Now that you have a good intuitive understanding of what a call and put option on a stock are, and how to use them, let's think about how to price them.
When building a model for options pricing, there tend to be a set of convenient assumptions made to simplify the process. First, there are no transaction costs or taxes. Secondly, you can buy or sell as much of any stock/option without altering the price. Thirdly, volatility (we'll get to that later) will be known and constant. I think that's all I'll need for this lesson.
Certainly, the value of the option is determined by the value of the stock at some given point in time. If the option is European, then the only relevant value is the price at the end of the time period.
Suppose we have a stock at price S0 currently, and there is a European call option expiring at time 1 with strike price K = S0. Now, in this particular oversimplified world, the stock can only take on two values at time 1: Su and Sd, standing for an "up" movement and a "down" movement. Since the option is at strike K = S0, the option will pay (Su - S0) at Su and zero at Sd. We'll denote these values by Cu and Cd respectively (representing the value of the call at an up movement and the value of the call at a down movement).
The price of the option can then be calculated as
Cu * P(up movement) * v
Where v is the present value factor to discount the payoff with interest back to time zero. We'll need an interest rate to do that, which we'll call r. We'll also need P(up movement) (the probability of an up movement) to calculate the price.
Through some mathemagic which I think I'll gloss over for now (but if you all want to see it, I'll happily spell it out), a probability which correctly calculates the price can be found using this formula:
P(up movement) = (e(r-d)*h-d)/(u-d)
Where r is the continuously compounded rate of return, d is the continuously compounded rate of dividend payments, h is the time period (in years), d is the multiplicative factor by which S can decrease over h, and u is the multiplicative factor by which S can increase over h.
There are also formulas for u and d, but if you need those I suggest reading this.
With all of that in place, there is now a formula for pricing a call option where the stock has only two movements, up or down. This model can be expanded to include more periods, use discrete dividends (say the stock pays $10 at time 0.75), handle American options, and do a variety of other nice tricks. However, the more important fact is that this is the backbone of the famed Black and Scholes model, which I will discuss in Lesson 3.
I hope you can see how this all gets very hairy mathematically very quickly. Some of the background I jumped over is done not by mathematical proof, but by economic logic, which may not sit well with some mathematicians. I have also tried to simplify a lot of this to be read by the general audience, whereas for the last four months I've been studying the specifics of this, and more complex models, extensively. If any of you readers are curious about this topic on a deeper level, feel free to post here, IM me in world, or email me at guardian.market@gmail.com. I can't guarantee I'll know the answer (it's a big world out there with options!) but I'll try to at least point you in the right direction.
