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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 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 ¥/$,

Binomial Option PayoffsValuing options prior to expiration

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

90¥//$

.5

100¥//$

.5

110¥//$

Binomial Option PayoffsValuing options prior to expiration

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

90¥//$

.5

.5

110¥//$

Binomial Option PayoffsValuing options prior to expiration

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

¥//$

.5

5¥//$

.5

10¥//$

Buy a $, Borrow ¥

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

How do you do that?

Buy a $, Borrow ¥

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¥)

Buy a $, Borrow ¥

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¥

Buy a $, Borrow ¥

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.

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

Using the Hedge Ratio to Value Currency Options

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¥.

The General Case of the Binomial Model

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

The General Case of the Binomial Model

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

The Black-Scholes Option Pricing 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)

The Black-Scholes Option Pricing Model

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

The Black-Scholes Option Pricing Model

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.

The Black-Scholes Option Pricing Model

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

The Black-Scholes Option Pricing Model

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

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