The Greeks and Volatility: Sensitivities Every Quant Must Know

Why the Greeks Matter

Knowing the price of an option is step one. Step two — and this is where the daily work lives — is understanding how that price changes when market conditions shift. A trader who owns a portfolio of options needs to know: "If the stock moves £1, how much do I make or lose? If volatility increases by 1%, what happens? What does the passage of time cost me?"

The Greeks answer these questions. Each Greek is a partial derivative of the option price with respect to one input. Together, they give you a complete map of an option's risk profile.

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Delta (( \Delta ))

[ \Delta = \frac{\partial C}{\partial S} ]

Delta measures how much the option price changes per £1 move in the underlying.

• Call delta: between 0 and 1. An at-the-money call has ( \Delta \approx 0.5 )
• Put delta: between -1 and 0. An at-the-money put has ( \Delta \approx -0.5 )

Interpretation: a delta of 0.6 means the option behaves roughly like holding 0.6 shares. This is why delta is the most important Greek for hedging: to delta-hedge an option, hold ( -\Delta ) shares of the underlying.

Fun fact: delta also approximates the probability that the option will expire in the money (under the risk-neutral measure). A delta of 0.7 means there is roughly a 70% chance of the option paying out.

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Gamma (( \Gamma ))

[ \Gamma = \frac{\partial^2 C}{\partial S^2} = \frac{\partial \Delta}{\partial S} ]

Gamma is the rate of change of delta — a second derivative. It tells you how quickly your hedge needs adjusting.

• High gamma near the strike (especially close to expiry) means delta changes rapidly
• Gamma is always positive for long options
• P&L from gamma: ( \frac{1}{2} \Gamma (\Delta S)^2 ) — gamma makes you money when the stock moves a lot (in either direction)

Gamma is the convexity of options — analogous to convexity in bonds. It is the reward for buying options and the cost of selling them.

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Theta (( \Theta ))

[ \Theta = \frac{\partial C}{\partial t} ]

Theta measures time decay — how much value the option loses each day just by existing.

• Theta is negative for long options (they lose value over time)
• Decay accelerates as expiry approaches (theta is proportional to ( 1/\sqrt{T} ))
• At-the-money options have the highest theta

There is a beautiful relationship between gamma and theta. For a delta-hedged portfolio:

[ \Theta + \frac{1}{2} \sigma^2 S^2 \Gamma = rC ]

High gamma (good) comes at the cost of high theta (bad). No free lunch.

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Vega (( \nu ))

[ \nu = \frac{\partial C}{\partial \sigma} ]

Vega measures sensitivity to volatility changes. (Technically "vega" is not a Greek letter — it is a finance invention, which is either charming or annoying depending on your temperament.)

• Vega is always positive for long options (higher volatility = higher option value)
• Longer-dated options have more vega exposure
• Vega is the key Greek for volatility trading

When traders say they are "long vol," they mean they have positive vega — they profit when implied volatility increases.

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Rho (( \rho ))

[ \rho = \frac{\partial C}{\partial r} ]

Rho measures sensitivity to the risk-free interest rate. It is usually the least important Greek for equity options (rate changes are small relative to stock moves and volatility changes). But for long-dated options and interest rate derivatives, rho matters a great deal.

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Greeks Summary Table

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Volatility: The Fifth Input

Of the five Black-Scholes inputs, volatility is the only one you cannot directly observe. This makes it the most interesting — and the most modelled.

Historical vs Implied

Historical (realised) volatility: computed from past returns. Backward-looking.

[ \sigma_{\text{hist}} = \sqrt{\frac{252}{n-1} \sum_{i=1}^{n} (r_i - \bar{r})^2} ]

(The 252 annualises daily volatility — there are roughly 252 trading days per year.)

Implied volatility: extracted from option prices. Forward-looking — it represents the market's expectation.

The difference between the two is often tradeable. If you believe realised volatility will be higher than implied, buy options. Lower? Sell them.

The Volatility Smile

In the real world, implied volatility varies by strike price. For equity options, deep out-of-the-money puts typically have higher IV than at-the-money options — the volatility smile (or skew).

This is because the market assigns higher probability to extreme downside moves than the lognormal distribution suggests. The 1987 crash taught the market this lesson permanently.

Volatility Surface

IV also varies by expiry, creating a two-dimensional volatility surface. Building, calibrating, and interpolating this surface is a core quant task. Models like SABR and local volatility are designed to produce a smooth, arbitrage-free surface.

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Volatility Clustering

Financial returns exhibit volatility clustering: big moves tend to follow big moves. GARCH models (Generalised Autoregressive Conditional Heteroscedasticity — a truly magnificent name) capture this by modelling volatility as a time-varying process.

This matters for risk management: after a volatile day, tomorrow's risk is higher than usual.

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Greeks in Python

from scipy.stats import norm
import numpy as np

def bs_greeks(S, K, T, r, sigma):
    d1 = (np.log(S/K) + (r + sigma**2/2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    
    delta = norm.cdf(d1)
    gamma = norm.pdf(d1) / (S * sigma * np.sqrt(T))
    theta = -(S * norm.pdf(d1) * sigma) / (2*np.sqrt(T)) - r*K*np.exp(-r*T)*norm.cdf(d2)
    vega = S * norm.pdf(d1) * np.sqrt(T)
    
    return {"delta": delta, "gamma": gamma, "theta": theta/365, "vega": vega/100}

greeks = bs_greeks(S=100, K=100, T=0.25, r=0.05, sigma=0.2)
for name, val in greeks.items():
    print(f"{name}: {val:.4f}")

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Building These Skills

The Greeks and volatility connect calculus, probability, and market intuition in one package. Quantt covers all of this with interactive modules — compute Greeks, visualise payoff profiles, explore the volatility surface, and write the Python code that makes it all work.

For supplementary material, the CBOE's options education is practical and market-focused, and Natenberg's "Option Volatility and Pricing" is the classic practitioner reference.