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Currency Option ValuationPowerPoint Presentation

Currency Option Valuation

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Currency Option Valuation. Option valuation involves the mathematics of stochastic processes. The term stochastic means random; stochastic processes model randomness. Myron Scholes and Fischer Black. Binomial Option Payoffs Valuing options prior to expiration.

Currency Option Valuation

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Currency Option Valuation

Option valuation involves the mathematics of stochastic processes. The term stochastic means random; stochastic processes model randomness.

Myron Scholes

and Fischer Black

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 ¥/$,

Now, assume two equally likely possible payoffs: 90¥/$ or 110¥/$, at the expiration of the contract

90¥//$

.5

100¥//$

.5

110¥//$

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

90¥//$

.5

.5

110¥//$

Right! You don’t exercise your option. Hence:

¥//$

.5

5¥//$

.5

10¥//$

Next, let’s replicate the call option payoffs with money market instruments and then find its value.

How do you do that?

Right. You BUY one $ at a cost of 100¥ (S) and you borrow 90¥ at 5%

(90/1.05 = 85.71¥)

The yen value of the $ at the end of the year will be either 90 or 110, but you have a liability of precisely 90¥. Hence, your expected payoff is 10¥. Which, as you have probably noted, is a multiple of your option payoff (10¥/2=5¥)

You BUY one $ at a cost of 100¥ (S) and you borrow 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) and you borrow 90¥ at 5%

100¥ - 85.71¥ cost of the bank loan

= 14.29¥

So, how do you scale down the “buy a dollar, borrow yen strategy until it is the same as the payoff on a call?

Of course, if you can do that, you can also value the call option.

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

What next?

What next?

You buy .5 of one $ at a cost of 50¥ and you borrow .5 of 90¥ or 45¥ at 5% or 42.86¥

The difference between 50¥ and 42.86¥ is 7.14¥

Hence, the yen value of a one-dollar call option is 7.14¥.

We can replicate our basic tree multiple times, where the up or down movement represents some function of E[S], or the expected mean

At the limit, the distribution of continuously compounded exchange rates approaches the normal distribution (which is described in terms 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

Call = [S*N(d1)] - [e-iT*X* N(d2)]

Where:

Call = the value of the call option

S = The spot market price

X = the exercise price of the option

i = risk free instantaneous rate of interest

- = instantaneous standard deviation of S
T = time to expiration of the option

N(.) = f(the standard normal cumulative P distribution)

Call = [S*N(d1)] - [e-iT*X* N(d2)]

d1 = [ln(S/X) + (i + (s2/2))T]/ (sT1/2)

d2 = d1 - sT1/2

e-iT = 1/(1+i) T

Discounts the exercise or strike price to the present at the risk-free rate of interest

At 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 rate remains a random variable. Hence, we need the expected value of ST - X, given that it expires in the money.

In Black-Scholes, N(d1) is the probability that the call option will expire in the money

S* N(d1) is the expected value of the currency at expiration, given S>X.

X* N(d2) is the expected value of the exercise price at expiration

e-iT discounts the exercise price to PV

Option Price