New on FinoLib · Volume II: Capital Markets
I have added a new article to FinoLib, my finance education website: Black-Scholes Model and the Option Greeks Explained, part of Volume II: Capital Markets. This page gives you the core of the topic in expert but plain language; the full article goes much deeper.
4.3 The Option Greeks — Measuring Sensitivity
The idea in plain words
The Greeks are simple answers to “what if?” questions about an option’s price. Delta asks what happens if the stock price moves. Gamma asks how quickly delta itself changes. Theta asks what happens as a day passes. Vega asks what happens if volatility changes, and rho asks what happens if interest rates change. A trader who hedges delta away is really betting on how much the market actually moves compared with what the option price assumed, with gamma and theta pulling against each other every day.
Key points
■The base call has delta 0.618, gamma 0.0191, theta −$0.016 a day and vega $0.38 per volatility point.
■Delta alone misprices big moves: for +$15 it predicts $9.27 against an actual $11.11.
■A five-day hedged short of 1,000 calls lost $352 when the stock swung at about 56% volatility against 20% sold.
A closer look
Because an option’s price depends on several inputs simultaneously — the underlying price, time, volatility, and interest rates — traders use a family of sensitivity measures, nicknamed the Greeks, to isolate how the option’s price responds to a change in each input individually, holding the others constant.
Dealers who sell options delta hedge: they hold delta shares per option sold and adjust as delta moves, leaving a position that profits or loses on how much the stock moves, not which way.
Decision rule
Compare your expected daily move with the break-even σ × S × √(1/365) ($1.05 for a $100 stock at 20%): above it, hold long gamma (bought options, delta hedged); below it, short gamma earns theta. Rehedge when delta leaves a band set by your trading costs, not on a fixed clock; more frequent hedging lowers variance but pays more spread. Ignore the comparison around earnings and other gap events.
The costliest mistake
Hedging delta and calling the book riskless. The dealer short 1,000 calls against 618 shares is delta-neutral, but a $15 overnight gap lifts each call $11.11 while the shares gain 0.618 × 15 = $9.27: a loss of about $1,837, nearly a fifth of the $9,925 premium, in one move; a desk short a million such options loses about $1.8 million. Limit gamma, not just delta, and cut short options before gap events.
4.4 Black-Scholes — The Intuition Behind the Formula
The idea in plain words
The Black-Scholes formula prices an option by asking what it would cost to build the same payoff yourself, using a mix of the stock and some borrowed money. Because that mix copies the option exactly, what you expect the stock to return drops out of the price. For a $100 stock with a $100 strike, one year to go, a 4% interest rate and 20% volatility, the call is worth $9.93 and the put $6.00. A binomial tree, which takes small up-or-down steps, converges on the same answer.
Key points
■d₁ = 0.30, d₂ = 0.10, N(d₁) = 0.6179, N(d₂) = 0.5398 give C = 61.79 − 51.87 = $9.93.
■A one-step tree gives $11.73 by replication and by risk-neutral pricing; 1,000 steps give $9.92.
■Since the 1987 crash, out-of-the-money index puts trade at higher implied volatility: the skew.
A closer look
The Black-Scholes model, published in 1973, was the breakthrough that gave options a defensible, mathematically derived fair price rather than a guess. Its full derivation involves stochastic calculus, but its core intuition is accessible without the mathematics: the model imagines that a trader can construct a replicating portfolio — a precise combination of the underlying asset and risk-free borrowing or lending — whose payoff exactly matches the option’s payoff in every possible future scenario. If such a portfolio can be built and continuously rebalanced, then by the law of one price, the option must be worth exactly what that replicating portfolio costs to build today.
Decision rule
Price a European option with Black-Scholes at its own strike’s implied volatility, never one historical volatility for all strikes; use a binomial tree when early exercise matters (American puts, calls before a dividend). Judge cheap or rich by implied against expected volatility, not dollar price. Across scheduled events, treat model prices as rough guides.
The costliest mistake
Pricing the wings at the at-the-money volatility. Selling three-month 90-strike index puts at the flat-20% price of $0.58 when the skew values them at $1.26 gives away 1.26 − 0.58 = $0.68 per share, $68 per contract, over half the fair value, in the crash tail where losses are largest. Read each strike’s volatility off the market surface.
What the full article covers
In the full article I take you through these sections:
■4.3 The Option Greeks — Measuring Sensitivity
■4.4 Black-Scholes — The Intuition Behind the Formula
It includes worked examples, a look under the hood at the mechanics, a clear decision rule, the one mistake that costs people the most, a recap, a reveal-on-click quiz and the official sources behind every figure.
Volume II: Capital Markets
Read the full article on FinoLib
The complete article, Black-Scholes Model and the Option Greeks Explained, with every worked example, diagram, quiz and source, is free to read in Volume II, Capital Markets on finolib.com.
Explore the Finance Education Series: Vol. II — Capital Markets · Vol. I — Global Economy · Vol. III — Finance Encyclopedia · Vol. IV — Finance Careers · Vol. V — Personal Finance