Option valuation and Greeks

"Options are like the spicy condiment of finance: a little changes everything — and you better know how hot it is."

You already learned about option payoff profiles (so you know what a call and a put do at expiry) and the pricing logic behind forwards and futures (so you remember forward price = spot carried forward with financing and dividends). Now we level up: how to value options before expiry and how to measure their sensitivities — the Greeks. This is where intuition meets math and traders become part-thermostat, part-hedge barber.

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What is option valuation (and why we care)

Option valuation determines the fair price of an option today based on: the current underlying price, strike, time to expiry, interest rates, dividends, and volatility. Unlike simple payoff diagrams (which tell you what happens at expiry), valuation gives the time value component and lets you trade, hedge, and risk-manage positions before expiry.

Put simply: option price = intrinsic value + time value, and most of the mystery sits inside time value, which is largely driven by volatility.

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How does pricing actually work? Two big frameworks

1) Risk-neutral valuation (aka the intuition that saves lives)

Under risk-neutral pricing you compute the expected discounted payoff where the expected growth rate of the underlying is the risk-free rate (not the actual expected return). We used this idea with forwards/futures pricing: replace expected drift with r and discount. For options, you sum expected payoff across states and discount — or use models that embed this logic.

2) Black-Scholes-Merton (continuous-time classic)

For European options on non-dividend-paying stocks, Black-Scholes gives a closed-form price. Key assumptions: lognormal returns, constant volatility, continuous trading, no arbitrage. The formula (for a call) is:

C = S * N(d1) - K * e^{-rT} * N(d2)
where
d1 = [ln(S/K) + (r + 0.5*sigma^2)T] / (sigma * sqrt(T))
d2 = d1 - sigma * sqrt(T)

N( ) is the standard normal CDF. For puts, use put-call parity (we covered payoffs; recall put-call parity ties calls, puts, spot and forwards).

Note: If dividends exist, replace S with the present value of forward adjustments or use forward price F in pricing: C = e^{-qT}S N(d1) - K e^{-rT} N(d2) or use forward-based forms where F = S e^{(r-q)T}. This connects directly to what you learned about forward prices.

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The Greeks — your sensitivity toolkit

Greeks measure partial derivatives of option price with respect to inputs. Think of them as dials on a control panel.

• Delta (Δ) — sensitivity to underlying price. For a call, Δ in (0,1); for a put, Δ in (-1,0).

  • Intuition: roughly how much option price moves if stock moves $1.
  • Use: hedging. Owning a call with Δ = 0.6 means short 0.6 shares to delta-hedge.

• Gamma (Γ) — sensitivity of delta to underlying price (second derivative). Always positive for standard options.

  • Intuition: curvature. High Gamma = delta changes fast, you must rebalance hedges frequently.

• Vega — sensitivity to volatility (not a Greek letter but accepted). Positive: higher implied volatility raises option prices.

  • Intuition: options are volatility bets. Vega is largest for at-the-money, longer-dated options.

• Theta (Θ) — sensitivity to time decay (price change per small passage of time). Usually negative for long options.

  • Intuition: each day the option loses time value; Theta is the cost of optionality.

• Rho (ρ) — sensitivity to interest rates. Small for many equity options but relevant for long-dated options and other underlyings.

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Quick reference table

Greek	What it measures	Shape/Sign	Practical use
Delta Δ	dC/dS	call: (0,1), put: (-1,0)	Hedge ratio, directional exposure
Gamma Γ	d²C/dS²	positive	Hedge rebalancing frequency
Vega	dC/dσ	positive	Volatility exposure, pricing shifts
Theta Θ	dC/dt	usually negative	Time decay, carry cost
Rho ρ	dC/dr	sign depends	Rate sensitivity, longer horizons

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Examples & intuition (no killjoy math required)

1. Delta-hedging thought experiment:

Imagine you sell 100 calls with Δ = 0.5. You're short 50 effective shares. To be delta-neutral, buy 50 shares. Now if the stock jumps, your net delta increases (because calls' delta rises via Gamma), so you must re-hedge. That rebalancing cost is what Gamma exposes you to.

2. Vega and earnings:

Before earnings implied vol is high. Buying calls before earnings is expensive because vega hurts you if volatility falls after the event. You might sell volatility instead: sell premium and collect Theta, but be ready for big stock moves.

3. Link to equity valuation:

If you value stocks using discounted cash flows, volatility impacts option-derived claims on equity (e.g., corporate debt/equity option models). Higher volatility inflates option values; for firms with debt, equity is a call on assets.

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Common mistakes in practice

• Confusing implied volatility with historical volatility. Implied is the market's forward-looking price of risk embedded in option quotes.
• Treating Black-Scholes outputs as gospel. It's a model: check assumptions, and remember for American options or options on assets with jumps, use binomial/CFA/pricing engines.
• Ignoring transaction costs and discrete hedging. Gamma risk becomes expensive when you must trade frequently.
• Neglecting correlation and portfolio-level Greeks. Single-option Greeks are fine, but real hedges happen across positions.

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Quick heuristics traders love

• ATM, long-dated -> high Vega, low Theta per day relative to vega exposure.
• Deep ITM calls behave like the underlying (Delta near 1); deep OTM calls behave like lottery tickets (Delta ≈ 0; high relative Gamma if near ATM).
• Short premium = collect Theta, pay Vega and risk tail events.

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Closing: key takeaways (so you actually remember this)

• Option valuation extends payoff thinking by pricing time value using risk-neutral logic; Black-Scholes is the archetypal closed-form example.
• Greeks are your dashboard: Delta (direction), Gamma (curvature), Vega (volatility), Theta (time decay), Rho (rates).
• Hedging is about controlling Greeks, not gambling on exact forecasts. Delta-hedging and Gamma management determine real-world performance.

"If options are the spice, Greeks are your recipe card — follow them and you might make a gourmet dish. Ignore them and you’ll set off the smoke alarm."

Go try this: pick a quoted option, compute its Delta and Vega (many brokers show these), and ask: what would I do if volatility falls 5% or the stock jumps 10%? If your answer involves more than 'pray', you're learning.

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Version notes: This builds on payoff profiles and forward/futures pricing and ties to equity valuation where option thinking explains claims on firm value and volatility effects.