Published November 1st, 2007
in Black Scholes.
When participating in a forum about option trading, I have noticed that some of the participants did not know about a few basics relations in pricing options. The question I saw was a long the line of “The difference between Call 100 and Call 90 looks very big to me”.
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.
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Published August 28th, 2007
in Black Scholes.
For those who do not know, the delta is the sensitivity of an option to a movement of the underlying. This can be extended to a portfolio in which the delta reflects the sensitivity of the portfolio to the underlying. It is usually the case that the delta is given to us by the software we use to trade.
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.
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Published June 18th, 2007
in Black Scholes.
I am currently reading a very interesting book by Jean-Philippe Bouchaud called “Theory of financial risks”. It is very recommended if you are mathematically inclined. The book tries to build option pricing from a different angle than the classical replication argument. These include assumption of continuous trading, constant volatility throughout the life of the option and log-normal returns.
In the last post, I have already displayed some graphs showing how the returns do not adhere to a normal distribution but to a skewed distribution with fatter tails. The point of the authors is that if in the Black and Scholes world, perfect replication is possible and hence no risk is involved, why do these instruments exist in the first place? The replication argument is so elegant that sometimes we forget to ask ourselves the most important questions. I used to ponder the same question myself from time to time. The answer probably lies somewhere in the middle. The markets are not in total chaos but nor are they complete.
The book develops a model for pricing options from a different angle, where there is no perfect hedge. The Black and Scholes world is just a pathological case i.e the exception and nothing more.
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