underlier terminal spot-price pdf --> option value

In any analysis of derivative valuation, we are interested in the the possible valuationS of a security at a given time. Suppose an IBM $190 option expires 22 Dec 2014, we want to know something about the possible price level on that day. We use a random variable ST to Denote S(t=T) i.e. the underlyer price at time=T. ST might be 180, 200, or 230 or whatever. (Actually IBM is quoted to 2 decimal places;-) However, as a continuous random variable, ST can be any value between 0 and 10x current price, or higher.

To keep things simple, we first look at the likelihood of ST falling below 150, between 150~200, 200~250, and beyond 250. By intuition, the probabilities of hitting these 4 "buckets" or ranges must add up to 100%.

That's too coarse. Let's divide into $1 buckets from 0 to $2000. We end up with 2000+1 ranges (including a special "above $2000" bucket). Say our smart computer model estimates that chance of ST falling into $200-$201 is 5 bps or 0.05%. So we draw a vertical bar of height=5; width=1/10,000 over the 200-201 range. Suppose the 201-202 probability is 3 bps, we draw a bar of that height. Iterating over our 2001 ranges, we get 2001 bars. Total area of the bars add to 1.0 [1]. Your first histogram! When the range size becomes infinitesimal, histogram becomes a pdf curve -- the beautiful lognormal bell curve.

Other posts in this blog discuss how to derive the exact pdf (prob density function) of the random variable ST from the Basic assumption

Once we have a pdf of ST, current value of an European call (before expiration) is tractable. Since the terminal value is a hockey-stick payoff function, we multiply the pdf by a piecewise linear function, and find area under the curve. This is abstract. Let's use histogram to illustrate.

Suppose our smart computer simulates 10,000 trials. 5 outcomes should fall into 200-201. Payoff = $200.5-$190 = $10.5. Similarly, 3 outcomes fall into 201-202, with payoff =$11.5. Roughly half the outcomes probably fall below $190 -- worthless. If we compute the average payoff, we might get something like $11.11. This depends on the sigma used in the 10,000 simulations and time to expiry. We assume 0 dividend and 0 drift.

[1] In fact, the bar of 5 consists of 5 minibars, each 0.0001 wide and 1.0 long. There are exactly 10,000 minibars in the histogram representing 10,000 trials.