narayan murth
TRANSCRIPT
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Currency Option ValuationOption valuation involves the mathematics
of stochastic processes. The termstochasticstochastic
means random; stochastic processes model
randomness.
Myron Scholes
and Fischer Black
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Binomial Option PayoffsValuing options prior to expiration
Given: You are a resident of Japan. You want
to buy a European call on one (1) US$.
The current spot rate (S) is 100 /$
The contract has an exercise price (X) = to the
expected future spot exchange rate (E[S]),
which is also 100 /$,
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Binomial Option PayoffsValuing options prior to expiration
Now, assume two equally likely possible
payoffs: 90/$ or 110/$, at the expiration of
the contract 90//$
110//$
100//$
.5
.5
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Binomial Option PayoffsValuing options prior to expiration
What do you do if the yen price of $ is 90?
90//$
110//$
.5
.5
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Binomial Option PayoffsValuing options prior to expiration
Right! You dont exercise your option. Hence:
//$
10//$
.5
.5
5//$
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Buy a $,
Borrow
Next, lets replicate the call option payoffs
with money market instruments and then
find its value.
How do you do that?
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Buy a $,
Borrow
Right. You BUY one $ at a cost of 100 (S) andyou borrow 90 at 5%
(90/1.05 = 85.71)The yen value of the $ at the end of the yearwill
be either 90 or 110, but you have a liability ofprecisely 90. Hence, your expected payoff is
10. Which, as you have probably noted, is amultiple of your option payoff(10/2=5)
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Buy a $,
Borrow
You BUY one $ at a cost of 100 (S) and youborrow 90 at 5%
(90/1.05 = 85.71)What you probably overlooked is the present
value of buying one $ at a cost of 100 (S) andyou borrow 90 at 5%
100- 85.71cost of the bank loan= 14.29
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Buy a $,
Borrow
So, how do you scale down the buy a dollar,borrow yen strategy until it is the same as thepayoff on a call?
Of course, if you can do that, you can also valuethe call option.
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Using the Hedge Ratio to
Value Currency Options(also called the option delta)The Hedge Ratio indicates the number of call
options required to replicate one unit (in this
case, one $) of the underlying asset.
Hedge Ratio =
spread of option prices/spread of possible
underlying asset values
Hence 0-10/0-20 = 10/20 = .5
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Using the Hedge Ratio to
Value Currency OptionsWhat next?
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Using the Hedge Ratio to
Value Currency OptionsWhat next?
You buy .5 of one $ at a cost of50and you
borrow .5 of 90 or45at 5% or 42.86
The difference between 50and42.86is7.14
Hence, the yen value of a one-dollar calloption is 7.14.
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The General Case of the
Binomial ModelWe can replicate our basic tree multiple times,
where the up or down movement represents
some function of E[S], or the expected mean
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The General Case of the
Binomial Model
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The General Case of the
Binomial ModelAt the limit, the distribution of continuously
compounded exchange rates approaches the
normal distribution (which is described interms of a mean (expected value, in this
case E[S]) and a distribution (variance or
standard deviation)This makes it equivalent to Black-Scholes
model
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The Black-Scholes Option
Pricing ModelCall = [S*N(d1)] - [e
-iT*X* N(d2)]
Where:
Call = the value of the call optionS = The spot market price
X = the exercise price of the option
i = risk free instantaneous rate of interestW = instantaneous standard deviation of S
T = time to expiration of the option
N(.) = f(the standard normal cumulative P
distribution)
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The Black-Scholes Option
Pricing ModelCall = [S*N(d1)] - [e
-iT*X* N(d2)]
d1 = [ln(S/X) + (i + (W8A(W8
d2 = d1 - W8
e-iT = 1/(1+i) T
Discounts the exercise or strike price to thepresent at the risk-free rate of interest
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The Black-Scholes Option
Pricing ModelAt expiration, time value is equal to zero and
there is no uncertainty about S (call option
value is composed entirely of intrinsic value).CallT = Max [0, ST - X]
Prior to expiration, the actual exchange rateremains a random variable. Hence, we needthe expected value of ST - X, given that it
expires in the money.
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The Black-Scholes Option
Pricing ModelIn Black-Scholes, N(d1) is the probability that
the call option will expire in the money
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The Black-Scholes Option
Pricing ModelS* N(d1) is the expected value of the currency at
expiration, given S>X.
X* N(d2) is the expected value of the exerciseprice at expiration
e-iT discounts the exercise price to PV
Option Price
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