文档介绍：Weighted Monte-Carlo Methods
for Multi-Asset Equity Derivatives:
Theory and Practice
Marco Avellaneda
Courant Institute of Mathematical
Sciences
New York University
Summary
° Statement of the Calibration Problem for Multi-Asset
Equity Derivatives
° Weighted Monte Carlo simulation (max-entropy)
° Application to Arbitrage Pricing of Basket Options
° Comparison between WMC and Steepest Descent Method
° Comments on Correlation Skew and the statistics of
Implied and Historical Correlations
1
Calibration Problem for Multi-Asset
Equity Derivatives
Given a group, or collection of stocks, build a stochastic model for the joint
evolution of the stocks with the following properties:
• The associated probability measure on market scenarios is risk-neutral: all traded
securities are correctly priced by discounting cash-flows
• The associated probability measure is such that stock prices, adjusted for interest
and dividends, are martingales (local risk-neutrality)
• The model simulates the joint evolution of ~ 100 stocks
• All options (with reasonable OI), forward prices, on all stocks, must be fitted
to the model. Number of constraints ~500 to ~1000 or more
• Efficient calibration, pricing and sensitivity analysis in real-time environment
Example: Basket of 20 Biotechnology
Stocks ( Components of BBH)
Ticker Price ATM ImVol Ticker Price ATM ImVol
ABI 55 GILD 46
AFFX 64 HGSI 84
ALKS 106 ICOS 64
AMGN 40 IDPH 72
BGEN 41 MEDI 82
CHIR 37 MLNM 92
CRA 55 QLTI 64
DNA SEPR 84
ENZN 81 SHPGY 47
GENZ 56 BBH 32
2
Implied Volatility Skews
Multiple Names, Multiple Expirations
AMGN Exp: Oct 00 BGEN Exp: Oct 00
75 68
66
Im pliedVol
Im pliedVol
70 64
62
65 60
58
BidVol
60 BidVol
56
54
55
52
50
50 50 55 60 65 70 75 80 85 AskVol
60 65 70 75 80 85 90 95 AskVol
Implied