Suppose you have a stock that is priced at 1$ and it has a 50% on either going up to 2$ or going down to 0.5$. Thus, we assume there are only two states with equal probablity. Let us now assume there is a Call option written on that stock with strike price of 1$. Thus, in the up state it makes 1$ and in the down state it makes 0. Now comes the million dollar question.. how much is the option worth? If you take the expected value approach, you’d say it should be worth 0.5$ since that is the expected profit ()

There are a few problems with the expected value approach. We could also say that the stock should be worth 1.25$ in the future by the same expected value approach. This adds more confusion to the proper way to price the option.

Let us assume that we go with the first approach and we price it to be worth 0.5$. In that situation we could sell two call options and buy one stock. We have no initial cost. If the stock goes up, we make nothing, but if the stock goes down, we profit 0.5$ since the options expire worthless. That is in essence an arbitrage, since we have no risk at all and we stand to gain a profit (although in only one state of the world). This tells us that something is wrong with this approach. The question is, is their a proper way of doing this?

Let us assume that there is no interest in this world of ours for simplicity. This means that as time passes, we demand no compensation for our money if we take no risk. We know that the option pays 1$ in the upstate, and nothing in the downstate. We can construct a portfolio that replicates the option buy noticing that buying 2/3 of the stock and selling short 1/3 of a bond with face value of 1$ with exactly replicate the option price (do it yourself to make sure). The total cost would be 1/3$. What is that magical 1/3$ number we got? I claim that is should be the fair price of an option!

As a good exercise, you should try to figure out why this should be the fair price. But do not worry. In the next post, I’ll explain more about why this should be the fair price and touch upon what is known as state prices.

I haven’t been updating this blog for about 2 months I think due to personal issues I had. I was also unable to answer any emails you had. But I am currently back in business!

I am currently testing to see whether there is any merit in using the implied risk neutral distribution as a good predictor to future prices in the Israeli Stock Market.

It had occured to me that some people are not aware of some of the basic pricing rules that YOU SHOULD know.  

The Bull Spread

Consider two Call Options. One with a 100 strike and the other with a 110 strike. We know that the 100 Call would be worth more than the 110 strike. You should notice C(100)-C(110) < 110-100 = 10. That is, the price difference would always be less than the difference of the strikes.
As always with options, relations like these are based upon the no arbitrage rule. Let's take a scenario in which we buy the C(100) and sell the C(110). You should notice that the maximum amount of money we can do with this spread is when the underlying passes the 110 mark. That is when we earn 10$. Now that we know the maximum of money we can earn, how much would you pay for such a contract? Would you pay more than 10$ for a portfolio that will earn you no more than 10$? That is the reason that such a spread will always be less than the difference of strikes(as an aside note, it would be less than 10$ divided by the interest rate for that period)

Hedge Relation

Is it possible that an option on a stock would be worth more than the stock? The answer is no. And yet again, this is true because of arbitrage relations. If such an event were to happen, you could make riskless money. How so? You would sell the option and buy the stock with that money(since the option is worth more than the stock, you’d also be left with some money in your pocket).
When the option expires, it either expires worthless(in which case you end up with the stock you bought) or you have to sell the stock by the amount of the strike price.

In the next part, I’ll discuss what is known as the butterfly spread and how does it impact the pricing of options.

I am sorry I didn’t get the chance to update the Blog. I was working with partners of mine on a project totally not connected to this blog. I did not realize it would take so much time from me. It was VERY long and I am proud of the outcome. It is an Aggregator of blogs in the Hebrew language. Hebrew readers can take a look at here . I would appreciate if you can give me feedback on the functionality of the website.

I am officially back updating the Blog and as always giving the most information I can to educate people about option trading so they don’t get ripped off by seminars.

Consider the following scenario: Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more likely?

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

If you answered 2, you were wrong. Kahneman and Tversky coined this the conjunction bias although it is just a matter of probability rules we were (hopefully) taught in highschool. The probability of one event happening is greater than the probability of two events happening in conjunction. More intuitively, imagine the group of all bank tellers. Wouldn’t you agree that the group of bank tellers includes also the group of active feminists? so the group of active feminists inside the group of bank tellers is smaller.

Conjunction bias in trading

What does it have to do with trading you ask? it has everything to do with trading. Especially in volatile days when investors and traders make the wrong decisions just because the above bias. For example, recall that a month ago the market was VERY volatile and it seemed all the world markets would collapse.
If I were to ask you back then what is more likely:

1. A drop in the market of more than 3%
2. A drop in the market of more than 1%

If you answered 1, then you were wrong. This is because although the market was very volatile and it seemed that a drop of more than 3% was very likely, for it to drop more than 3% it has to first drop 1%! hence option 2 is ALWAYS more likely.

How to apply it

To apply it, you have to be aware it exists. For instance, consider options. If people think a 3% drop is more likely, put options deep in the money should be expensive relative to at the money options. This happens mostly in volatile situations as described above. As a good exercise, you could think of other situations where anomalies in prices occur due to the above bias.

Before you start yelling at me and tell me how can I say they are not good investment if I myself trade them, let me explain what I mean by that. It is true that I trade options, but only for short term trading. I usually get rid of my options in the same day. Let us consider the situation where a person thinks a stock should rise in value in the future. Instead of buying the stock itself, he wants to buy a call option on the stock. A person might do this because of the leverage an option provides.

Some Option Drawbacks

It is true that options provide leverage that can boost profits. The problem is that, unlike a stock, they can lose value pretty quickly and even if the move you waited to happen has happened, you might not profit from it.

Losing Value With Time

Unlike a stock, the option loses value with time. Intuitively, this is because the option has less probability to finish inside the money as time progresses.

Losing Value If Implied Volatility Changes

Lower implied volatility means less chance that the option will finish inside the money, hence causing it’s price to decline.

The above should be a reminder on how NOT to trade options. You don’t buy an option to sit back and look how the option rises in value.

Some Exceptions

Exceptions naturally occur when you use options as hedging tools in conjunction with stocks. Buying a stock and buying a
put is a fairly good idea for the risk averse investor.

I have come across a news article in Hebrew on some predictions of analysts to the upcoming year. As always it serves for good laughs and fun. As you already know, I consider those prediction to equivalent of witchcraft.

One Economist says that 2008 would be a very tough year because of the slow growth in the US economy and the sub-prime crisis. Classic after the fact analysis. Who ever heard about the sub-prime crisis before it happened?

Another Economist, called the “optimistic” one by the reporter, predicts that the local stock market would rise between 12%-17%. I have really no idea where these people get those numbers. It is pretty pretentious if you ask me.

The delta in its mathematical form is just the derivative of the option with respect to the underlying. The question to ask of course is how good is this number? if we are dealing with a big portfolio errors in the delta can be damaging in our assessment of exposure to movements of the underlying. The problem with the traditional delta that comes from the black and scholes formula, is that it assumes that trading is continuous and not discrete as it really is.

Discrete Delta

We can evaluate what is known as a “discrete” delta which can give us a better estimate of the true sensitivity. Here is an example. Suppose a stock is traded at $100 and a call option on that stock trades at $5. We note that when the stock goes up to 101, the price of the option goes up to $5.5. The delta in our case would be:

This is the essence of finding a “discrete” derivative. This would prove more accurate than the black scholes formula. This could be extended to even other “greeks” such as the gamma and vega.

Sorry for not updating the site in the last week. I am working on another program and of course you could download it for your enjoyment. It is some kind of a trading platform. I am also moving in the next hours the website to another database and I’ve been told the site would have to be down for 24 hours. So don’t panic if you can’t get into the site since it would be temporary. I’d make it up with a nice post about hedging risks though so stay tuned.