Notes On Stochastic Rates, Dividends, Volatilities in an Equity Framework

Abstract

Part 1 - Black-Scholes formulas are consistent with a diffusion in the respective forward neutral probability.Part 2 - Mixing Forward Neutral Probabilities for different maturities, we construct a probability more natural for non european payoffs, different from the Risk Neutral measure, that we call the Pricing Probability.Part 3 - Writing the Radon-Nikodym derivative for changing measure to the Pricing Probability, and the modified HJM relationship.Part 4 - We use a simple Ho & Lee model to get a sense of how the path-dependent hybrid terms impact the Stock diffusions. Due to hybrid effects, BS vols term structure is usually increasing.Part 5 - The Pricing Probability seems to be the natural Probability to be used to price Equity-Linked Structures with any kind of Hybrid Risks - ie all non Strictly European Structures. The alternative - working in the Risk Neutral Probability - would necessitate to recalibrate all volatility, correlation parameters from the Black / Black Scholes world (Forward Neutral Probabilities) to the Risk Neutral world.Part 6 - Some examples of Interest Rate models.Part 7 - Stochastic Dividends diffusions. Ho & Lee example.Part 8 - Cross-Currency framework. Ho & Lee example.Part 9 - Risky Curve Diffusions in the Survival Neutral Probabilities.Analogy with the Currency Quanto World.A quick look at Counterparty Risk and CVA adjustments.Part 10 - Multi-Underlying Framework. Ho & Lee example. Due to hybrid effects, BS correlations term structure is usually increasing.Part 11 - Stochastic Volatility Framework.A statistical bias between vol of realized vols and vol of implied vols.Variance Swaps on Spots, Variances Swaps on Forwards, European Options.Part 12 - Natural parametrizations of volsurfaces coherent with vol dynamics and risks, based on vol, volvol and correl spot/vol breakevens.Historical backtests of implied vs realized breakevens.Generalization with 6 parameters per Maturity.Part 13 - Correlation Skew and Stochastic Volatility.We show that we can calibrate a basket skew independently of individual skews without introducing a spot-dependent correlation: this is typically the case when individual vols move more with the basket spot than with individual spots.