FINM7405 Chap.5 Options Pricing and Greeks
Options Pricing and Greeks
Options create nonlinear rights whose terminal payoff must be distinguished from current price and investor profit. This chapter begins with call and put payoffs, moneyness, bounds and put-call parity. It then interprets Black-Scholes as the cost of a dynamic replicating portfolio and uses delta, gamma, theta, vega and rho to describe local risk.
Binomial trees price backward from terminal payoffs, while implied volatility reverses the pricing equation. Strategy payoffs and option VaR show why a delta-neutral book can still carry curvature, volatility and jump exposure.
Parity and bounds come before the model because they follow from cash flows and no-arbitrage.
If a call-plus-bond package and a put-plus-stock package deliver the same terminal value, a price difference supports an offsetting buy-sell strategy. Black-Scholes then adds a continuous replication framework under idealised trading and volatility assumptions. Its risk-neutral weights are pricing devices rather than real-world forecasts.
The Greeks organise local risk.
Delta gives the immediate stock hedge, gamma explains how that hedge changes, theta records time decay, vega measures implied-volatility sensitivity and rho captures rates. A binomial tree makes the replication and early-exercise comparison visible at discrete nodes. Full revaluation becomes important when payoff curvature, barriers, changing volatility or jumps make a linear delta-normal risk approximation unreliable.
What this chapter covers
- 01
Call and put payoff geometry
- 02
Put-call parity and arbitrage bounds
- 03
Black-Scholes and risk-neutral pricing
- 04
Delta, gamma, theta, vega and rho
- 05
Binomial trees, strategies and option VaR
Worked example · free
A Black-Scholes call
The marks shown separate the option-pricing steps and are not a published university marking scheme.
- +1Calculate d1 = 0.35.
- +1Calculate d2 = 0.15.
- +1Use N(d1) approximately 0.637 and N(d2) approximately 0.560.
- +1Insert the discounted strike into the call formula.
- +1Obtain a value of approximately $10.45 and state the assumptions.
Key terms
- Put-Call Parity
- A no-arbitrage equality linking European calls, puts, stock and the present value of the strike.
- Implied Volatility
- The volatility input that makes a pricing model reproduce an observed market option price.
- Delta
- The first-order change in option value for a small change in the underlying price.
- Gamma
- The change in option delta for a change in the underlying price.
Options Pricing and Greeks FAQ
What is the difference between option payoff and profit?
Payoff is the contract value at maturity, such as max(S minus K, 0) for a call. Profit also deducts the premium and financing. An in-the-money option can still produce a trading loss if the payoff does not cover the premium.
Why is a delta-neutral position still risky?
Delta neutrality is local. Gamma changes delta after spot moves, vega changes value when implied volatility moves, theta changes value through time and jumps can occur before the hedge is rebalanced.
When is a binomial tree preferable?
A tree is useful when discrete exercise decisions, dividends or contract features make a closed-form result less suitable. It works backward from terminal payoffs and can compare immediate exercise with continuation at each node.
Exam move
Draw terminal payoff before calculating and separate payoff, premium and profit. Prove put-call parity with terminal cash flows in both stock-price states rather than recalling only the symbols. For Black-Scholes, label every input, calculate d1 and d2 in order, and explain that risk-neutral probabilities are pricing weights.
Make a Greek table that states the driver, local direction and hedge implication for delta, gamma, theta, vega and rho. Practise updating delta with gamma after a spot move and describe why discrete rebalancing leaves jump and transaction-cost error. Work one binomial tree backward from payoff, checking early exercise when the contract is American.
Compare Black-Scholes, trees and Monte Carlo by contract feature instead of ranking them universally. Close every option-risk answer with the residual volatility-surface, jump, liquidity or execution exposure that a current delta hedge does not remove. Use a three-panel routine for every contract: payoff diagram, no-arbitrage relationship and local risk measures.
The diagram fixes long or short direction and separates strike from breakeven. The parity or bound check identifies prices that cannot coexist, while the Greeks explain how a valid price changes before expiry. For binomial work, label up and down stock nodes, terminal payoffs, risk-neutral probability and discounting separately; at American nodes compare exercise value with continuation value explicitly.
For Black-Scholes work, estimate whether the call price should rise or fall when each input changes before using the formula. That directional forecast catches swapped signs and implausible outputs. Build a Greek exposure table at position level by multiplying per-option sensitivities by contract quantity and multiplier.
Then shock spot and implied volatility together, update delta using gamma, and explain why the hedge trade itself can incur slippage. Contrast long-gamma and short-gamma rebalancing to see who buys after rises and sells after falls. Finally, distinguish model price from executable price by discussing bid-ask spreads, volatility-surface dynamics, discrete hedging and jumps.
The strongest conclusion identifies which approximation is local and which market feature breaks it.
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