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The Smokin' Cracktuary
04-06-2008, 07:16 PM
I am trying to figure out their use of B-S in the solution part b)

they have 130[FN(d1)-KN(d2)]*exp(-r) K=F=Exchange rate

This has me a bit confused. I see that this essentially means at the money call with strike & initial value = 100.

But why is the exp(-r) outside the brackets?

Ok. It may have just hit me. But I'll run it by you anyway.

For currency options q=foriegn risk-free. Here we are just assuming that equals the domestic risk free rate, hence the strike and the initial value will both be discounted by exp(-r).

When doing this problem I was looking for the foriegn risk free rate, and I (obviously mistakenly) assumed it was the 5.5% that the counterparty could borrow. Mainly due to lack of any other options.

I am not sure why they think it's ok to just assume that the foriegn risk free rate is also 5%.

Another thing that chaps my a\$\$ about this problem is that the estimate the default probability from the spread this guy must borrow at, which is all well and good accept that if this jerk is from a foreign country, how do we know that borrowing rate spread isn't off his domestic risk-free rate?

Implying that we don't really know what his credit spread is, and therefore cannot estimate his default probability.

I suppose this is just one of those places that you have to go and assume, and be sure to tell the graders that we assume, that the borrowing rate given for this guy is based on domestic rates and/or the foreign country's risk free rate is the same as ours.

Or we could assume that this is a pure financial transaction by two Americans just playing the exchange market, hence the borrowing rate would be domestic. However, we'd still have to assume that the foreign currency's risk free rate is the same 5% as ours.

Which just seems stupid when asking a question about a currency futures contract. How effing hard would it be to just say that, and prevent idiots like me from making stupid assumptions about this guy being from another country and being able to borrow at the risk free rate in his homeland.

Ok. I am done.

Other than that, everything is going well. Thanks for asking.

BTW. I am in my 8th hour of studying today (18th of the weekend). Hopefully that helps explain my cheery disposition.

Laurelinda
04-06-2008, 10:58 PM
Hey Smokin', you're right, it's weird. For one thing, the problem is set up to be about a forward contract, not a call option. If they had asked for the valuation of a normal forward contract and not a currency call option, q would have been the domestic risk free rate. It honestly looks like a screw up, to me, and you'd have to state your assumptions for the grader the way you suggest. :shrug:

Car'a'carn
04-06-2008, 11:45 PM
You both are off a bit. The call option formula is:

c \ = \ S_0exp{-r_fT}N(d_1)-K exp{-rT}N(d_2)

Since

F \ = \ S_0 exp{(r-r_f)T}

we get the formula in the solution.

The Smokin' Cracktuary
04-07-2008, 11:12 AM
You both are off a bit. The call option formula is:

c \ = \ S_0exp{-r_fT}N(d_1)-K exp{-rT}N(d_2)

Since

F \ = \ S_0 exp{(r-r_f)T}

we get the formula in the solution.

Yeah. I forgot the notion here. I was thinking they were just using:

F \ = \ S_0

Which in this case it does, which is why I thought that, but (F) does have it's own notation meaning as you mentioned.

In any case, what I said does still stands regardless. You don't need to know that

F \ = \ S_0 exp{(r-r_f)T}

to do this, or any, problem.

F \ = \ S_0 exp{(r-r_f)T}

Drops out from that fact (i.e.- r_f=q). Just as with a dividend paying stock will decrease in value at a rate of (q) while expected to grow at the risk-free rate (r), making the effective growth:

S_0 exp{(r-q)T}

This could be called the "forward stock price at time = T".

This is equivilent to saying that our currency is expected to grow at the domestic risk-free rate, and the foreign currency will grow at a rate the foreign risk-free rate, making the value of our money in relation to thiers (i.e. factoring in expected exhange rate change) at time = T equal to:

S_0 exp{(r-r_f)T}

Which is the forward price at time = T.

It is just the concept that the value of the underlying variable is growing at the risk-free rate, but decreasing in value at some other rate for some reason. Making the effective expected growth the difference.

It is a by product of the fact that r_f and q are effectively doing the same thing in the formula.

In any case, what I said stands, and the only reason their solution looks the way it does is because the foreign currency risk-free rate is assumed to equal the domestic risk-free rate, making:

F \ = \ S_0

At the very least they would have added a line calculating (F) in the solution if this weren't the case.

Which seems to be a stretch to think you can make the assumption that the risk-free rates are equal when using B-S for the intended purpose, given that r_f is a primary input.

Car'a'carn
04-07-2008, 11:28 AM
So what if the problem states that the foreign risk free rate is US risk free rate plus 1%?

Hint:

Solution is the same.

The Smokin' Cracktuary
04-07-2008, 12:02 PM
So what if the problem states that the foreign risk free rate is US risk free rate plus 1%?

Hint:

Solution is the same.

Except that they would put in a line calculating what F equals, as I said. Now the solution says is F=K=.76923, which is only true because the risk free rates are assumed equal.

So no, the solution would actually not be the same since F would not equal K.

Car'a'carn
04-07-2008, 12:09 PM
Except that they would put in a line calculating what F equals, as I said. Now the solution says is F=K=.76923, which is only true because the risk free rates are assumed equal.

Or because the problem says to calculate at the money option.

So no, the solution would actually not be the same since F would not equal K.

You still have plenty of time to review Hull.

The Smokin' Cracktuary
04-07-2008, 01:42 PM
Or because the problem says to calculate at the money option.

You still have plenty of time to review Hull.

This works best with an example.

Calculate a T-year at the money call option on stock worth \$50 and a constant dividend yield of 2%. Risk-free rate = 5%

Strike = 50 = K (at-the-money)

You have two options to account for the dividend.

1.) Use: [d_1] = (ln(S_0/K) +(r-q+SD^2/2)T)/SD(T^1/2)

Initial stock value equals S_0 and q is accounted for by reducing the risk-free rate in d1.

or,

2.) Use: [d_1] = (ln(S_0exp^(-qT)/K) +(r+SD^2/2)T)/SD(T^1/2)

Assumed initial stock price equals initial stock price minus expected dividends until time T,
S_0exp(-qT) and the risk-free rate is unadjusted in the formula for d1. Making the call price.

S_0exp^(-qT)N[d_1] - Kexp(-rt)N[d_2]

Here they use the fact that F = S_0exp(r-r_f)T

The just factor out exp(-rT) leaving,

exp(-rt)(S_0exp^((r-q)T)N[d_1] - KN[d_2])

or

exp(-rt)(FN[d_1] - KN[d_2])

Clearly the stike price (K) doesn't depend on the amount of the dividend (q). It will be \$50 regardless.

(F) however, clearly does depend on (q). Here,

F = 50exp(.03*5) = 58.09

Say we change (q) to .03, then:

F = 50exp(.02*5) = 55.25

Looks different to me. Hmmm, the strike price is still at the money at \$50, but (F) doesn't equal \$50. Hmmmm. But that would suggest that (F) doesn't always equal (K) for an at-the-money option. In fact that would suggest that it depends on the expected dividend yield.

How on Earth can (F) ever equal (K)?

Oh!!!!! I know, I know!!!

Lets make the dividend yield (q) equal the risk-free rate (r).

So here (r) = (q) = 5%. That means,

F = 50exp(0*5) = 50

Well \$50. That sure seems to me like it equals \$50. What a coincidence, that's what (K) equals. I know (K) equals \$50 because it is an at-the-money option and the current stock price is \$50. That's totally wierd. It seems that the only way (F) could ever equal (K) is if the dividend yield equals the risk-free rate. I did not know that.

Now, how could I apply this same concept to currency options?

Hint:

[tex]r_f = q

The Smokin' Cracktuary
04-07-2008, 01:57 PM
Man. I don't understand this at all. I need to be reading the Hull right now.

TiderInsider
04-08-2008, 03:03 PM
Don't lose sight of the forest. Tell this tree to f itself and start studying something else.

Laurelinda
04-09-2008, 12:35 PM
Well \$50. That sure seems to me like it equals \$50. What a coincidence, that's what (K) equals. I know (K) equals \$50 because it is an at-the-money option and the current stock price is \$50. That's totally wierd. It seems that the only way (F) could ever equal (K) is if the dividend yield equals the risk-free rate. I did not know that.

Smokin', I just reread the problem. Part (b) says calculate the value of an at-the-money call, so both of us were thinking, hey, S0 has to equal K, but who knows what F is? F could be anything, depending on the foreign risk-free rate. But then I saw that one of the premises of the problem setup is: "Your company enters into a one year forward contract to sell 100 U.S. dollars
for 130 Euro in New York. You are given the following: contract is initially at-the-money". So in other words, we've been told ahead of time that F = K as well as S0 = K. So yeah, because F = S0, r = rf. But we didn't have to know that r = rf.

Maybe that's what you said in your last post above...I got kind of lost reading it. :oops: At any rate, I think this makes sense to me now. :tup:

The Smokin' Cracktuary
04-09-2008, 02:16 PM
Smokin', I just reread the problem. Part (b) says calculate the value of an at-the-money call, so both of us were thinking, hey, S0 has to equal K, but who knows what F is? F could be anything, depending on the foreign risk-free rate. But then I saw that one of the premises of the problem setup is: "Your company enters into a one year forward contract to sell 100 U.S. dollars
for 130 Euro in New York. You are given the following: contract is initially at-the-money". So in other words, we've been told ahead of time that F = K as well as S0 = K. So yeah, because F = S0, r = rf. But we didn't have to know that r = rf.

Maybe that's what you said in your last post above...I got kind of lost reading it. :oops: At any rate, I think this makes sense to me now. :tup:

Got it.

So, short answer: (r) must equal (rf) because the forward contract is at the money.

That was the missing piece.