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1 Monte Carlo Simulations on Asian Basket Multi Digital Options Analytical Finance I MMA- 707 Seminar Group Members Artem Rybakov Moazam Riaz Presented to: Jan Röman Shoaib Hashmi 2009 Simon Porras 2007 2 The Task Build an application in Excel/VBA to solve option prices for an Asian-BasketMulti-Digital option. That is; an option with maturity in 6 months on a basket of N (say 10) shares. 3 Concepts & Definitions 4 Asian Basket Multi Digital Options • Exotic options, composed of many (say 5 , 10) underlying stocks. • Types of Asian Digital Options Price Average price of each underlying over the period Strike Fixed at the beginning Floating Price Price Floating Strike Strike Price at the time of maturity Average over the life of an option 5 Payoffs Illustration of one simulation... Asian Part Digital Part Underlying Average Price of last month Strike Condition Stock 1 A1 K1 A1>K1 Stock 2 A2 K2 A2>K2 Stock 3 A3 K3 A3<K3 Stock 4 A4 K4 A4>k4 Stock 5 A5 K5 A5>k5 Stock 6 A6 K6 A6<k6 Stock 7 A7 K7 A7<k7 Stock 8 A8 K8 A8<k8 Stock 9 A9 K9 A9<k9 Stock 10 A10 k10 A10>k10 Pay off One Simulation If 5 or more stocks > fixed strike 1 in this case 6 Quantitative Justification: Strong Law of Large Numbers Motivation… • Fundamental Theorem of Arbitragefree Pricing. • Brownian Motion Assumption for Underlying Stock Price. Price Path Sampling 7 General Procedure to obtain Expected Payoff of Option 1. 2. 3. Generate a “large” number of Price Paths. Calculate the payoff for each simulation. Estimate the expected value of the option as: E [ A]= X1 X2 Xn n 8 The Strong Law of Large Numbers Proof… 9 Contd… 10 Monte Carlo Simulations And Its Applications 11 Underlying Asset Pricing Formula St S0 e ( 2 2 ) t Z ( t ) 2 ( ) t ( Z ( t ) Z ( t 1)) St e 2 S t 1 S t S t 1 e ( S t S t 1 e 2 2 ( ) t Nt ( 0, t ) 2 2 ) t t Nt ( 0,1) 12 13 14 Data, Results And Comaprisons 15 Volatility & Risk Free Interest Rates n Historical volatility where… u i 1 n 2 i n -1 ( u i ) 2 i 1 n(n - 1) ui ln( k St ) S t 1 St is a time t stock price St-1 is a time (t-1) stock price N is a number of prices in the interval the volatility is calculated on k is an amount of time frames in 1 year (253 trading days) The risk free interest rate has been chosen in correspondence with coupon interest rate on long-term government bonds. r=7%. 16 MICEX 17 Comparisons of Results from Real Data Russian Market Simulated prices C= 0,020026 (volatile market) C= 0,067265 (stable market) Price obtained from real data C= 0,9656 (discounted payoff without uncertainty) 18 Stockholmbörsen • ABB • ATLAS COPCO A • ELECTROLUX B • ERICCSSON B • HENNES & MAURITZ • SEB • SCANIA • SVENSKA HENDELSBANKEN • SWED BANK • TELE 2 19 Price Dynamics Stockholmborsen (Real Market Data) 20 Price Dynamics (One Simulation Results…) Stockholmborsen 21 No. of Underlying above Strike (100 Simulations) Stockholmborsen 22 Price Dynamics (One Simulation Results…) Russian Market 23 No. of Underlying above Strike (100 Simulations) Russian Market 24 Questions? 2009 2007 25 Tack så mycket 2009 2007