This is the boss fight. Five lessons rebuilt an option’s price from first principles — why a payoff is not a price, how replication and no-arbitrage pin the value, the binomial tree and its risk-neutral shortcut, the Black–Scholes formula and the Greeks that move it, and the implied volatility that turns the whole machine inside out to read the market’s fear. Now we find out whether the pricing engine actually runs in your head. No formula sheet, no hints, and no take-backs: every answer locks the instant you submit it, and the wrong options are the exact misconceptions that wreck real pricing intuition. Read every choice twice.
How this exam works
This is a graded exam. Questions arrive one at a time. Once you submit an answer it is final — there is no going back, no second try, and a wrong answer simply fails that question. Your score stays hidden until the very end, where you need 70% to pass. Slow down and read every option before you commit.
Why does naive 'expected value' pricing — multiply each payoff by its real-world probability and discount — fail to give an option's correct price?
Select an answer to continue.
You made it to the end of the pricing machine. Whatever the score reads, the chain you just stress-tested — no-arbitrage and replication, the binomial tree and its risk-neutral shortcut, Black–Scholes, the Greeks, and the implied-vol smile — is the literacy every quant and options trader leans on. Here is the whole topic in one glance.
Big picture
The Options-Pricing Machine
- Options Pricing
- No-arbitrage & replication
- Payoff (at expiry) is not the price (today)
- Naive expected value fails: unknown odds + risk premium
- Replicate with Δ shares + cash; Δ = (Cu−Cd)/(Su−Sd)
- Real probabilities cancel → risk-neutral valuation
- Binomial model
- q = (e^(rT)−d)/(u−d); V = e^(−rT)[q·Vu+(1−q)·Vd]
- q is a pricing device, not a forecast
- No-arbitrage bound: d < e^(rT) < u
- CRR: u = e^(σ√Δt), d = 1/u; converges to Black–Scholes
- American = max(continuation, exercise); early exercise mainly puts
- Black–Scholes
- C = S0·N(d1) − K·e^(−rT)·N(d2)
- d1, d2 = d1 − σ√T; N(d2) = risk-neutral prob ITM
- N(d1) = delta; price = PV(get stock) − PV(pay strike)
- Assumes GBM, constant vol & rate, no dividends, frictionless, European
- The Greeks
- Delta = N(d1); gamma peaks ATM
- Theta negative (decay), vega positive for long options
- Rho positive for calls, negative for puts; signs flip when short
- Long options = long gamma / short theta; delta-hedging
- Implied vol & the smile
- Invert BS numerically (vega/Newton) → market vol forecast
- Implied vs realized → variance risk premium
- Constant-vol fails → smile; equity SKEW vs FX SMILE
- Volatility surface / term structure; VIX = 30-day fear gauge
- No-arbitrage & replication