Download An Asian Basket Multi Digital option

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
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Quantitative Justification:
Strong Law of Large Numbers
Motivation…
• Fundamental Theorem of Arbitragefree Pricing.
• Brownian Motion Assumption for
Underlying Stock Price.
Price Path Sampling
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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
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The Strong Law of Large Numbers
Proof…
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Contd…
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Monte Carlo Simulations And Its
Applications
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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)
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13
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Data, Results And
Comaprisons
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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%.
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MICEX
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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)
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Stockholmbörsen
• ABB
• ATLAS COPCO A
• ELECTROLUX B
• ERICCSSON B
• HENNES & MAURITZ
• SEB
• SCANIA
• SVENSKA HENDELSBANKEN
• SWED BANK
• TELE 2
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Price Dynamics
Stockholmborsen (Real Market Data)
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Price Dynamics (One Simulation Results…)
Stockholmborsen
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No. of Underlying above Strike (100 Simulations)
Stockholmborsen
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Price Dynamics (One Simulation Results…)
Russian Market
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No. of Underlying above Strike (100 Simulations)
Russian Market
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Questions?
2009
2007
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Tack så
mycket
2009
2007