Quick comment: Since I haven’t posted for two weeks, I'll post twice as much this week. Thank you to everyone who complained and made me finally write something.
Call Vol = Put Vol for Strike K
Put call parity (for European options!) is an equation. Equations are amazing because they can have all sorts of mathematical operations done to both sides to deduce properties. Keep in mind that only the theoretical, strict definitions of arbitrage have equations. That makes put-call parity a great starting point for talking about implied volatility (Hull).
The following (slightly different) is found in Hull 5E, Chapter 15:
Buying a call and selling a put is the same as owning the stock and subtracting the carrying cost. Put-call parity holds in both the Black-Scholes world and the real world since it is based on a no arbitrage argument. Please take a look to the top of the post.
What is happening up there?! The first line is a formulation of put call parity for Black-Scholes option prices. The second line is the formulation for the market prices. The third line is a manipulation of the top two lines, specifically the first line less the second line. The forth line is just the third line rearranged. The outcome shows that the dollar pricing error in Black-Scholes is the same for calls and puts!
What's with the word implied?
The volatility that, given a particular pricing model, yields a theoretical value for the option equal to the current market price is implied because it must be inferred from the model. More simply, to imply the volatility means you have to guess and check to find it. There are a couple of ways to guess and check more efficiently than random.
Bisection Method: Guess in the middle and see if you are too low / too high. Then guess in the middle of the half that is better.
Newton-Raphson: Guess in the middle, guess again and see how much closer you are. If you increased the volatility by 1 point and got 2 dollars closer and you were 6 dollars off, you should try increasing the volatility by another 2 points.
Brent's Method: (If you would like to explain this clearly, please email me and I'll post it.)
Different Vols for Different Strikes
- Distribution adjustment - Since the underlying does not have to act lognormally (please see below posts for more on this), some parts of the distribution become more probable than others. This type of correction is very prominent in currency options because of the discrete nature interest rate moves.
- Leverage adjustment - "As a company's equity declines in value, its leverage increases. This makes even lower stock prices more likely. As the company's equity increases in value, its volatility decreases in value, making higher stock prices less likely."
- Reflexivity - Bad news triggers more bad news. The financial markets are reflexive. An example of positive reflexivity is giving a company a good credit rating. That good credit rating lowers their cost of borrowing. Since their cost of borrowing is less, the company's value increases. Since the company's value increases, it's credit rating increases. Understandably, the effect gets smaller and smaller with each iteration, but the positive feedback is clear. This is a possible explanation for fat tails commonly noticed in financial data.
- Volatility clustering - "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes" - Mandelbrot. This is different from above two in that the direction does not matter.
- Supply and Demand - Before the crash of 1987, the smile was not prominent in option prices. It is possible that investors are crashophobic and willing to pay a certain premium to have a back up. Since it is unclear how to arbitrage the smile (more on this in the next post), premium just builds up in a certain strike. Supply and demand also constantly updates the market's perception of the probability of the options ending up in different strikes.
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