Generalities: Generalities


Market Skews: Market Skews Dominating fact since 1987 crash: strong negative skew on Equity Markets Not a general phenomenon Gold: FX: We focus on Equity Markets


Skews: Skews Volatility Skew: slope of implied volatility as a function of Strike Link with Skewness (asymmetry) of the Risk Neutral density function ?


Why Volatility Skews?: Why Volatility Skews? Market prices governed by a) Anticipated dynamics (future behavior of volatility or jumps) b) Supply and Demand To “ arbitrage” European options, estimate a) to capture risk premium b) To “arbitrage” (or correctly price) exotics, find Risk Neutral dynamics calibrated to the market


Modeling Uncertainty: Modeling Uncertainty Main ingredients for spot modeling Many small shocks: Brownian Motion (continuous prices) A few big shocks: Poisson process (jumps)


Slide6: To obtain downward sloping implied volatilities a) Negative link between prices and volatility Deterministic dependency (Local Volatility Model) Or negative correlation (Stochastic volatility Model) b) Downward jumps 2 mechanisms to produce Skews (1)


2 mechanisms to produce Skews (2): 2 mechanisms to produce Skews (2) a) Negative link between prices and volatility b) Downward jumps


Model Requirements: Model Requirements Has to fit static/current data: Spot Price Interest Rate Structure Implied Volatility Surface Should fit dynamics of: Spot Price (Realistic Dynamics) Volatility surface when prices move Interest Rates (possibly) Has to be Understandable In line with the actual hedge Easy to implement


Beyond initial vol surface fitting: Beyond initial vol surface fitting Need to have proper dynamics of implied volatility Future skews determine the price of Barriers and OTM Cliquets Moves of the ATM implied vol determine the D of European options Calibrating to the current vol surface do not impose these dynamics


Barrier options as Skew trades: Barrier options as Skew trades In Black-Scholes, a Call option of strike K extinguished at L can be statically replicated by a Risk Reversal Value of Risk Reversal at L is 0 for any level of (flat) vol Pb: In the real world, value of Risk Reversal at L depends on the Skew


A Brief History of Volatility: A Brief History of Volatility


A Brief History of Volatility (1): A Brief History of Volatility (1) : Bachelier 1900 : Black-Scholes 1973 : Merton 1973 : Merton 1976 : Hull&White 1987


A Brief History of Volatility (2): A Brief History of Volatility (2) Dupire 1992, arbitrage model which fits term structure of volatility given by log contracts. Dupire 1993, minimal model to fit current volatility surface


A Brief History of Volatility (3): A Brief History of Volatility (3) Heston 1993, semi-analytical formulae. Dupire 1996 (UTV), Derman 1997, stochastic volatility model which fits current volatility surface HJM treatment.


A Brief History of Volatility (4): A Brief History of Volatility (4) Bates 1996, Heston + Jumps: Local volatility + stochastic volatility: Markov specification of UTV Reech Capital Model: f is quadratic SABR: f is a power function


A Brief History of Volatility (5): A Brief History of Volatility (5) Lévy Processes Stochastic clock: VG (Variance Gamma) Model: BM taken at random time g(t) CGMY model: same, with integrated square root process Jumps in volatility (Duffie, Pan & Singleton) Path dependent volatility Implied volatility modelling Incorporate stochastic interest rates n dimensional dynamics of s n assets stochastic correlation


Local Volatility Model: Local Volatility Model


From Simple to Complex: From Simple to Complex How to extend Black-Scholes to make it compatible with market option prices? Exotics are hedged with Europeans. A model for pricing complex options has to price simple options correctly.


Black-Scholes assumption: Black-Scholes assumption BS assumes constant volatility => same implied vols for all options.


Black-Scholes assumption: Black-Scholes assumption In practice, highly varying.


Modeling Problems: Modeling Problems Problem: one model per option. for C1 (strike 130) s = 10% for C2 (strike 80) σ = 20%


One Single Model: One Single Model We know that a model with s(S,t) would generate smiles. Can we find s(S,t) which fits market smiles? Are there several solutions? ANSWER: One and only one way to do it.


Interest rate analogy: Interest rate analogy From the current Yield Curve, one can compute an Instantaneous Forward Rate. Would be realized in a world of certainty, Are not realized in real world, Have to be taken into account for pricing.


Volatility: Volatility Local (Instantaneous Forward) Vols read Dream: from Implied Vols How to make it real?


Discretization: Discretization Two approaches: to build a tree that matches European options, to seek the continuous time process that matches European options and discretize it.


Tree Geometry: Tree Geometry Binomial Trinomial Example: To discretize σ(S,t) TRINOMIAL is more adapted 20% 5% 20% 5%


Tango Tree: Tango Tree Rules to compute connections price correctly Arrow-Debreu associated with nodes respect local risk-neutral drift Example


Continuous Time Approach: Continuous Time Approach Call Prices Exotics Distributions Diffusion ?


Distributions - Diffusion: Distributions Diffusion Distributions - Diffusion


Distributions - Diffusion: Distributions - Diffusion Two different diffusions may generate the same distributions


The Risk-Neutral Solution: The Risk-Neutral Solution But if drift imposed (by risk-neutrality), uniqueness of the solution


Continuous Time Analysis: Continuous Time Analysis


Implication : risk management: Implication : risk management Implied volatility Black box Price Perturbation Sensitivity


Forward Equations (1): Forward Equations (1) BWD Equation: price of one option for different FWD Equation: price of all options for current Advantage of FWD equation: If local volatilities known, fast computation of implied volatility surface, If current implied volatility surface known, extraction of local volatilities, Understanding of forward volatilities and how to lock them.


Forward Equations (2): Forward Equations (2) Several ways to obtain them: Fokker-Planck equation: Integrate twice Kolmogorov Forward Equation Tanaka formula: Expectation of local time Replication Replication portfolio gives a much more financial insight


Fokker-Planck: Fokker-Planck If Fokker-Planck Equation: Where is the Risk Neutral density. As Integrating twice w.r.t. x:


FWD Equation: dS/S = (S,t) dW: FWD Equation: dS/S = (S,t) dW Equating prices at t0:


FWD Equation: dS/S = r dt + s(S,t) dW: FWD Equation: dS/S = r dt + s(S,t) dW


FWD Equation: dS/S = (r-d) dt + s(S,t) dW: TV + Interests on K – Dividends on S FWD Equation: dS/S = (r-d) dt + s(S,t) dW Equating prices at t0: ST ST