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Table 10.4: American Options: Check each Node for Early Exercise

Table 10.4: American Options: Check each Node for Early Exercise

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Dividend Flows, Foreign Exchange

and Future Options



34



• 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



35



Option Pricing under the Normal

Distribution



36



• 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



37



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



39



Option Pricing under the Normal

Distribution



40



• 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



41



• 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



42



Option Pricing under the Normal

Distribution



43



• 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



44



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Table 10.4: American Options: Check each Node for Early Exercise

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