course2 question

[retaker]

You are given a one-year option to buy a particular foreign currency at the current exchange rate of $1.35. You have the following information.

Current price of currency = $1.35
Exercise price = $1.35
Standard deviation of the exchange rate change = .2
Dollar interest rate = .05
Foreign currency interest rate = .07

What is the value of the option?

[Gandalf]

That can't possibly be enough info. Maybe they want you to make assumptions corresponding to Black-Scholes pricing of stock options?

(You can get the expected value of the exchange rate based on the two interest rates, but valuing the option requires knowing or assuming more than the mean and standard deviation.)

[retaker]

They give you the standard deviation = .2

[Gandalf]

I see standard deviation = .2. I have trouble understanding what the standard deviation of the exchange rate change = .2 means. Does it mean the standard deviation of the ending US currency rate per 1 of foreign currency is .2? If not, what?

However, no matter what it means, it is true in general and for the problem you stated in particular that it takes more than the mean and standard deviation to specify the distribution of ending values, and hence the value of the option.

E.g., consider a stock whose ending value is distributed with mean 30 and standard deviation 1, and the valuation of an option to buy that stock at 31.

If the distribution is p(29) = .5, p(31) = .5, the option is worthless.
If the distribution is p(28 ) = .125; p(30) = .75, p(32) = .125, the value of the option is .125.

[retaker]

I am assuming that they meant the standard deviation of the continuously changing exchange rate so that you can use Blacky Scholes?


This is what they said the answer is:

P=1.35/1.07 P/PV(EX) =1.05/1.07

d_1 = ln(1.05/1.07)/.2 + .2/2

d_2 = ln(1.05/1.07)/.2 - .2/2

Call = P*N(d_1) - PV(EX)*N(d_2) = .09

I thought that to find the PV of the price, since the price is actually an exchange rate, you would use the interest rate parity in reverse, ie:

PV(EX) ie. (prior exchange rate) = ($1.35/1.05)/(1 foreign/1.07)
= 1.35*(1.07/1.05)

[Gandalf]

I haven't read any of the Course 2 texts, and had to look up formulas in a book by Panjer to see how to apply Black-Scholes to a common stock, so this may be off the wall.

However, suppose I tried to apply the common stock formulas to this. I would be trying to get the price for an option to buy 1 share of common stock at time 1. That's analogous to 1 unit of foreign currency at time 1.

What is my price today for 1 unit of foreign currency at time 1? The way to have 1 unit of foreign at time 1 is to buy 1/1.07 today, which costs me 1.35 * (1/1.07) = 1.2617. That's P.

The option lets me acquire the foreign currency for 1.35 of my currency in one year. I need 1.35/1.05 of my currency today to have 1.35 of my currency then. That's PV(EX) = 1.2857.

Using those, with your values for d_1 and d_2 which I think are certainly right, gives the option value of 0.09.

I'm sure I've never seen any study material discussing application of Black-Scholes to exchange rates, so trying to do this problem as if the unit of foreign currency is equivalent to a share of stock may be absurd.

[retaker]

"The option lets me acquire the foreign currency for 1.35 of my currency in one year. I need 1.35/1.05 of my currency today to have 1.35 of my currency then. That's PV(EX) = 1.2857."

I like this part, but don't they already give you the price of the foreign currency at the current time to be $1.35? Why do you need to discount back? Even if you do need to figure out the price of the foreign currency in the previous period, wouldn't you use the interest rate parity:
($1.35/1.05)/(1/1.07) ?

[Gandalf]

If there's not a similar example in the text, or at least a discussion of options on exchange rates, I would be amazed to see a question like this on the exam.

It's hard for me to put it into Course 2 terms, since (1) I haven't read the Course 2 texts (2) my understanding of Black-Scholes is very weak anyway.

I reasoned that Black-Scholes is based on markets making investors neutral about purchasing the real financial instrument or an option to acquire the same quantity at a set strike price. With stock options, that would usually be one share on a given future date. To buy the real instrument, you would also buy one share today.

With this foreign currency, the option is per 1 unit of foreign currency at time 1. If you buy 1 unit now, you will have too much then. You need to buy 1/1.07 now to have 1 then.

So I'm not discounting, interest-rate paritying, or anything else the price. The price is 1.35 per 1 unit. The price I'm paying is 1.35/1.07 because I'm buying 1/1.07 units. (E.g., if they had told you that at inception, $135 local buys 100 foreign, you wouldn't consider using P = $135, because you don't want 100 foreign. My "solution", which may or may not be right, says you don't want 1 foreign, either.)

I can't tell from what you typed what the authors of the problem thought the numeric value of P and PV(EX) should be.

Good luck.

[retaker]

Thanks