$, so its value still depends on carry and remaining maturity.

The quantities $d_1$ and $d_2$ are standardized measures of "moneyness," expressing how far in- or out-of-the-money the option is, adjusted for time and volatility. These quantities serve as the arguments to the cumulative normal distribution function, converting the lognormal distribution of stock prices into a tractable form.

Consider $d_2$ first:

$$d_2 = \frac{\ln(S/K) + (r - \frac{\sigma^2}{2})\tau}{\sigma\sqrt{\tau}}$$

where:

• $S$: stock price
• $K$: strike price
• $r$: risk-free rate
• $\sigma$: volatility
• $\tau$: time to expiration

To understand $d_2$, we need to recall how the stock price evolves under geometric Brownian motion. The numerator $\ln(S/K) + (r - \frac{\sigma^2}{2})\tau$ is the expected log-return of the stock under the risk-neutral measure, relative to the strike. The term $r - \sigma^2/2$ appears rather than just $r$ because of the difference between arithmetic and geometric means. When a variable follows geometric Brownian motion with drift $\mu$, its expected logarithm grows at rate $\mu - \sigma^2/2$, not $\mu$. This correction factor, sometimes called the Itô correction or convexity adjustment, arises from the nonlinear relationship between a variable and its logarithm.

Recall that under geometric Brownian motion with drift $r$, the expected log-price is:

$$\mathbb{E}[\ln S_T] = \ln S + (r - \frac{\sigma^2}{2})\tau$$

where:

• $\mathbb{E}[\ln S_T]$: expected log stock price at time $T$
• $\ln S$: log of current stock price
• $r$: risk-free rate
• $\sigma$: volatility
• $\tau$: time to expiration

So the numerator equals $\mathbb{E}[\ln S_T] - \ln K = \mathbb{E}[\ln(S_T/K)]$. This is the expected value of the log of the ratio of the terminal stock price to the strike. When this is positive, the stock is expected to end up above the strike on average. The denominator $\sigma\sqrt{\tau}$ is the standard deviation of this log-return. The volatility of the log-price after time $\tau$ is $\sigma\sqrt{\tau}$, which reflects the diffusion of prices over time. Thus $d_2$ measures how many standard deviations the expected log-price exceeds the log-strike, which is exactly the input you'd need to find $\mathbb{P}(S_T > K)$ using the normal distribution.

The quantity $d_1$ exceeds $d_2$ by $\sigma\sqrt{\tau}$. Completing the square in the lognormal density shows that $\mathbb{E}^{\mathbb Q}[S_T\mathbf1_{\{S_T>K\}}]=Se^{r\tau}N(d_1)$. This is the stock receipt weighted across all outcomes, with zero contribution when exercise does not occur. To obtain the expectation conditional on exercise, divide by $N(d_2)$ when that probability is positive.

The technical explanation involves a change of measure: for this non-dividend stock, $N(d_1)$ is the probability of the call event $S_T>K$ under the stock-numeraire or share measure. The density change weights outcomes by the terminal stock price, increasing the probability of this upper-tail event. It decreases the probability of the complementary lower-tail event $S_T<K$; the comparison is not the same for put exercise.

As we learned when studying option basics, put-call parity is a fundamental relationship linking European call and put prices:

$$C - P = S - Ke^{-r\tau}$$

where:

• $C$: call option price
• $P$: put option price
• $S$: stock price
• $K$: strike price
• $r$: risk-free rate
• $\tau$: time to expiration

This equality does not require the Black-Scholes stock-price distribution. It follows from matching European calls and puts on a non-dividend stock under the stated frictionless trading and financing assumptions. The long-call, short-put portfolio pays $S_T-K$. Buying a share and borrowing $Ke^{-r\tau}$ today, with $K$ repaid at expiration, has the same payoff. No arbitrage therefore equates their current values.

Let's verify that the Black-Scholes formulas satisfy put-call parity. This verification serves two purposes: it confirms that our formulas are internally consistent, and it demonstrates the elegant symmetry properties of the normal distribution.

Starting with the Black-Scholes call and put formulas:

$$\begin{aligned} C &= S N(d_1) - K e^{-r\tau} N(d_2) \\ P &= K e^{-r\tau} N(-d_2) - S N(-d_1) \end{aligned}$$

where:

• $C, P$: call and put option prices
• $S$: stock price
• $K$: strike price
• $r$: risk-free rate
• $\tau$: time to expiration
• $N(\cdot)$: cumulative standard normal distribution
• $d_1, d_2$: moneyness terms

We compute the difference $C - P$ to verify the relationship. The key mathematical fact we will use is the symmetry of the standard normal distribution: $N(x) + N(-x) = 1$ for any value of $x$. This follows from the fact that the normal distribution is symmetric around zero, so the probability of being less than $x$ plus the probability of being greater than $x$ (which equals the probability of being less than $-x$) must sum to one.

$$\begin{aligned} C - P &= [S N(d_1) - K e^{-r\tau} N(d_2)] - [K e^{-r\tau} N(-d_2) - S N(-d_1)] && \text{(substitute formulas)} \\ &= S [N(d_1) + N(-d_1)] - K e^{-r\tau} [N(d_2) + N(-d_2)] && \text{(group terms)} \\ &= S \cdot 1 - K e^{-r\tau} \cdot 1 && \text{(symmetry: } N(x) + N(-x) = 1\text{)} \\ &= S - K e^{-r\tau} && \text{(simplify)} \end{aligned}$$

where:

• $C$: call price
• $P$: put price
• $S$: stock price
• $K$: strike price
• $r$: risk-free rate
• $\tau$: time to expiration

The Black-Scholes formulas are internally consistent and satisfy the no-arbitrage constraint of put-call parity. This verification is more than a mathematical exercise. It confirms that the formulas respect the fundamental arbitrage relationships that must hold in any well-functioning market. Any pricing model, no matter how sophisticated, must satisfy put-call parity. If a proposed option pricing formula violated this relationship, we would know immediately that it contains an error.

Let's implement the Black-Scholes formula in Python and explore its behavior. These educational functions use scalar inputs with $S,K,\sigma>0$ and positive remaining time for the pre-expiry formulas, with a separate expiry-payoff branch. They assume finite, numerically usable inputs and are not general-range pricing routines. The validator explicitly rejects zero volatility, which would require a separate deterministic-limit formula. Negative remaining maturity is rejected; zero remaining maturity returns the expiry payoff.

At these example inputs, the implementation agrees with put-call parity to floating-point precision.

Understanding the intermediate values $d_1$ and $d_2$ helps build intuition for how the formula works:

The negative $d_2$ value tells us there's less than a 50% risk-neutral probability that this option finishes in-the-money. The value of $N(d_2)$ calculated above represents the precise probability of the stock price exceeding the strike at expiration under the risk-neutral measure.

The key parameters for the Black-Scholes implementation are:

• $S$: Current stock price. Higher prices increase call values and decrease put values.
• $K$: Strike price. The reference price for the option's payoff.
• $T$: Time to expiration in years. Longer times generally increase option values.
• $r$: Risk-free interest rate (annualized). Higher rates favor calls over puts.
• $\sigma$: Volatility (annualized). Controls the dispersion of terminal stock prices; with the other inputs fixed, higher volatility increases both call and put values.

Let's visualize how option prices change with stock price, volatility and remaining maturity, holding the other inputs fixed. These examples build intuition for the Greeks introduced later in this chapter.

The Greeks are partial derivatives of the option price with respect to various inputs. They quantify how sensitive an option's model price is to small changes in those inputs. Understanding the Greeks is essential for risk management and hedging. Each Greek measures a different dimension of local, model-based sensitivity; this set does not capture every risk, including model misspecification or liquidity risk.

The Greeks derive their importance from the dynamic nature of hedging. As we learned when studying the derivation of the Black-Scholes PDE, perfect replication within that model requires continuous adjustment of a hedge portfolio. Delta gives the model's stock hedge ratio, while the other Greeks help assess how that hedge and the option value respond as inputs change. These sensitivities do not eliminate errors from discrete rebalancing or departures from the model's assumptions.

Delta ($\Delta$) measures the local rate of change of the option price with respect to the stock price, holding the other inputs fixed. Multiplying delta by a stock-price change gives a first-order estimate of the option-price change, not an exact repricing for a finite $1 move. Delta answers the practical question: if the stock moves a little, how much does your option position change in value?

$$\Delta_{\text{call}} = \frac{\partial C}{\partial S} = N(d_1)$$

where:

• $\Delta_{\text{call}}$: delta of the call option
• $C$: call option price
• $S$: stock price
• $N(d_1)$: cumulative normal distribution value at $d_1$

The result that call delta equals $N(d_1)$ comes directly from differentiating the Black-Scholes formula. The derivative of the first term $S \cdot N(d_1)$ with respect to $S$ is not simply $N(d_1)$ because $d_1$ itself depends on $S$. However, when we carefully apply the chain rule, the additional terms cancel out, leaving just $N(d_1)$. This cancellation is not coincidental; it reflects a deep property of the Black-Scholes formula related to the self-financing nature of the replicating portfolio.

$$\begin{aligned} \Delta_{\text{put}} &= \frac{\partial P}{\partial S} \\ &= N(d_1) - 1 \\ &= -N(-d_1) \end{aligned}$$

where:

• $\Delta_{\text{put}}$: delta of the put option
• $P$: put option price
• $S$: stock price
• $N(d_1)$: cumulative normal distribution value at $d_1$
• $N(-d_1)$: cumulative normal distribution value at $-d_1$

Call deltas range from 0 to 1, while put deltas range from -1 to 0 in this non-dividend model. A spot-at-the-money call often has delta near 0.5 for short remaining maturity and modest carry, but its exact delta is $N((r/\sigma+\sigma/2)\sqrt{\tau})$. Delta gives a first-order estimate for a small stock move. A deep in-the-money call approaches delta 1, while a deep out-of-the-money call approaches delta 0.

Delta also has an interpretation as a hedge ratio: to hedge a short call position, you should hold $\Delta$ shares of stock. If you sell a call with delta 0.5, you need to buy 0.5 shares to neutralize your exposure to small stock price movements. This hedge ratio must be continuously updated as the stock price moves and as time passes, which is why delta hedging is dynamic rather than static.

Gamma ($\Gamma$) measures the local rate of change of delta with respect to the stock price, holding the other inputs fixed. Multiplying gamma by a small stock-price change approximates the change in delta. This second-order sensitivity helps assess how stable the delta hedge will be. High gamma means that even small moves in the stock price can significantly change the delta, requiring frequent rebalancing of the hedge.

$$\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{N'(d_1)}{S \sigma \sqrt{T-t}}$$

where:

• $\Gamma$: gamma of the option
• $V$: option price (call or put)
• $S$: stock price
• $N'(d_1)$: standard normal probability density at $d_1$
• $\sigma$: volatility
• $T-t$: time to expiration

The function $N'(x)=e^{-x^2/2}/\sqrt{2\pi}$ is the standard normal probability density. Gamma measures the stock-price derivative of call delta $N(d_1)$, which is the call exercise probability under the share measure. It is not the derivative of the risk-neutral exercise probability $N(d_2)$.

Gamma is the same for calls and puts with the same strike and expiration. This equality follows from put-call parity: since $C - P = S - Ke^{-r\tau}$, and the right-hand side is linear in $S$, the second derivative with respect to $S$ must be identical for calls and puts.

High gamma means delta changes rapidly with the stock price, requiring frequent hedge rebalancing. As expiration approaches, gamma becomes concentrated near the money, but its exact maximum depends on the variable being changed. At fixed $S,\tau,r,\sigma$, gamma peaks across strikes at $K=S\exp((r+\sigma^2/2)\tau)$. At fixed $K,\tau,r,\sigma$, its peak across stock prices is $S=K\exp(-(r+3\sigma^2/2)\tau)$. Long gamma gives positive local price curvature; total profit also depends on time decay, financing and the trading strategy.

Theta ($\Theta$) measures the local rate of change of the option price as calendar time $t$ advances, holding the expiration date $T$ and the other inputs fixed. Since remaining maturity is $T-t$, this is the negative of the derivative with respect to remaining maturity. Theta can have either sign: a negative value indicates time decay, while a positive value indicates a local increase in price as time passes. With time measured in years, the formula below gives theta per year.

$$\Theta_{\text{call}} = \frac{\partial C}{\partial t} = -\frac{S N'(d_1) \sigma}{2\sqrt{T-t}} - rKe^{-r(T-t)}N(d_2)$$

where:

• $\Theta_{\text{call}}$: theta of the call option
• $S$: stock price
• $N'(d_1)$: standard normal PDF at $d_1$
• $\sigma$: volatility
• $T-t$: time to expiration
• $r$: risk-free rate
• $K$: strike price
• $N(d_2)$: cumulative normal CDF at $d_2$

The formula consists of two components, each with a distinct economic interpretation:

• $-\frac{S N'(d_1) \sigma}{2\sqrt{T-t}}$: Represents the erosion of value due to the resolution of uncertainty (time decay at fixed volatility). As time passes, there is less time for favorable price movements to occur. This term is always negative, which reflects the fact that options lose optionality value over time.
• $- rKe^{-r(T-t)}N(d_2)$: Represents the financing contribution from deferring the strike payment. For positive rates, the call's deferral benefit shrinks as expiration approaches. This contribution vanishes at zero rates and reverses sign at negative rates. In put theta, the financing term has the opposite sign because puts involve receiving rather than paying the strike.

Theta is often negative for long option positions, but time decay is not universal. A sufficiently deep in-the-money European put can have positive theta when interest rates are positive, because receiving the strike sooner can outweigh the loss of optionality. Near expiration, at-the-money options can have especially large negative theta, making time decay a practical concern for their holders.

The Black-Scholes PDE links theta and gamma through $\Theta + \tfrac{1}{2}\sigma^2 S^2\Gamma = r(V-S\Delta)$, where $V$ is the option value. The right-hand side accounts for financing the delta-hedged position. Positive gamma therefore does not, by itself, force theta to be negative; the option value, delta, stock price, and interest rate also enter the relationship.

Vega ($\nu$ or $\mathcal{V}$) is the local derivative of option value with respect to volatility expressed as a decimal. Despite being called "vega," this Greek is not named after an actual Greek letter, but the convention is so established that the name persists. Vega is useful when a position expresses a view on volatility. Future realized volatility must be estimated, while implied volatility can be inferred from an option's quoted market price when that price lies within the model's admissible range.

$$\nu = \frac{\partial V}{\partial \sigma} = S \sqrt{T-t} \cdot N'(d_1)$$

where:

• $\nu$: vega of the option
• $V$: option price
• $S$: stock price
• $T-t$: time to expiration
• $N'(d_1)$: standard normal PDF at $d_1$

A change in decimal volatility of $0.01$, such as $20\%$ to $21\%$, is one percentage point. Traders commonly quote vega as the raw derivative divided by $100$. Multiplying that quotation by a one-percentage-point change gives a first-order price estimate, not an exact finite-change gain.

Vega is the same for calls and puts and is positive in the stated pre-expiry domain: $S>0$, $T-t>0$ and $N'(d_1)>0$. Thus a small volatility increase raises both vanilla option prices with the other inputs fixed.

Vega is often concentrated near the money, but the location of its maximum depends on which inputs are held fixed. At fixed stock price, remaining maturity, interest rate and volatility, it peaks across strikes at $d_1=0$, or $K=S\exp((r+\sigma^2/2)(T-t))$, not necessarily at $K=S$. Increasing maturity raises the $\sqrt{T-t}$ factor but also changes $N'(d_1)$, so vega need not increase over every maturity interval.

Trading volatility can involve exposure to changes in implied volatility or to differences between realized and implied volatility. The examples here isolate vega exposure, with the other inputs held fixed. For the non-dividend-paying European call and put here, with $S,K>0$, positive volatility and time remaining before expiry, a long option has positive vega; holding the other inputs fixed, a rise in implied volatility raises its model value. A short option has the opposite exposure. Total profit also depends on stock moves, time decay, financing and other changes.

Rho ($\rho$) is the local derivative of option price with respect to the interest rate expressed as a decimal. Its contribution to risk depends on the size of the rate change and the contract's maturity. Comparisons with other Greeks require specified input changes because the derivatives use different units.

$$\rho_{\text{call}} = \frac{\partial C}{\partial r} = K(T-t)e^{-r(T-t)}N(d_2)$$

where:

• $\rho_{\text{call}}$: rho of the call option
• $K$: strike price
• $T-t$: time to expiration
• $r$: risk-free rate
• $N(d_2)$: cumulative normal CDF at $d_2$

Call rho is positive because higher interest rates reduce the present value of the strike payment while leaving the current stock price unchanged. A call holder benefits from this because they pay the strike in the future, and higher rates make that future payment cheaper in today's dollars.

$$\rho_{\text{put}} = \frac{\partial P}{\partial r} = -K(T-t)e^{-r(T-t)}N(-d_2)$$

where:

• $\rho_{\text{put}}$: rho of the put option
• $P$: put option price
• $K$: strike price
• $T-t$: time to expiration
• $r$: risk-free rate
• $N(-d_2)$: cumulative normal CDF at $-d_2$

Put rho is negative because the put holder receives the strike at exercise, and a higher rate reduces its present value. Rho contains a remaining-maturity factor, but discounting and $N(d_2)$ also depend on maturity. Its magnitude need not grow proportionally or increase over every maturity interval.

Interest-rate sensitivity reflects the cost of carrying a hedged option position. Longer-dated equity options can be more sensitive to rate changes than near-term options, although the exact maturity dependence is not universal. Assess the exposure using a specified rate change and the position's rho.

Let's implement and calculate the Greeks for our example option: