Remembering Curreny Option Duality

[Time_value_of_Sex]

Regular Option duality intuitively makes sense to me because you can just think about what security your selling and buying and the payoff of the option you are considering.

Currency option duality, however, really does not make sense to me. For clarity, currency option duality says this:


C$ (Y, K, T) = Xo K PY ( $, 1/K, T)


Is there an easy way to think about this without just memorizing it?

[sjb554]

There is nothing easy about this ever!! I promise! The duality you describe is extremely helpful...but also confusing. The only solution is problem after problem after problem. you will have 2 to 4 questions about this.

[mathmajor]

Originally Posted by sjb554

There is nothing easy about this ever!! I promise! The duality you describe is extremely helpful...but also confusing. The only solution is problem after problem after problem. you will have 2 to 4 questions about this.

I wouldn't say 2-4... maybe 1-2. I never really learned this and just sort of guessed on the exam. If you learn it well, it will be an easy point or two.

[appana]

It's actually not different at all. If you think about "normal option duality", a call option for one person is a put option for another.

If you translate this into currencies, the Xo and K just appear because you need to convert from one currency to the other.

If you remember the rules for exchange options, currency option duality is just a special case.

[SliceApproximateIntegrate]

At some point (hopefully), you'll have that aha! moment when you realize that stock, currency, and exchange options follow the exact same rules. You'll have way less that you feel you have to memorize, and it's awesome. Until then, keep staring/asking questions until it makes sense. (I had my oh! I get it! moment in ch. 14 of ASM with gap options...)

But to break down this particular question (in hopefully a way that makes more sense) ...

The left hand side of the equation with the call means that at time T, if the spot price is greater than the strike price (ie X > K) , you'll pay $K in exchange for $Xo.

The put on the right hand side means that at time T, if 1/X < 1/K (or equivalently, X > K), then you'll sell 1/Xo and receive 1/K.

So we have the same conditions for when these options have a payoff (I mean that they both only payoff if X > K at time T), so we should have some equivalence otherwise uh oh ARBITRAGE!!! So let's show that before I get some crazy ideas on how to make risk free money.

In the call, we pay K and receive Xo. In the put, we pay 1/Xo and receive 1/K. If we multiply our payoff for the put by Xo*K (ie buy Xo*K puts), then we pay Xo*K/Xo = K and receive Xo*K/K = Xo. So the call is indeed equivalent to Xo*K puts as desired.

I hope that made more sense? QED, maybe?

[waterborne]

As sjb554 said, expect 2-4 questions on this. There may be 1-2 questions that specifically ask about this but expect a couple more where this is an implicit part of the problem. I just memorized the formulas and knew when to use them and it served me well.

[Time_value_of_Sex]

@ Slice

Yes, I think I might have to read it a few more times before I can explain it to myself, but thanks for showing me a way to reason through it myself.

By the way, why are you going right from MFE to C without finishing M?

[SliceApproximateIntegrate]

So, one of my coworkers took MLC in May. He said he didn't good about the test because he studied from an old MLC book, and the syllabus change was significant. Another one of my coworkers (who also just passed MFE in April) figured it would make sense to let the SOA iron out what they wanted the test to be and let authors adjust their manuals to the new syllabus.

Not that you can tell from my previous post, but for someone who is trying to become an actuary, I'm kinda a slacker. I finished my once through the MFE material 8 days before my exam day. That was a really fun week, you might guess.

Anyway, he's a MUCH better student than me. He's THAT guy who convinces you to join the study group and go over practice questions, meet at a library and do a practice exam in real time, is well ahead of schedule on studying, asks questions here on AO, etc. Plus, he's really good for asking questions like "What manual should I get?" "When do we have to register by?" "Should I get rid of my iPhone in exchange for Boost mobile?"

The answer to the latter is clearly no, heck no. So basically, he's jumping off the C bridge, and I follow people off of bridges.

[waterborne]

Given the choice between C and MLC I would definitely go with C. It is a bad time to take MLC as the SOA clearly stumbled with the last exam.

[MusaAli]

Here's how i do it:
I just focus in what EXCHANGE is taking place. like, a dollar denominated call on euro means dollars are given for a euro. and a euro denominated put on dollar means a dollar is sold (given) for euros. So u see? the same exchange! so they are the same option, BUT to calculate the dollar price of the call from the eurp price of the put, u have to make two changes:

Firstly, let's suppose the stike is 1.3 dollars. so in the call, u give 1.3 dollars for the euro. but in the put, u'd be selling only ONE dollar ryt? so we need to make that 1.3 dollars too. so we multiply the euro price by 1.3 (the dollar strike). Now both portfolios give 1.3 dollars.

But the price of the put is still in euros. so we convert it to dollars by multiplying it with the the current conversion rate. and tada! u got the dollar price of the call.

I know it looks longer this way, but if u read it slowly once or twice and understand the simple reasoning, and then solve a problem or two, you'll get the hang of ti =)