THAT COSTS WHAT! PROBABILISTIC LEARNING FOR VOLATILITY & OPTIONS

Transcription

1 THAT COSTS WHAT! PROBABILISTIC LEARNING FOR VOLATILITY & OPTIONS MARTIN TEGNÉR (JOINT WITH STEPHEN ROBERTS) 6 TH OXFORD-MAN WORKSHOP, 11 JUNE 2018

2 VOLATILITY & OPTIONS

3 S&P 500 index S&P 500 [USD] today $

4 S&P 500 index S&P 500 [USD] Lehman 08 today $ Volatility Volatility: measure of variability of returns

5 S&P 500 index S&P 500 return Lehman 08 today Volatility Volatility: measure of variability of returns

6 Historical vs. forward-looking volatility: Stock vs. option prices!

7 S&P 500 index S&P 500 [USD] today $ strike: $1550 maturity: 2 years Call option: The right to buy a stock at future maturity date for strike-price agreed today (sell put)

8 S&P 500 index S&P 500 [USD] today payoff: $439 strike: $1550 payoff: $0 maturity: 2 years At maturity: payoff = (price strike) +

9 Black-Scholes option pricing Stock price under real-world measure P ds t = µs t dt + S t dw t Volatility parameter is a constant

10 Black-Scholes option pricing Stock price under real-world measure P ds t = µs t dt + S t dw t Volatility parameter is a constant Invariant under change to pricing measure Q~P ds t = rs t dt + S t d W t C t (T,K)=E Q e rt (S T K) + S t = s

11 Black-Scholes option pricing Under Black-Scholes, historical volatility is the same as option-implied, forward-looking volatility a constant parameter

12 EUROPEAN OPTION MARKET

13 S&P 500 call options S&P 500 [USD] quoted strike-maturity points ( )

14 S&P 500 call options S&P 500 bid/ask [USD] strike [USD] maturity [years] Bid-ask prices

15 S&P 500 call options implied volatility strike [USD] maturity [years] Implied volatility: market s estimate of future volatility

16 MODEL Historical volatility S&P 500 [USD] Lehman 08 $ today Volatility European implied volatility options strike [USD] 2000 Implied volatility Fwd./market bet maturity [years] PRICING/HEDGING/RISK Derivatives/exotics OTC market

17 LOCAL VOLATILITY MODEL

18 Local volatility model Stock price, Q-dynamics ds t = rs t dt + (t, S t )S t dw t, t 2 [0,T? ] Local volatility function our modelling target :[0,T? ] R +! R

19 Local volatility model local vol PDE! call price strike maturity strike maturity Risk-neutral pricing: function-to-function mapping (, ) 7! C(, )

20 Why local volatility? A model can reproduce a set of option prices if within range attainable by model For local volatility, any set attainable as long as consistent in the sense of no static arbitrage Uniqueness in the limit of infinite data In practice, need calibration approach to fit model to observed data

21 Why local volatility? PDE!

22 PROBABILISTIC APPROACH

23 Bayesian framework We take a Gaussian process to define a functional prior distribution over local volatility Likelihood: data is noisy observations of true fair price Infer posterior distribution over local volatility from observed data

24 Inference Calibration ~ sampling from posterior Provides best point-estimate(s) in terms of model-tomarket error and notion of uncertainty in estimate Rich probabilistic representation of calibrated model

25 Markov chain Monte Carlo

26 Posterior local volatility local volatility strike [USD] maturity [years] local volatility maturity: 0.13 year ±2SD MAP data loc strike

27 Posterior local volatility local volatility strike [USD] maturity [years] local volatility maturity: 0.99 year ±2SD MAP data loc strike

28 Posterior local volatility local volatility strike [USD] maturity [years] local volatility maturity: 3 year ±2SD MAP data loc strike

29 Posterior uncertainty 1. Nonlinearity of pricing operator Price sensitivity to changes in local vol across T and K 2. Availability of observed data Density of prices across T and K

30 Posterior option prices Feed local volatility back into model for posterior over prices and implied volatility

31 Posterior option prices call price maturity: 0.13 year ±2SD MAP data implied volatility maturity: 0.13 year ±2SD MAP data strike strike

32 Posterior option prices call price maturity: 0.48 year ±2SD MAP data implied volatility maturity: 0.48 year ±2SD MAP data strike strike

33 Posterior option prices call price maturity: 0.99 year ±2SD MAP data implied volatility maturity: 0.99 year ±2SD MAP data strike strike

34 Posterior option prices call price maturity: 3 year ±2SD MAP data implied volatility maturity: 3 year ±2SD MAP data strike strike

35 PART 2: VOLATILITY DYNAMICS

36 S&P 500 option market

37 Volatility dynamics Local volatility provides good static fit of option market; need to re-calibrate on regular basis Our framework provides way of inferencing dynamical behaviour by introducing temporal dependency in the Gaussian process prior Posterior over local volatility (call prices, implied vols etc.) with dependency across time

38 Calibration

39 Prediction The Gaussian process readily provides predictive distribution over local volatility (prices, implied vol etc.) Since we model temporal dependency, may predict (local, prices, implied) volatility We use the posterior to make one-week ahead predictions

40 Prediction

41 Prediction

42 Prediction Volatility Lehman Realised volatility: historical measure from returns

43 Prediction VIX-index: 30-day forward volatility measure implied by market 1-week ahead predictions

44 Concluding remarks We present an approach for nonparametric modelling of local volatility The approach is probabilistic, gives notion of uncertainty Using Gaussian processes, it is flexible and scales with data

45 Concluding remarks Connection to classical inverse calibration with regularised optimisation, but gives explicit link to probabilistic framework Including time dimension in input variable, gives means of studying temporal behaviour of (local) volatility over time, as well as for predicting future surfaces Computational methods (MCMC) expensive

46 Thank you for your attention!