Unit 7 of 14
Volatility and path dependent derivatives
- 8 lessons
- Videos 49 to 56
One constant volatility rarely fits a set of option prices, and many contracts depend on a price path. This unit covers volatility surfaces, further models and path-dependent and multiasset payoffs.
Implied volatility is unique inside the arbitrage bounds because a European price strictly increases in volatility, and static arbitrage restricts the surface in strike and maturity. Dupire's equation recovers local volatility from a smooth call surface, but differentiating noisy prices is unstable. The Heston model adds a variance process with an affine characteristic function, and jumps generally make a market incomplete. Barrier options use hitting times and reflection, Asian options an extra state variable and exchange options a change of numeraire. Rough volatility is beyond the course.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Implied volatility and arbitrage-free surfaces
Implied volatility and inverse pricing
How implied volatility inverts the Black-Scholes price, why positive vega and the arbitrage bounds give exactly one root, how inverse sensitivity limits accuracy, and how bracketed inversion finds the root.
Video 49
Static arbitrage and volatility surfaces
How call spreads and butterflies give strike monotonicity and convexity, why calendar consistency holds for carry-normalized prices, and how to read and interpolate a volatility surface without static arbitrage.
Video 50
Local volatility, stochastic volatility and jumps
Dupire local volatility
How a smooth call price surface determines a local volatility through the Dupire equation, how local variance follows from maturity and strike derivatives, and why recovering it from noisy quotes is ill-posed.
Video 51
Heston stochastic volatility
How the Heston model lets variance follow a mean-reverting square-root diffusion, what the leverage correlation and the Feller condition mean, and how the characteristic function reduces to Riccati equations.
Video 52
Jump diffusion and incomplete hedging
How Poisson jumps enter a stock model under a pricing measure, why the drift between jumps is compensated by the expected jump, how jumps make the pricing generator nonlocal, and what jump residual a delta hedge leaves.
Video 53
Path-dependent and multiasset options
Barrier options and first passage
How barrier options depend on a hitting time, how the reflection principle gives first passage probabilities, why matching knock-in and knock-out prices add to the vanilla, and how bridge crossing probabilities handle monitoring.
Video 54
Asian options and running averages
How an Asian option pays on an arithmetic or geometric average of fixings, why its path dependence calls for state augmentation, and how the exact geometric-average law bounds the arithmetic price.
Video 55
Multiasset derivatives and exchange options
How correlated diffusions give the effective volatility of a price ratio, how a change of numeraire prices the exchange option exactly, and why basket options rely on lognormal moment matching.
Video 56