imposing your view on the market: risk-neutral & real world … · 2016. 9. 27. · goutham...
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Imposing Your View on the Market:Risk-Neutral & Real World Models
Goutham Balaraman, Ph.D.
Numerix
Overview
• Risk Neutral Models & Real World Models– Risk Neutral Black Scholes vs Real World Black Scholes
• Risk Neutral Model Examples
– Handling Volatility Skew
– Heston Model
• Real World Models
– Real World Heston
– Calibration Examples
• Hybrid Models & Extensions
• Some Applications
Risk Neutral vs Real World
• Used in risk estimation
• Path distributions are meant to model observed distributions.
• Used by asset managers / insurance
• Pricing trades in a way to avoid arbitrage
• Path distributions are not consistent with reality distributions
• Used by dealers in pricing
Risk Neutral Real World
Risk Neutral vs Real World
• Risk Neutral
• Real World
Constant Drift Risk Premium
Volatility
VolatilityRisk Free Rate
Risk Neutral Models
Implied Volatility Surface
Strike
Tenor
Implied
Volatility
Note: SP500 volatility surface as of 8/31/2016 (Bloomberg)
– Volatility varies across the Strike and Tenor
– Volatility surface is used to price OTC derivatives
– Stochastic volatility models to fit to the market option quotes
– How does one fit a model?
Stochastic Volatility Models
• Heston Stochastic Volatility Model
Asset PriceAsset Variance
Asset-Variance Correlation
Reversion Strength Long Variance Vol of Vol
Calibrating Parameters
• Price the different options for different strikes and maturities
• Find parameters such that the error between the model and market is minimized
• Choice of solvers
– Local Solvers
– Global Solvers
Sample Calibration Report
Name Avg Abs Error Theta Kappa Sigma Rho V0 Time (s)Scipy LM1 3.015253 0.12575 7.92E+00 1.887934 -0.36494 0.05539 4.65
Scipy LM2 7.0195 0.04818 -5.49E-01 0.197958 -0.99955 0.09057 20.5
Scipy LS1 3.015251 0.12575 7.92E+00 1.887949 -0.36494 0.05539 4.55
Scipy LS2 5.096414 3.13677 4.90E-06 -0.00025 -0.00001 1.5979 28.1
Scipy DE1 2.859113 0.12322 5.01E+00 0.950309 -0.57034 0.07844 99
Scipy DE2 2.876087 0.12225 5.00E+00 0.849266 -0.63771 0.07948 117
Scipy BH1 2.850972 0.12346 5.07E+00 0.995095 -0.56211 0.07901 134
Scipy BH2 2.863356 0.1234 4.79E+00 0.904397 -0.59371 0.07922 109
Local
Glo
bal
Application - Valuing Structured Notes
Real World Models
Real World Heston Model
Constant Drift Risk Premium
Calibrating to Time Series
• Calibrate using Maximum Likelihood Estimator
Index Volatility Distribution Index Distribution
Heston Model - Matching Moments
Params
kappa : 5.0
xi : 2.02815683102
rho : -0.713871038544
volsquared0 : 0.01825
theta : 0.0596195733347
constant drift : 0.13020657511
Given Model
Mean 0.17 0.17
Variance 0.04 0.04
Skew -1.0 -1.0
Kurtosis 2.5 2.5
SP500 1-Year Returns
Heston Model - Matching Percentiles
Application - Estimating Tail Risk
0 250 500 750 1,000 1,250
Numerix1000 Paths
Numerix300 Paths
New Academy1000 Paths
CTE70 FROM DIFFERENT TRIALS MILLIONS
CTE70 with Error Estimate s
Down 20% Base
Hybrid Model Framework
IRHW1F
IR(S)BK1F
IRHW2F
IR(S)BK2F
IRLMM
INFJY (HW)
INFJY (BK)
CRBK1F
EQBS
EQDupire
EQHeston
EQBates
EQLSV
FXBS
FXHeston
CMDTYBS
CDMTYS1F
CMDTYGS2F
CMDTYHeston
CRSBK1F
RATES INFLATIONCREDIT
CRBEQUITIES
EQBFX
FXBCMDTY
IRSV-LMM
FXDupire
FXBates
FXLSV
INFIMM
INFSV-IMM
Hybrid Real Model Simulation
Baseline / Trend GrowthReturn 6.50 6.50 6.50 9.40 1.40 3.10 3.65
Risk 17.00 18.00 26.00 27.50 1.25 5.00 10.00
Correlation
US Stocks 1.00
Dev xUS Stocks 0.83 1.00
EM Stocks 0.75 0.75 1.00
Prvt Mkts 0.75 0.65 0.63 1.00
Cash -0.05 -0.09 -0.05 0.00 1.00
Core Bonds 0.28 0.13 0.00 0.31 0.19 1.00
LT Bonds 0.32 0.16 0.01 0.32 0.11 0.94 1.00
Severe Recession
Return -43.50 -43.50 -43.50 -40.60 0.40 15.60 23.65
Risk 42.00 43.00 51.00 52.50 1.25 15.00 20.00
Correlation
US Stocks 1.00
Dev xUS Stocks 0.98 1.00
EM Stocks 0.98 0.98 1.00
Prvt Mkts 0.98 0.98 0.98 1.00
Cash -0.05 -0.09 -0.05 0.00 1.00
Core Bonds -0.22 -0.37 -0.50 -0.19 0.19 1.00
LT Bonds -0.18 -0.34 -0.49 -0.18 0.11 0.98 1.00
Base Scenario – Real World Scenario
Recession Scenario – Real World Models
Summary
• Risk Neutral Models– Stochastic models can fit to the volatility skew
– The model parameters are calibrated to the market observables
– The model parameters are sensitive to the choice of solvers
• Real World Models
– Has a constant drift or risk premium term
– Can be calibrated to fit historical observations, or future projections
– They can be used to evolve and estimate a portfolio through time
• Hybrid Models– Available for both Risk Neutral & Real World models
– Incorporate correlations between different observables
– Can be used to simulate the dynamics of correlated markets in a realistic fashion
