Monday, July 31, 2006

Derivatives Primer

Extra-article comment:
Hello Friends. I am publishing this draft because I want to give to the loyal following a hint of the glut of articles for the blog I am working on, I haven't finished any because I am researching on many different subjects, I have been busier, and the articles are though, those that require extra effort and that I am proud of. Since this is a work in progress, I am not announcing it, but allow you to see it. Expect potentially substantial changes before I delete this notice

The Options Strategies Thread.

Part #1: A primer on Derivatives pricing.

Before describing the workings of the Angelwarewest-Asatru Skald-Knapper Tech-Chicagrafo money pump, it has become clear that the effort wouldn't provide much value to the audience if I don't explain the highly technical fundamentals of Stock Options and Derivatives first. This may be basic for some people, if that is the case, please, try anyway to read this article to post a comment that may be useful for the rest of the audience.

Words of caution are necessary: The Stock Market attracts the very brighest of the people able to understand the sophisticated concepts and issues of financial analysis; it is very tough already to be "smarter" than the market just picking with a modicum of consistency which stocks will go up and down; with options, where you multiply the risks by an order of magnitude and the technicalities are perhaps daunting to even bright investors, the chances of losing everything by getting into too deep waters are very high. It worries me deeply that I have friends and acquaintances in the message boards who regularly trade with options without thorough understanding of derivatives concepts. That is like playing with nitroglicerin. I guess that beginning with the fundamentals I will facilitate understanding of the dangers about hedging or speculating with options. If you think you know a lot about this subject, I reiterate the invitation to keep reading. If something surprises you, please, leave a comment. I myself discovered quite a number of misconceptions I had when I undertook the task of seriously studying options.

First, Derivatives:

I guess that those who trade options want to make money off them. Just as it is important to understand when a company may be over/under valued, it is important to be able to assess the right price of derivatives. In the case of options, it is also very important to understand their "wasting asset" nature: In the case of shares, if you are right about your evaluation of a company, you can go long/short as long as it takes for the market to agree with you; but with options, you not just have to be right, the market has to agree with you quickly enough, otherwise your options will be accumulating losses until they are either worthless if you went long on them, or costing you a fortune of you went short. Then, it is also very important to be able to accurately understand how the options time-decay. As if that wasn't enough complication already, in the case of shares, there is only one price, only one bid/ask pair; but in the case of options there are three more dimensions: Strike price, expiration, and role (whether to go long or short, they are not symmetrical), forcing you to compare among different competing views on the same underlying.

Replicating Portfolio

A Stock Option Contract is a "proxy" for the underlying, it is an instrument that allows you to control 100 shares. If you can do the same that an option allows you to with shares and cash, then the price of that setup (portfolio) is the price of the option.

There is a grave misconception rampant: The pricing on a derivative does not has to do with the expectations of the underlying to go up or not; it only has to do with how expensive it is to replicate a portfolio that simulates the gain curve of the derivative. Let me explain it this way: If AMD is going to go up almost certainly, with almost no chances of going down, that will not affect the price of the options, the price will be affected only if the share price itself, the underlying, goes up as a result of the expectations. That is, the expectations may only indirectly affect the derivatives prices if they are priced into the underlying.

To replicate a portfolio that will behave like the derivative you must take into account the cheapest interest rate, which is the risk-free, or the government bonds because goverments can not declare bankruptcy.

Without going further, a dummy example: Suppose a stock price currently traded at $100 will be priced either $125 (with probability 65%) in six months, or $80 (35% prob); and the stock will not have any other prices but $125 or $80. Suppose that the "risk-free" interest rate is 5% yearly.

Let's price a $100 strike price six months to expiration european-style call option on those shares.

No options-pricing calculator that I know of will tell us a price for that option because the shares are weird in the sense that they will have only two possible prices in two years, but we still can find a methodology to price them. Let's analyze the value of the option at expiry:
  • If the stock moves to $125, the value at expiration of the option will be $25.
  • If it goes to $80, its value will be 0.
Suppose that it is possible to replicate those cases with a portfolio of shares and cash. The price of the option would be the cost of replicating it with shares and cash. So, let's find how many shares and cash replicate that option:

X is the amount of shares, and Y is the amount of cash.

From the first case, that the price goes to $125, we have:

(1)X*$125 + Y*SQRT(1.05) = $25 (the cash will grow 5% per year, or SQRT(1.05) in six months)

From the second case,

(2) X*$80 + Y*SQRT(1.05) = $0.

Substract (2) from (1) and

(3) X*$45 = $25, thus X = 5/9 shares. And substituting, Y = -5/9*$80/SQRT(1.05) = -$400/(9*SQRT(1.05)) ~= -$400/(9*1.0247) ~= -$43.37

That means that going long on 5/9 shares and borrowing at risk-free $43.37 I will have a portfolio that will give the exact returns that the call option would give, thus, the call option price should be ~ 5/9*$100 - $43.37 ~ $12.19. Notice that it is irrelevant how probable the up or down case are.

Another example: A put option, strike $200, one year expiry:
Case it goes up to $125, the value of the put would be $75, thus (1) X*$125 + Y*1.05 = $75;
Case it goes down to $80 (2) X*$80 + Y*1.05 = $120
X*$45 = -$45 <==> X = -1, Y = $200/1.05 ~= $190.48:

Shorting one share and putting $190.48 to grow at 5% per year the portfolio gives the same returns on both cases, the portfolio has an initial value of $190.48 - $100 = $90.48, that is the appropriate price of the put option.

Risk-Neutral approach:

Do you see that as long as the replicating portfolio gives the same returns as the option in every case, then the probabilities of every case don't matter? If they don't matter, we can assign "probabilities" that suit our calculations better. Let's say that the expected return of one share is the risk-free rate, that is, $100 initially will become $105, thus, being p the "probability" of the stock going to $125, $125*p + (1 - p)*$80 = $105 <==> 45*p = 25 <==> p = 5/9. "Plugging" the probabilities into the put option cases, E(Pp) = 5/9*$75 + 4/9*$120 = $(375 + 480)/9 = $95, with the risk-free discount of the investment, dividing by 1.05, we have $90.48, the same price (!).

We have seen so far that

1) The derivatives are agnostic with regards to the expectations on the underlying,
2) the risk-free rate is essential to price the derivatives,
3) we may assign fictitious probabilities for the cases just assuming that the underlying will appreciate according to the risk-free rate, and those probabilities allow to price the options.

The next step to make the pricing suitable for real life options is to assign (fictitious) probabilities to the infinite number of cases for the evolution of a stock price. The intelligent reader may construct a procedure, although laboriously, to price options in cases in which the next price is a bifurcation: for every two contiguous final prices, you can price the option for the previous step, and use those values to price the preceding step, until the current period, that's the "binomial" method.

But in real life we don't have bifurcations but jumps of random magnitude. For practical purposes, we can assume infinite end prices. In that scenario, one case is a sequence of jumps and bumps. Just as it happens in many scenarios in which you have binomials and want to transcend to the continous analysis, the binomials transform into "Normals" (Gaussian Bells). Here, the number of "paths" or cases that end in a specific price may be modelled as following a "Normal Random Variable" distribution, but of course, not normal in the price of the stock, because there can not be negative prices, but in the logarithm of the relative price variation.

A stock traded initially at $10 that ends up at $100, appreciated 10 times, the (natural) logarithm of this relative variation would be 1, if it crashed to $0.1, 1/100 of the original price, -10, if it remained flat, 0. Taking the logarithm of the relative variation also conveys the true nature of investment: exponential.

To complete a description of infinite cases that we will take into account to price options, we may need a fifth ingredient (the others are Stock Price, Strike Price, Time to Expiration, Risk-Free rate), the volatility of the variations of the stock price

With those five ingredients, Fischer Black and Myron Scholes cooked "The Pricing of Options and Corporate Liabilities" in the excellent year of 1973 (because it brought "Yours Truly" to this world!), just a month (May, of which I am harvest of) after the Chicago Board Options Exchange opened. Both epoch-defining moments eventually enabled the whole market for options to be as liquid and developed as it is today.

The complication with the Black-Scholes model is that it requires differential calculus to be understood. As a matter of fact, if you don't have solid understanding of that subject, you would do better staying away from options because you won't be able to detect when the theoretical price is not adequate, won't be able to compare among different strike prices and expirations, and worst of all, you will be utterly incompetent to choose the right expiration times for the options that suit best your strategy.

If you object that the market will price the options for you, then you are very wrong. Not even with a stock so profusely traded such as AMD there isn't enough liquidity in the options market, thus the spreads are high and the prices are susceptible to distortion: The number of investors who regularly trade with options is insignificant compared with plain shares, and the options trader has not only a company to choose, but literally hundreds of options, so the already small liquidity is spread hundreds of times thinner. I don't want to deviate from the main subject, but I have made a bit of a living lately scavenging for minor distortions here and there in options pricing; thus I have first hand knowledge of how easily ignorant option players are losing their money, but not only that, I know that I am making mistakes due to incomplete understanding that the real "Pro"s are taking advantage of.

The next article will get into Black-Scholes to finish the Part #1

P.S.: This approach to explain options pricing was imitated from the book "Investment Mathematics" by Andrew Adams, Philip Booth, David Bowie, and Della Freeth; "Wiley Finance". I don't dare recommend this book, because I don't know nothing about books that cover this subject; while I was looking for a way to explain things about options, I just took the first book on the subject that crossed my path, which was this, and replicated the approach because it made sense to me. I learned in a very dis-advisable way: Googling every concept until I understood enough. If possible, don't make the same mistake.


Anonymous said...

Sorry to sound stupid, but what is this main point of this.

Anonymous said...

I second the first motion.


Eddie said...

The main point of this article, of course, is to teach the fundamentals of options pricing.

If you already knew what is explained in the article, good for you, disregard it, or try to improve it with a comment. If you didn't know, and don't care, you didn't have to read it. If you care, and it helped you, then this article is fulfilling its purpose.

Anonymous said...

I read, I tried, I failed I'll get a real job to make my money.!

Anonymous said...

howling2929 said...

Good I took the "Managing potfolios" and "managing derivatives" electives on the MBA. By December I should be ready to reply to this type of thing. For now, I will stick to comments on company strategy, were i got my A (strat I) and an A+ (strat II).

Sorry to the audiencce, the promised article on the three sides of the coin is taking longer than expected...