Noobie question

Tony Optionaro · Feb 27, 2020

Hi, my professor asked this at my Uni on a exam today, and I'm not sure if I did it the right way.
So the question goes: Price an European ATM call option using the following distribution (picture). Here's the known variables. Stock is trading at 14,5 hence ATM option strike K=14,5 as well. r= 0,0014, T= 0,5 (six month maturity), no dividends. The distribution is the probable odds of the stock price for the next two months from now. We didn't get to use the standardized norm.distribution sheet as a tool. I got the price by calculating the EV and using it to derive the volatility of the stock, and also using a rough estimation of N(d1)=0,5 because ATM option should be close to delta of 50. However, getting the exact number required memorizing the normal standardized distribution sheet, and I don't think I was supposed to solve this using the B&S-model.
So what am I missing here? Am I supposed to solve this just using the PCP?
Thanks.

Tony Optionaro · Feb 27, 2020

I wasn't able to post the pic for some reason, so the distribution went like this:
Stock price (Probability)

14 (10%)
14,5 (16%)
15 (41%)
15,5 (25%)
16 (8%)

Tony Optionaro · Feb 27, 2020

There is no way of getting an accurate number if you can’t use B&S. You could calculate top and bottoms bounds for the call, but that would be just.. stupid. If you calculate the IV, you would still need the N sheet to calculate N(d1), you can estimate delta to be 50, but again, no use if you can’t use B&S. So some guru here, prove me that either I’m wrong or my Prof is high on PCP (no pun intended).

pablosss · Feb 27, 2020

Drop the University and the teacher

TommyR · Feb 28, 2020

I would like to offer one idea. In probability what you usually try to do is write one random variable in terms of another random variable. So for example the s and p could be written as the sum of all its parts but you don't know what the probability of it being a particular value is. you know the identity will hold over all values it takes. If you hear people discussing the probability of a recession being 40% or the distribution of the returns of a stock being normal you know its bullshit. So in options the idea is that you are trying to write the payout of the option in terms of the stock so that they always agree no matter which value the stock takes.

ironchef · Feb 28, 2020

TommyR said:
So in options the idea is that you are trying to write the payout of the option in terms of the stock so that they always agree no matter which value the stock takes.

I assume as MM, that is what you do?

And, aren't BSM, binomial attempts to do that? And, if you do not assume a distribution function of the stock, how do you do that?

Thanks.

Tony Optionaro · Feb 28, 2020

TommyR said:
I would like to offer one idea. In probability what you usually try to do is write one random variable in terms of another random variable. So for example the s and p could be written as the sum of all its parts but you don't know what the probability of it being a particular value is. you know the identity will hold over all values it takes. If you hear people discussing the probability of a recession being 40% or the distribution of the returns of a stock being normal you know its bullshit. So in options the idea is that you are trying to write the payout of the option in terms of the stock so that they always agree no matter which value the stock takes.

Thanks Tommy!

Thing is, the prof wanted one exact price. I think he just forgot to provide us with the N sheets..

Kevin Schmit · Feb 28, 2020

Tony Optionaro said:
the distribution went like this:
14 (10%)
14,5 (16%)
15 (41%)
15,5 (25%)
16 (8%)

You don't need "N sheets" to approximately price an ATM call under BSM. You should know that the price of an ATM call ~= phi(0) * sqrt(T) * v * S. where phi(0) = .3989

sqrt(T) * v (root time vol) from the discrete distribution above, in logs, is ~.05, which you should be able to derive on a hand calculator in a few seconds, with pencil and paper in less than a minute.

Alternatively you can price it under the Universal Law of Asset Pricing (terminal expectation discounted back to the present).

Either answer is correct, under different measures and numeraires.

Also, if you rely on TommyR for advice, you're doomed. You're going to fail the course.

ironchef · Feb 29, 2020

Kevin Schmit said:
You don't need "N sheets" to approximately price an ATM call under BSM. You should know that the price of an ATM call ~= phi(0) * sqrt(T) * v * S. where phi(0) = .3989

sqrt(T) * v (root time vol) from the discrete distribution above, in logs, is ~.05, which you should be able to derive on a hand calculator in a few seconds, with pencil and paper in less than a minute.

Alternatively you can price it under the Universal Law of Asset Pricing (terminal expectation discounted back to the present).

Either answer is correct, under different measures and numeraires.

Also, if you rely on TommyR for advice, you're doomed. You're going to fail the course.

What is phi(0) and why is it = .3989?

ph1l · Feb 29, 2020

ironchef said:
What is phi(0) and why is it = .3989?

It appears to be the probability density function value of a standard normal distribution (mean == 0, standard deviation == 1).
http://mathworld.wolfram.com/StandardNormalDistribution.html