by Helmut Schmidhofer
When a clock has stopped, it shows the correct time only twice a day. Option pricing formulas are similar. They show correct values only occassionally. Mostly, they are either ahead or behind the time.
The formula developed by Fischer Black and Myron Scholes in 1973 is by far the most popular. Together with Merton, Scholes received the 1997 Nobel Prize in Economics for his work (Black had died by then). Check the mathematics here
You can see from the above spreadsheet that five inputs, S, E, T, R and V, are required to calculate C and P. All of the input values except V are known. The trick is to have a better view of V than the market has.
Use the spreadsheet to develop an understanding how the parameters affect the values. Change only one parameter at a time and study the effect it has on C and P.
In the following profit-loss diagrams, the stock price S is always 100 and strike prices E are as noted (think of them as % values), T is 30 days, R is 8% and V is 30%.
Commissions and the bid-ask spread have been ignored. Actually, because 'spreads' are option strategies, I prefer the term Bid-Ask Discrepancy, BAD. Some of the strategies may not be viable when commissions and BAD are too large.
When you combine the synthetic short put (covered call) discussed in the previous lesson with a natural long put, the result is a net profit that correspondes with the carry cost of the underlying stock at the risk-free rate R. See what happens when you set R to 0?
Question: If you are long and worried in an optionable stock, why not collect the risk-free carry cost by writing calls and buying puts, and put your worries to rest? Say your worries are justified and the stock drops to 90. Instead of taking a 10 point loss, you keep 3.76 received for the short call and you make a profit of 10-3.10 on the long put for a net gain of 0.66 (see also equity collar).
The long and short straddles, also discussed in the previous lesson, serve as a good example of taking advantage of mispriced options...
The Black-Scholes formula ensures that at the current volatility, V, there is no arbitrage, i.e. the product of negative outcomes times their probabilities equals the product of positive outcomes times their probabilities.
However, when the true volatility is greater than V, the long straddle is profitable. Vice versa, if the true volatility is smaller, the risky short straddle is profitable.
If you are averse to the risk of a short straddle, the butterfly (sometimes called sandwich) and condor spreads let you profit from shrinking volatility while limiting the potential loss.
The maximum profits are 3.76+3.1-1.76-1.25 = 3.85 and 1.76+1.25-0.7-0.37 = 1.94 respectively, with the condor's smaller profit holding over a wider range.
The maximum risks are 3.85-5 = -1.15 and 1.94-5 = -3.06 respectively.
The Black-Scholes model assumes a normal (Gaussian) distribution of future stock prices. However, real distributions often display skewness and kurtosis.
It stands to reason that in a trending market prices will cluster on the side of the trend, and in a drifting market prices will be more evenly distributed across a narrow range.
How to tailor options to take advantage of these differences is discussed next.
Back to Option Basics Proceed to Option Strategies 2
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