Saturday, May 26, 2012

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

\frac{dS}{S} = \mu \,dt+\sigma \,dW\,
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.

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