Option problems

knowledgebone · Dec 10, 2003

Does anyone here know any probability problems/questions that involve options. For example, how much would you pay for the option to roll the dice one more time, pick another lottery ticket, etc. I have been looking for problems like this on the internet, but I am coming up short. Anything you guys can provide would be much appreciated. I just want to know how to set them up and solve them? If my question is not clear, let me know. Thanks guys.

Knowledgebone

StillTrading · Dec 17, 2003

I'm not sure if I'm following exactly what your asking about, but still if your looking to price options, particularly American Options I'd do a little research on binomial tress (either the Cox, Ross, Rubenstein or Jarrow and Rudd models) or monte carlo simulations. Another thing to note, is that probability is meaningless in terms of options analysis. Expectancy is much more important. Let me give a brief example, which comes from Nassim Taleb's book fooled by randomness. Lets say there is a .9 probability that a trade will make $1 in profit and .1 probability that the trade will lose $100. While there is a strong probability that if this is done once that you will profit, but the expectancy over time is negative -9.1 ( = .9 * 1 + .1 * -100), hence a losing strategy. I guess this roughly equates to selling OTM options naked. Where you might do that for a very long time profitably its very likely to cause a blow up somewhere down the road. Hope this helps a little.

sle · Dec 17, 2003

Here are a few that I ask at the interviews:

a) You are offered to play the following game - you throw a coin and your payout is number of heads/number of throws when you stop the game. You can exit at any time, but you can choose to play the game forever. How much would you pay to play this game?

b) Todays stock price is $80, the interest rate is 0. How much would you pay for a perpetual american digital option paying $1 when the stock hits $100?

c) What's the price of the down-and-out option on EUR/USD where the barrier and the strike are 1.2?

When you get though these, i can add a few more.

StillTrading · Dec 17, 2003

Quote from sle:

Here are a few that I ask at the interviews:

a) You are offered to play the following game - you throw a coin and your payout is number of heads/number of throws when you stop the game. You can exit at any time, but you can choose to play the game forever. How much would you pay to play this game?

b) Todays stock price is $80, the interest rate is 0. How much would you pay for a perpetual american digital option paying $1 when the stock hits $100?

c) What's the price of the down-and-out option on EUR/USD where the barrier and the strike are 1.2?

When you get though these, i can add a few more.

ok for giggles I'll bite on A (b an c require thinking and Im not about to start doing that today

). Id pay < .50 which is what I'd expect to make if i played the game forever.

sle · Dec 17, 2003

Quote from StillTrading:

ok for giggles I'll bite on A (b an c require thinking and Im not about to start doing that today ). Id pay < .50 which is what I'd expect to make if i played the game forever.

If you play the game forever, you expect to make $1 dollar - any barrier will be hit eventually, right? However, the answer is correct for a different reason - to hedge such an option I would buy X shares of stock so it would pay $1 dollar when the stock hits $100, which is 1/100 shares and todays price of that (barring transaction costs) would be 50c.

StillTrading · Dec 17, 2003

it seems like your talking about question b right? Because the payout of question a number of heads / number of throws def. approaches 1/2 the longer you play.

StillTrading · Dec 17, 2003

Quote from sle:

any barrier will be hit eventually, right?

Im not sure I agree with that assertion. And if that assertion is not true, how does that affect what you would be willing to pay for the perpetual option?

sle · Dec 17, 2003

Quote from StillTrading:

it seems like your talking about question b right? Because the payout of question a number of heads / number of throws def. approaches 1/2 the longer you play.

No, I was talking about question a. The answer to question B is kind-of tricky, i will post it later, but I can tell you the price for question b: 79c .

sle · Dec 17, 2003

Quote from StillTrading:

Im not sure I agree with that assertion. And if that assertion is not true, how does that affect what you would be willing to pay for the perpetual option?

Well, as for assertion - in conditions of non-zero volatiltiy (i should have added that to the conditions), any barrier will be hit if you hold the options perpetually. The only other condition i would add is that we are talking about default-free world. But even the assertion was not true, the hedging argument still stands, so it still would be 50c .