🎓 Prepared by students from Boğaziçi University

What is Option Pricing?

Option pricing is the practice of calculating the fair value of financial derivatives — call or put options on stocks. The value depends on the underlying stock price, time to expiration, volatility, strike price, and interest rates. The Black-Scholes model is the most famous formula.

Short answer

Option pricing uses formulas like Black-Scholes to determine the theoretical value of a call or put option based on the stock price, volatility, time remaining, strike price, and risk-free rate.

Call option value vs stock price at expiration

x: Stock price at expiry ($) · y: Call option profit ($)

Try it: interactive calculator

Call price (approx)

0currency

= 100*2.71828^(-0.05*1)*(0.5 - 0.01*1*(100-100)) - 100*2.71828^(-0.05*1)*0.5

Stock price S₀100 currencyStrike price K100 currencyInterest rate r5 %/yearTime T1 yearsVolatility σ20 %/year

Step-by-step worked examples

Stock price $100, strike $100, 1 year, volatility 20%, rate 5%. Estimate call value.

Using Black-Scholes (at-the-money):
d₁ ≈ 0.55, d₂ ≈ 0.35, N(d₁) ≈ 0.71, N(d₂) ≈ 0.64
C ≈ 100(0.71) − 100·e^(−0.05)·0.64 ≈ $10.45

Stock $100, call strike $90 (in-the-money), 6 months, vol 30%, rate 5%. Value?

ITM call is worth more; intrinsic value ≥ $10.
Black-Scholes with σ=30% (higher vol) ≈ $15–18.
Approx: $16.

Same stock & call but only 1 week left (T=0.02 years). How does value change?

Shorter time → less time value → option worth less.
With T=0.02: C ≈ $10 (mostly intrinsic, time value eroded).

Flashcards

Quick quiz

Q1.Stock $100, call strike $100. After 6 months, stock rises to $120. Intrinsic value of call?

Correct answer: C. Intrinsic = max(stock − strike, 0) = max($120 − $100, 0) = $20.

Q2.Higher volatility → option value?

Correct answer: C. Higher volatility = greater chance of large move → more valuable option.

Q3.Call option with 1 day left vs 1 month left (same strike/stock)?

Correct answer: B. 1-month has more time value; longer time = more chance of profitable move.

Q4.Black-Scholes model assumes…

Correct answer: B. B-S is an idealized model; real markets have costs and changing volatility.

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Common mistakes

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Confusing option price with stock price. — Correct: Option price (premium) is separate; it's the cost to buy the right to trade the stock.

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Thinking the option's strike price is its value. — Correct: Value = premium (B-S formula). Strike is just the exercise price.

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Ignoring volatility in pricing. — Correct: High volatility increases option value because more upside/downside is possible.

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Assuming put price = call price. — Correct: Put-call parity relates them: C − P = S − Ke^(−rT), but they're usually unequal.

FAQ

What is option pricing?

Determining the fair value (premium) of a call or put option using formulas like Black-Scholes, which account for stock price, volatility, time, strike, and rates.

Why does volatility increase option value?

Higher volatility = higher chance of large stock moves → more upside potential → buyers pay more.

What is intrinsic vs time value?

Intrinsic = profit if exercised now (e.g., stock $120 − strike $100 = $20). Time value = additional premium from expiration time remaining.

Is Black-Scholes perfect?

No — it assumes constant volatility, no transaction costs, and European exercise. Real markets are messier.