Sunday, January 11, 2009

Sunday Reflection: A Theoretical Model for the Value of Derivatives

I was developing a model for understanding prices of options based on the betting system at a race track.

Consider European options (because they are easier conceptually and can only be exercised at expiration) on a stock selling at $50 with an excise price of $50. Assume a call is $5 and a put option is $2.

Also assume that the standard deviation for the stock is $2.

Race tracks can guarantee their profits because they will adjust the ratios in such a way that no matter who wins the race, the race track will take profits on all bets. But for simplicity sake, let's say that we can analyze this system as a snap shot in time first (an assumption we can remove later to strengthen our system).

First, let's say that for every call option you sell a put option (another assumption we can relax later).

So you sell 1 call for $5 and 1 put for $2 for a total premium of $7.

Note that with a standard deviation of 2, the following probabilities are likely (majorly oversimplified, but if you want to crack open a spreadsheet to do this properly, be my guest). Each set is one interval or between standard deviations:

E1 - P(X>56) = Negligible
E2 - P(56>X>54) = 2%
E3 - P(54>X>52) = 13.5%
E4 - P(52>X>50) = 34%
E5 - P(50>X>48) = 34%
E6 - P(48>X>46) = 13.5%
E7 - P(46>X>44) = 2%
E8 - P(X>44) = Negligible

However, because of the options, the relative values of each event are:

E2 - Ct = 50 - 56 = -6, Pt = 0, prem = +7, VE2 = -6 + 0 + 7 = +1
E3 - Ct = 50 - 54 = -4, Pt = 0, prem = +7, VE3 = -6 + 0 + 7 = +3
E4 - Ct = 50 - 52 = -2, Pt = 0, prem = +7, VE4 = -6 + 0 + 7 = +5
E5 - Ct = 0, Pt = 0, prem = +7, VE5 = +7
E6 - Ct = 0, Pt = 48 - 50 = -2, prem = +7, VE6 = -2 + 0 + 7 = +5
E7 - Ct = 0, Pt = 46 - 50 = -4, prem = +7, VE7 = -4 + 0 + 7 = +3

So the total expected value is:
E1 = 0
E2 = +1 x 2% = 0.02
E3 = +3 x 13.5% = 0.405
E4 = +5 x 34% = 1.7
E5 = +7 x 34% = 2.38
E6 = +5 x 13.5% = 0.675
E7 = +3 x 2% = 0.06
E8 = 0

E(R) = SUM(E1:E8) = 5.24

Notice a few key things... There are no events with a negative return (you are guaranteed to make profits) and for selling two options, you are expected to make $5.24 per pair.

Now, let's slowly start to remove some key assumptions and point out a few key points.

1. You can't actually sell (a large volume) of options (in pairs) to make this kind of profit. Usually, call options will be MUCH more popular than puts.

I simulated the above results and instead of having them in pairs, I tried saying that if a call is $5 and a put is $2, then a call is 2.5x more popular than puts ($5/$2). A HUGE assumption, but it turns out the expected value is still positive (although lower... Where as you are expected to profit $5.24 per $7 pair of premiums you sell, I worked it out to about E(R) of $2.10 per $20 of premiums you sell - in other words, still profitable, but less so... which makes sense). It doesn't really much ratio of sales of Call to Put you use as long as you calculate the Ct and Pt properly to understand your potential loses in all scenarios. But, obviously, as you start relaxing your assumptions to reflect "real life" your margins get smaller to reflect investor behaviour. GIGO.

Also to note, in the model where you sell more of one option than another and the underlying asset of that option suffers from extreme deviations from the expected mean, you can lose a LOT of money because the premiums from one type of option don't nearly cover the losses from the other.

2. For a stock trading at $50 (or there abouts) with an excise price of $50 (or there abouts) the Call will cost $5 and the Put will cost $2. Going back to Put Call parity, reducing arbitrage and finding the intrinsic / time value of options, you can adjust these values accordingly, however, they will directly affect your expected value. As you can see, the premiums carry directly to the "bottom line" (expected returns) proportional to the number of options sold.

3. You don't have to know the exact price of the stock nor be certain if it will be above or below that price any degree of certainty. You only need to know approximately where it will be (know the standard deviation).

4. You can make even more money if you take the premium and invest it in at the risk free rate (reinvestment in a T-bill) over the holding period.

5. You didn't actually invest any money. As long as you have money to cover "potential losses" you are fairly safe.

6. Every time you sell an option, you can raise the price to increase the spread to reflect demand. Of course, if the price rises too fast, you'll limit the multiplication factor to leverage your expected return.


Azrael 38 said...

Hey Josh, I was google'ing again (this time not for MY name xD) and I ran across your blog. Pretty amazing experience in the middle east I bet! Anyways, I was reading through your blog and I ran across this article. Before your analysis of probabilities and whatnot, you mentioned that the call premium is $5 while the put premium is $2. After analysis of probabilities, you expect a profit of $5.24 per pair sold. However, I would believe no such opportunity will ever arise due to call-put parity (or w/e I understand it to be). With a $5 call and $2 put, I would personally be arbitraging by buying a lot of shares, selling a call, and buying a put for a net profit (GUARANTEED, 100%) of $3 per share for orders executed this way. Talk to you on MSN soon.

Joshua Wong said...

Good call. This was an idea I was writing about when I was reading the CFA and I was trying to price options. Turns out that this is a very rudimentary (incomplete) version of the Black-Scholes model.

Also, if people have good information: as you say, there should be call-put parity to prevent arbitraging.

I was trying to say that, but may not have articulated well. I was just excitedly writing about how I would price an option if I had to.