Basic Option Strategies, Part 2 - Options Pricing

I really like that you guys are doing this.<p>Former options guy myself and was thinking about doing some similar writeups on options pricing.<p>i like the graphs of you drew of the strips with different times to expiraiton. One thing that is generally counterintuitive for people is how there is translation of the graph on the x axis over time (theta and carry rates).<p>also generally, I think introducing black scholes is good, but doesn't really "stick" for most people for a while. it takes a while to develop an intuition about the components of "C = SN(d1) - KN(d2)" means.<p>I think teaching these intuitions would really sit well for the HN crowd. Also explaining option pricing as the value of the cash flows of the hedge portfolio with constant or regular hedging (like physics, start with assumptions like GBM, zero transaction cost, no bid-ask spread -- then if you ever feel like it later, address how hedge strategies can dramatically change option pricing). Natenberg does a simplied version of this.<p>Also, Cox Ross Rubenstein is a great way to teach euro AND american options -- just change the rules at each node to get different rules and you can demonstate convergence to Black Scholes with the right parameters.

Imagine a drunk man walking down a dark, rainy road one night after his car breaks down. He gets out, and begins a "random" walk along the road, looking for help. Neither you (the trader), nor he (the stock), can see anything - it's raining, and it's dark. The man stumbles along the road and your job is to predict his future path, and in return, you get paid money.<p>A lot of money.<p>Here's the problem: the guy is both drunk and blind (and probably just a little stupid) - he can't see anything and hence his movements are erratic. How in the hell can you predict where this idiot is going to end up? You have bills to pay, derivatives to price and insurance to sell. So you latch onto the closest thing that'll work - volatility (anchoring bias).<p>You use his volatility and assume that, dependant upon his past movements, and in turn his apparent level of drunkenness, you can, more or less, predict the possible range of his future random walk. This is a very useful model for predicting where he will be within the next 30 seconds. It works very well, and you make a lot of money.<p>It feels good.<p>Now you become confident. You start projecting it out just that little bit further, putting on more precise predictions with tighter spreads, and levering up your bets - because, of course, everyone else is competing with you and driving down your alpha. You have now mistaken past movement for the actual risk of movement - they are not the same thing.<p>Unfortunately for you, of course, the man has broken down on the edge of a steep cliff. He continues his random walk, blissfully unaware of his impending doom. You continue your bets on his volatility. I mean, why wouldn't you? You're king of the fucking world after all - in fact, not only are you rich, but you also have a Nobel Prize in Economics from a Swedish Bank!<p>Oops - too late. Your man has just fallen off the cliff, and you, and your savings, along with him. You yell inefficient markets, beta sucks balls, VAR is a trap, the CAPM is a lie and modern portfolio theory is fucking stupid. The last thing we hear, before that final, brutal, resounding thud is the faint line: "It was all a fucking lie."<p>You have just met real risk. It has not been a pleasant experience. Welcome to the real world.

Suggestion: In practice, for stock-options, volatility is strike-dependent ie implied volatility has a skew (lower strikes have higher implied volatility - the leverage effect). Also, the volatility is dependent on time to expiry of the option.<p>Also, as I've mentioned in the HN thread for your previous article, single-name stock options which are exchange-traded have an American exercise type. The valuation of these cannot be done using Black-Scholes if they are (a) long-dated (ie time to expiry is quite long) or (b) they are deeply in the money/out of money or (c) have an underlying which has a significant dividend yield (in which case you'd want to own the stock rather than the option).<p>Puts behave different than calls if the exercise-style is American (puts have a limited upside - so you wouldn't wait too long if the underlying stock has fallen far enough - your payoff is not likely to be larger).<p>You may want to discuss this and option greeks the next time.

The bible on this subject, fyi:<p><a href="http://www.amazon.com/Options-Futures-Other-Derivatives-Edition/dp/0131499084/ref=pd_vtp_b_3" rel="nofollow">http://www.amazon.com/Options-Futures-Other-Derivatives-Edit...</a>

The math looks off on the 50% vol 30 day option. No way a $50-strike option on a stock at $25 would be pricing at $20. Unless you mean 50% non-annualized vol, which would not be very conventional.

I really like this series of posts--it's a great introduction to options.<p>The little aside in one of the notes made me curious: what <i>is</i> the logic behind trading options on their expiry date? I can't think of any obvious reason to do that, but I'm not terribly experienced with finance. That's why I'm curious about it, I suppose.

I think the author of the article hasn't really grasped the concept of option pricing. You are not calculating an 'expected value'.<p>The whole point of the theory is there is a correspondence between a 'no-arbitrage argument' and calculating an expectation under the 'risk-free' measure. The mathematical operation of expectation E is only a tool. This link between no-arbitrage and martingale theory is called the 'Fundamental Theorem of Asset Pricing'.<p>There is a nice (basic) explanation of this idea in the book by Baxter and Rennie "Financial Calculus" where they compare the 'expected value' approach of a bookmaker and the 'no-arbitrage' approach.<p>For a more advanced explanation, you can have a look at the book by Delbaen and Schachermayer (2006).

I feel like this could have been longer. Volatility is really the only way options make money, no matter what your position is. Is volatility going to be covered in later parts as well?

I'm going to get flamed for this, but promulgating Black-Scholes as the "standard way" to price options is a little misleading. A better choice of word would be "conventional", but that's also inexact because the BSM is broken, since BSM makes a lot of unrealistic assumptions for the model to work. The point is that there is no One True Pricing Model... the price is what the market thinks the value is. Too many people who should know better get it all mixed up, with disastrous effects.