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CH 08 B 16e 06172019 - YouTube
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the second part of chapter 8 takes us
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into option valuation we're going to
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cover three methodologies the binomial
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model the black-scholes option pricing
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model and put call parity here's all the
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data for our sample stock option to
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begin we'll look at the binomial option
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pricing model has a lead-in to black
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Scholes note how the stock price is
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described the current price is $27 it
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can only go up by a factor of 1.4 1 or
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down by 0.7 t1 those are the only two
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possible values at the end of 6 months
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hence binomial the exercise price is $25
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risk-free rate 6% putting this into
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chart form adds the value of the option
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at both the up and down legs here's a
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zoomed view of the chart so you can more
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clearly see how the terminal stock and
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option values are computed if the price
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goes up by a factor of 1.4 1 then the
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ending up price is 3807 for $13 and
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seven cent pay off on the option if down
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the price is 1917 and the payoff is 0
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we're creating a portfolio by buying n
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sub s shares of stock and writing or
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selling one call option we're creating a
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hedge portfolio which will have the same
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payoff regardless of whether the stock
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price goes up or down we're buying the
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in chairs and selling a call so the C up
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or down is negative we need to find out
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how many shares we need to make the
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hedge work so we set the up and down
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portfolio values equal and solve for M
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in our example n is equal to point six
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nine one five we'll zoom in on this
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chart on the next slide but this
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expanded version from the mini case
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excel file includes all the given data
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and details how the hedged portfolio
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construction works given n equal to zero
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point six nine one five the ending value
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of both the up and the down portfolios
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is 13 26 the ending values on both legs
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people the value of the end shares of
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stock minus the option payoffs
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here's the zoomed in version where you
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can see the end of leg calculations a
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bit better we've succeeded in creating a
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hitch portfolio since both ending values
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are the same if the payoff either way is
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1326 at the end of six months then the
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portfolio of stock an option is riskless
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so the appropriate discount rate is the
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risk-free rate so what's the PV of that
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payoff amount discounted it's 6% per
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year for a half a year on a daily basis
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today our portfolio would be worth 12
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dollars and 87 cents
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alternately we could just use the PV
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formula to arrive at the current value
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of 1287 we now know the value of the
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portfolio at x 0 the portfolio is
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composed of the point six nine one five
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shares of stock in a written call
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rearranging the equation in the middle
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of the slide we can solve the current
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value of the call and we'll see the
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calculations on the next slide our
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result the call is currently worth about
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five dollars and eighty cents
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we found the pv of the portfolio to be
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twelve eighty seven we know the current
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price of the stock is twenty seven and
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we bought point six nine one five shares
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so that value point six nine one five
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times twenty seven 1867 subtracting the
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1287 yields the value of the call today
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five eighty we can modify our approach
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to valuing the option by creating what
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is called a replicating portfolio
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because it's designed to replicate the
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call options payoff as you can see on
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the slide the right hand side of the
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equation is the replicating portfolio in
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this case we buy the in shares of stock
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and borrow an amount equals to the
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present value of the hedge portfolios
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payoff 1287 if you borrow an amount
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equal to the present value of the hedged
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portfolios riskless pay off and purchase
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hen shares of stock the portfolio's pay
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off will replicate the call options
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payoff if this is not true then an
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arbitrage opportunity exists riskless
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arbitrage is the situation in which you
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have none of your own money invested and
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no risk yet a positive cash flow this
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continues the examples showing that
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net portfolio payoffs on both legs
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exactly match those of the option
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finishing up our example if the options
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price is not equal to the cost of the
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replicating portfolio an arbitrage
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opportunity exists and we'll see how
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that might work on the following slides
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we found the option value to be five
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dollars and 80 cents but if it's priced
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at six dollars we create the replicating
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portfolio for a net gain of twenty cents
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you borrowed to buy the stock so none of
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your own money is invested you have no
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risk since the payoffs to the
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replicating portfolio are equal whether
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the stock price moves up or down you
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gain 20 cents if you could do this with
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0 of your own money invested you'd
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replicate it as much as possible as fast
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as possible this would create price
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pressure on the option driving the price
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down to 580 we've only looked at a one
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period binomial model for simplicity the
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binomial model can be expanded to many
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periods with the time periods reduced to
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very small increments these are solved
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with a binomial lattice the idea that a
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stock price can only take on two values
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may have seemed completely unrealistic
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but consider in a second say how much
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can a stock price move a penny up or
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down as the time period becomes smaller
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and smaller the binomial option pricing
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model converges to the more widely used
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black Scholes option pricing model the
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black Scholes option pricing model like
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most theoretical models is based on some
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strong underlying assumptions an
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important one is that the stock pays no
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dividends during the life of the option
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since most dividends are paid quarterly
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and options are short-term instruments
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this is not that unrealistic but
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remember that after a theory is
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developed assumptions are relaxed and
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the model tested again there are
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versions of the black Scholes model that
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incorporate dividends paid and as is
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usual we assume away taxes and
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transaction costs the black Scholes
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model also assumes that investors can
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borrow at the risk-free rate and that
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investors receive full proceeds from
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short selling not the usual case and
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another important limitation is that
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black Scholes values European options
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those that can only be exercised at
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maturity most stock options in the
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United States are American in that they
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can be exercised at any time from
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purchase to expiration
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since the option to exercise that
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anytime has value we can assume that an
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American option is worth at least as
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much as the European one here's the
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formula to value a call option using the
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black-scholes option pricing model while
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these look complicated they're really
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not much more than plug and chug once
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you identify all the variables which
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we'll do on the next slide note
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especially in D 1 and II in D 1 requires
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either the use of an excel function or
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standard normal table which is appendix
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D in your book these value is shown here
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and it's a key on your calculator and
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there's an Excel function exp we'll be
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using this because as stated the
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assumptions the black Scholes model is a
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continuous model everything we've looked
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at relative to compounding and
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discounting so far has been discrete but
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we'll be using continuous discounting in
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the black Scholes model
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note that for short term instruments
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will use a short-term risk-free rate
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usually taken from the t-bill rates the
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time to expiration T is expressed in
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years as a fraction the natural log
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function is also available through an
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Excel function and as a key on your
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calculator Sigma is the standard
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deviation of the rate of return on the
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stock underlying the option and here's
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our sample data - value a call option
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well see as we plug values into the
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formulas that it is more tedious than it
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is difficult the first step is to
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compute d1 this is the calculation that
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requires the most care as there's so
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many components to it if working in
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Excel I strongly recommend computing
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each component separately and then
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combining them trying to do it all in
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one cell just offers too many
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opportunities for error in order of
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operations if nothing else in the
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placement of parentheses we found d1
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0.48 1/9
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from d1 we signed d2 0.135 5 given those
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values we can you
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the tables but probably not or the Excel
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function which we've done here note that
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Excel has two normal functions norm SD
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ist finds the value for a standard
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normal distribution with the norm V ist
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function no es in the middle you have to
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specify the mean and variance but you're
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taken as zero and one for standard
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normal we're ready to plug to the main
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equation here's where continuous
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discounting comes into play the term X e
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raised to the minus RT is the continuous
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version of discounting X at the
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risk-free rate for time T using our
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given values we find the value of the
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call has five dollars and six cents
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there's a video tutorial covering the
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calculation of a call price posted with
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this unit now that we've seen and worked
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through the black-scholes model let's
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look at how each of the variables
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impacts the call options value obviously
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as the stock price rises so does the
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value of a call on that stock as the
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strike price increases however the value
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of the call decreases a call with a
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strike price of $25 is worth more than
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one of the strike price of 35 remember
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the time has value so at the time the
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expiration increases so does the value
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of the option more time for the stock
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price to end up in the money as the
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risk-free rate increases it decreases
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the discounted value of the exercise
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price which increases the value of the
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call option the last one is important as
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the variance of the returns on the
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underlying stock increases so does the
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value of the option this is completely
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opposite to the impact on the value of a
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share of stock where volatility is a
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negative the reason is that a higher
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variance increases the probability of
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the stock price rising and with a call
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option there's no downside
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while we focused on call options we need
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to at least take a look at put options
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to understand both types a put gives the
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hunger the right to sell an asset you
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might use this in a protected put
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strategy suppose you're concerned that
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the price of the stock you own may go
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down but you don't want to sell it
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you could buy put with a strike price
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slightly below where the stock is
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trading
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price falls you exercise and limit your
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loss price doesn't fall you lose the
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premium you paid something like
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insurance put-call parity specifies the
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relationship between a put a call in the
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underlying stock price for our example
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assume we create two portfolios in one
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you hold a put option and a share of
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stock in the other you hold a call
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option and the present value of the
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exercise price you'll see how this works
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on the next slide this table shows the
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value of each portfolio at expiration
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when the ending stop price is above and
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below the exercise price note the
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results in both cases the value of the
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two portfolios is the same if the
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payoffs are equal the portfolio's must
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be equal which leads to put call parity
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but call parity can be used to solve for
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the value of a put given the other
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variables this ends our coverage of
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options and of unity
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