Stock price is $15 today. In a year, it has 50%-50% chances of going up or down. If it goes up, then 80% probability of the price being $25. If it goes down, then there is a 40% probability of price being $5 and a 60% probability of it being $0. Find the price of a call option at strike $25.

My approach:

I think I should, first of all, find the value of the stock on the up branch of the tree that happens with a probability of 0.2. I don't know though how to find such value.

I thought to include in the reasoning the formulas of $p,u,d$ of the tree, but it doesn't seem to help.

This is the tree I am talking about. Nevertheless, in this problem, the tree should have only one node so $n=1$, and since the maturity is 1 year, t should also be 1, $t=1$

  • 4
    Your question is?
    – Pete B.
    Sep 16 at 16:33
  • 1
    the "20%" case should be given to you - without it, there's not an answer. It's a big difference between the stock having a 20% change of being $15.01 and a 20% chance of being $100.
    – D Stanley
    Sep 16 at 17:50
  • You also need an interest rate, or discount factor of some sort.
    – D Stanley
    Sep 16 at 17:55
  • And, in the end, the value of the option is completely dependent on that value, since in the other three cases the option will be worthless.
    – D Stanley
    Sep 16 at 18:08
  • The options prices are given mostly by volatility and the real market - it "feels it". But if you want to calculate a theoretical value of the option with parameters given from your perspective, try a simplified formula like this one here: money.stackexchange.com/questions/97534/…
    – pedrouan
    Sep 22 at 12:11

If this is a theoretical homework problem then I have no clue what the answer is. In the real world, option pricing is based on six option pricing variables:

  • stock price

  • strike price

  • time remaining until expiration

  • dividend, if any

  • carry cost

  • volatility

In your example, the first five are known values today. Volatility is the wild card. Without it, you can't accurately price the option using an option pricing model to calculate fair value.

  • 1
    There are binomial option models that don't require an explicit "volatility" input, but model "volatility" as a discrete set of possible outputs and probabilities. The problem is that the question is missing one of those possible outcomes (the "20%" case if the stock goes up).
    – D Stanley
    Sep 16 at 17:48
  • Pardon my skepticism but IMO, that's all guesswork since the answer is based on the probability which in itself, is just another guess. Sep 16 at 18:00
  • You're right, and it's a purely academic exercise - in practice a singular vol is much more common, although there are binomial models that use different vols along the lattice based on time/moneyness.
    – D Stanley
    Sep 16 at 18:04
  • Although one could argue that even a single implied vol is purely a guess (but a guess based on market sentiment)
    – D Stanley
    Sep 16 at 18:04
  • Such theoreticals are above my pay grade. I suppose that they're useful for examinations but in my world, they don't contribute to any kind of a trading edge. As for the implied volatility, it's not a guess. It's an iterated derivative of option price. You could argue that option price is a function of sentiment but that's getting a bit far afield from from the meaning of implied volatility. Sep 16 at 18:48

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