Options and Black–Scholes

Build calls, puts and spreads, price them with Black–Scholes, and watch the Greeks move.

The right to buy at the strike. Pays off if the price rises.

Stock price today$100

Strike$100

Days to expiry90

Implied volatility25%

How much the market expects the price to swing, per year.

Interest rate4.00%

Cost to open$5.43

Max profit at expiryUnlimited

Max loss at expiry−$5.43

Breakeven105.4

Solid: profit or loss on expiry day. Dashed: what the position would be worth today if the price jumped there.

Opening this long call costs $5.43. At expiry it breaks even at a stock price of $105.43. Each day that passes changes its value by about −$0.03, and a one point rise in volatility moves it by +$0.20.

The Greeks

Traders don't just price an option. They measure how its price reacts to each thing that can change.

Delta: 0.556; Value change per $1 move in the stock
Gamma: 0.0318; How fast delta itself changes
Theta: −$0.03; Value lost or gained per day
Vega: +$0.20; Per one point of volatility

Leg	Strike	Price	Delta
Buy call	$100	$5.43	0.56

Delta across stock prices

Where delta is steep, gamma is high: the position's exposure flips quickly. Shorten the days to expiry and watch it sharpen.

Check the formula by simulation

Black–Scholes says an option's fair price is its average payoff in a world where the stock grows at the interest rate, discounted back to today. So simulate 50,000 expiry prices, average the payoffs, and see if the two agree.

Black–Scholes, options only$5.43

Monte Carlo estimate$5.42± $0.07 at 95%

Put–call parity

Call − put = $5.43 − $4.45 = $0.98

Stock − discounted strike = $100.00 − $99.02 = $0.98

If these ever differed, a trader could lock in a riskless profit.

Estimate as paths are added

The dashed band is the 95% error range. It shrinks with the square root of the number of paths: 100 times more paths, 10 times more precision.

Try this

Pick a long call and drag days to expiry down to 1. Watch the dashed curve fold into the solid hockey stick.
Raise implied volatility on a straddle. The cost jumps, because a big move is now more likely in both directions.
Sell an iron condor, then set volatility to 80%. The credit grows, and so does the chance the price escapes your range.

The math

C = S\,N(d_1) - K e^{-rT} N(d_2), \quad d_{1,2} = \frac{\ln(S/K) + (r \pm \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}

$C$: price of a call option
$S$: stock price today
$K$: strike price
$T$: years to expiry
$r$: risk-free interest rate
$\sigma$: implied volatility
$N(\cdot)$: standard normal cumulative probability

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