narayan murth

8/9/2019 narayan murth

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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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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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What do you do if the yen price of $ is 90?

90//$

110//$

.5

.5


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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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Value Currency OptionsWhat next?


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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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Binomial Model


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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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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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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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Pricing ModelIn Black-Scholes, N(d1) is the probability that

the call option will expire in the money


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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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