Download Option Price and Portfolio Simulation

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Known dividend to be paid before option
expiration
◦ Dividend has already been announced or stock pays
regular dividends
◦ Option should be priced on S0 – PV(dividends
anticipated before t)
= S0 – Div * exp(-r*time dividend paid)
◦ Example: Coca-Cola pays relatively stable quarterly
dividend around $0.31 per share on November 29th
each year. Assuming it is November 14, use
S0 – 0.31* exp(-r*15/365)
for current stock price in Black-Scholes model for
January option.
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Continuous Dividend Payments
◦ Index funds on basket of stocks (e.g. S&P 500
index): the many stocks pay out their dividends
throughout the year
◦ Merton Model:
 Assume continuous dividend yield k
C  Se N d1   Xe N d 2 
 kt
 rt
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ln S / X   r  k   2 / 2 t
d1 
 t
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Strategies if anticipate that stock price will
rise over period t
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Purchase calls
Purchase stock
Purchase stock and put to insure portfolio
Write put
Strategies if anticipate that stock price will
decline over period t
◦ Purchase puts
◦ Write calls
 Covered call: purchase stock
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Most financial models of stock prices
assume that the stock’s price follows a
lognormal distribution. (The logarithm of
the stock’s price is normally distributed)
This implies the following relationship:
Pt = P0 * exp[(μ-.5*σ2)*t + σ*Z*t.5]
P0 = Current price of stock
t = Number of years in future
Pt = Price of stock at time t Random Variable!!
Z = A standard normal random variable with
mean 0 and standard deviation 1 Random
Variable!!
◦ μ = Mean percentage growth rate of stock per
year expressed as a decimal
◦ σ = Standard deviation of the growth rate of
stock per year expressed as a decimal. Also
referred to as the annual volatility.
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Option price is the expected discounted
value of the cash flows from an option on a
stock having the same volatility as the stock
on which the option is written and growing
at the risk-free rate of interest.
The cash flows are discounted continuously
at the risk-free rate
The option price does not depend on the
growth rate of the stock!
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Simulate the stock price t years from now assuming
that it grows at the risk-free rate rf. This implies the
following relationship:
Pt = P0 * exp[(rf-.5*σ2)*t + σ*Z*t.5]
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Compute the cash flows from the option at
expiration t years from now.
Discount the cash flow value back to time 0 by
multiplying by e-rt to calculate the current value of
the option.
Select the current value of the option as the output
variable to determine its mean price and other
statistics.
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Want to guarantee that t periods from now
you will have at least I*z
◦ z is a number generally between 0 and 1that
guarantees a minimum value
◦ Want to invest in Stock with price S0 and Put for
stock with exercise price X
◦ A package of share + put costs S0 + P(S0,X)
◦ Buy a packages where
 a =I/(S0 + P(S0,X))
 Minimum $ return = aX which should be set to I*z
 Pick X to guarantee that S0 + P(S0,X)=X/z
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Variance-Covariance Method
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Historical Simulation
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Monte Carlo Simulation
◦ Using an assumed distribution for the asset return (e.g.
normally distributed), estimated mean, variances &
covariance, compute the associated probability for the
VaR
◦ Use sorted time series data to identify the percentile
value associated with the desired VaR
◦ Specify probability distributions & correlations for
relevant market risk factors and build a simulation
model that describes the relationship between the
market risk factors and the asset return. After
performing iterations, identify the return that produces
the desired percentile for the VaR.
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Use options to create other securities
◦ Bull spreads (written and purchased calls)
◦ Collars (stock, written call and purchased put)
◦ PPUP (Principal-Protected, upside potential: bond plus
at-the-money call)
◦ Butterfly
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Options can be replicated by a long or short
position in the underlying stock and a long or
short position in the risk-free asset (e.g. bond)