In plain English
A derivative is a side bet whose payoff depends on something else — a price, a rate, an index. Forwards lock a price in; options buy the right to change your mind.
The advanced view
Options are priced by replication: a dynamic mix of the underlying and cash reproduces the payoff, so the option must cost what the replicating portfolio costs. Risk-neutral valuation is the same statement re-expressed — discount expected payoffs under a probability measure where every asset drifts at the risk-free rate. The Greeks are the sensitivities of that replication.
Derivatives derive value from an underlying. Four building blocks: forwards (bilateral contracts to trade at a future date), futures (standardised, exchange-traded, daily settled), options (the right but not the obligation) and swaps (exchanging cash-flow streams, most often fixed for floating).
Option pricing rests on no-arbitrage: two portfolios with identical payoffs in every state must have the same price. The binomial model builds a tree of up/down moves and prices by replication, working backwards to today. Black–Scholes is the continuous-time limit, assuming geometric Brownian motion and constant volatility.
Black–Scholes call
C = S×N(d1) − K×e^(−rT)×N(d2) d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T) d2 = d1 − σ√T Put–call parity: C − P = S − K×e^(−rT)
The Greeks measure sensitivities: delta to spot, gamma to delta, theta to time decay, vega to volatility, rho to rates. Implied volatility is the volatility that makes model price equal market price — forward-looking, and the number traders actually quote.
Essential vocabulary
- Moneyness
- Relationship of strike to spot: ITM, ATM, OTM. Determines intrinsic value.
- Risk-neutral pricing
- Price as if investors are risk-neutral and discount at the risk-free rate. Works because replication does not depend on preferences.
- Volatility smile/skew
- Implied vol varies across strikes; OTM puts price higher, reflecting crash risk. A documented failure of constant-vol Black–Scholes.
- Hedging
- Offsetting positions to reduce risk. Delta hedging neutralises direction; gamma and vega risk remain.
Intuition
A derivative is a contract about a payoff, and every payoff can be drawn as a hockey stick. Once drawn, pricing follows from replication: if you can build the same payoff with cash and the underlying, arbitrage forces the two prices together. Put-call parity is that argument in one line.
Common pitfalls
- ×Ignoring the premium when quoting a break-even.
- ×Treating implied volatility as a forecast rather than a price.
- ×Forgetting that a hedge changes the distribution of outcomes, not the expected value, before costs.
Worked example — call break-even and parity
Step 1 of 4
- 1Strike 100, premium 6, spot 98
Why it works
No-arbitrage is again the engine. Put-call parity, C − P = S − Ke^(−rT), holds because the two sides deliver identical payoffs at expiry in every state of the world; if prices differ you can lock a riskless profit today. That single identity lets you back out a missing option price, an implied forward, or an implied dividend.
How it is used — check an option quote with parity
Step 1 of 4
- 1S = 100, K = 100, r = 5%, T = 1, call =
Deeper
Deeper: put-call parity and what the Greeks actually tell you
Put-call parity is the single most useful derivatives identity: C − P = S − K·e^(−rT). It lets you price a put from a call, synthesise a forward, and spot mispricing without any model. It holds by arbitrage, independent of Black–Scholes.
The Greeks are sensitivities, not predictions. Delta is the hedge ratio (∂V/∂S), gamma the curvature of delta, vega sensitivity to volatility, theta the daily bleed from time. A long option is long gamma and long vega but short theta: you pay time value for the right to be convex.
Must know cold
- ✓Put-call parity: C − P = S − K·e^(−rT).
- ✓A forward has linear payoff; an option has asymmetric payoff and costs a premium.
- ✓Option value rises with volatility and with time to expiry.
- ✓Delta hedging removes direction, not volatility risk.
Exercises
Try each one on paper before revealing the worked solution.
Exercise 1
S = 100, K = 100, r = 4%, T = 1, call = 10. What is the put worth?
Exercise 2
An airline hedges fuel with a collar: buys a call at 90, sells a put at 70, zero net premium. What has it done?