Table 10.4: American Options: Check each Node for Early Exercise
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Dividend Flows, Foreign Exchange
and Future Options
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• When the underlying asset pays out dividends or other cash flows
we need to adjust the RNP formula
• Consider an underlying stock index that pays out cash at a rate of
q per year. In this case we have
• When underlying asset is a foreign exchange rate then q is
set to interest rate of the foreign currency
• When the underlying asset is a futures contract then q = rf
so that RNP = (1-d) / (u-d) for futures options
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Option Pricing under the Normal
Distribution
• We now assume that daily returns on an asset be independently
and identically distributed according to normal distribution
• Then the aggregate return over days will also be
normally distributed with the mean and variance
appropriately scaled as in
• and the future asset price can be written as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Option Pricing under the Normal
Distribution
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• The risk-neutral valuation principle calculates option price as the
discounted expected payoff, where discounting is done using riskfree rate and where the expectation is taken using risk-neutral
distribution:
is the payoff function and rf is
• Where
the risk-free interest rate per day
• The expectation
is taken using the risk-neutral
distribution where all assets earn an expected return equal
to the risk-free rate
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Option Pricing under the Normal
Distribution
• In this case the option price can be written as
• where x* is risk-neutral variable corresponding to the
underlying asset return between now and maturity of
option
• f (x*) denotes risk-neutral distribution, which we take to
be normal distribution so that
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Option Pricing under the Normal
Distribution
38
• Thus, we obtain the Black-Scholes-Merton (BSM) call option price
• where Φ(*) is the cumulative density of a standard normal
variable, and where
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Option Pricing under the Normal
Distribution
• Interpretation of elements in the option pricing formula
•
•
•
is the risk-neutral probability of exercise
is the expected risk-neutral payout when exercising
is the risk-neutral expected value of the stock
acquired through exercise of the option
∀ Φ (d) is the delta of the option, where
is the first
derivative of the option with respect to the underlying asset price
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Option Pricing under the Normal
Distribution
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• Using the put-call parity result and the formula for cBSM, we can
get the put price formula as
• Note that the symmetry of the normal distribution implies
that
for any value of z
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Option Pricing under the Normal
Distribution
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• When the underlying asset pays out cash flows such as dividends,
we discount the current asset price to account for cash flows by
replacing St by
• Where q is the expected rate of cash flow per day until maturity of
the option
• This adjustment can be made to both the call and the put price
formula, and the formula for d will then be
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Option Pricing under the Normal
Distribution
• The adjustment is made because the option holder at maturity
receives only the underlying asset on that date and not the cash
flow that has accrued to the asset during the life of the option.
• This cash flow is retained by the owner of the underlying asset
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Option Pricing under the Normal
Distribution
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• We now use the Black-Scholes pricing model to price a European
call option written on the S&P 500 index
• On January 6, 2010, the value of index was 1137.14
• The European call option has a strike price of 1110 and 43 days to
maturity
• The risk-free interest rate for a 43-day holding period is found
from the T-bill rates to be 0.0006824% per day (that is,
0.000006824)
• The dividend accruing to the index over the next 43 days is
expected to be 0.0056967% per day
• Assume volatility of the index is 0.979940% per day
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Option Pricing under the Normal
Distribution
• Thus we have:
• from which we can calculate BSM call option price as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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