A lot of people and funds claim that they perform arbitrage in order to make a profit. Sometimes it is a little confusing because the term "arbitrage" is used by different individuals to mean quite different things (especially with terms such as stat arb). I think arbitrage can carry a few distinct meanings:
1. The real, good-old sure profit arbitrage
This refers to the option-pricing kind of arbitrage. The PnL is deterministic, since you are basically earning a sure profit by long-shorting the same security across different markets to exploit the price discrepancy. Examples would be a) trading the same contract listed on two exchanges when the prices diverge; and 2) synthesizing a contract (e.g. future) using something else (e.g. call and put).
2. Statistical arbitrage, looking at historical data
This refers to fitting a model to the historical performance. A classic example is pairs trading, in which the price of two assets are expected to return to the "normal" level when a deviation is observed. In a sense, this kind of arbitrage relies on the real world probability distribution as suggested by historical data.
3. Statistical arbitrage, looking at market data
This refers to fitting a model to the market prices. For example, one can try to see if the deep OTM options are over-priced or under-priced, by studying the Greeks. In a sense, this kind of arbitrage relies on the risk-neutral probability distribution as suggested by market data.
Thursday, February 28, 2013
Monday, February 25, 2013
Book Review: Volatility Trading by Euan Sinclair
I first came across this book a few years back. I didn't put much time and thought on it back then but now that I revisit it, I find that it is actually quite a practical book for anyone who wants to investigate volatility trading.
The book covers a lot of ground, from basic vol trading principles to vol forecasting to hedging and bet sizing. Perhaps you will find that it tries too hard to be comprehensive, so feel free to skip some sections (such as Ch.8 Psychology).
Nonetheless, there are some good parts. In Ch.2 the author talks about different ways to measure volatility (different calculation methodologies, data with different frequencies etc.). These go beyond the most common definition of volatility, and are essential for creating useful and relevant estimators.
Of particular interest is the section on forecasting volatility. Here Sinclair explains the merits and shortcomings of GARCH type models from a practical point of view. The volatility cone is introduced, and although some might find it too heuristic and ad hoc, it does provide a relatively simple and systematic way to bring implied vol and realized vol into comparison.
The chapter on vol surface dynamics is quite weak, which is possibly a price to pay by not including too much theory. However the chapter on hedging is definitely worth a read, as it is not so much a crash course on Greeks calculation, but rather an in-depth investigation on why, what and when to hedge when trying to trade volatility.
http://www.amazon.com/Volatility-Trading-CD-ROM-Wiley/dp/0470181990
The book covers a lot of ground, from basic vol trading principles to vol forecasting to hedging and bet sizing. Perhaps you will find that it tries too hard to be comprehensive, so feel free to skip some sections (such as Ch.8 Psychology).
Nonetheless, there are some good parts. In Ch.2 the author talks about different ways to measure volatility (different calculation methodologies, data with different frequencies etc.). These go beyond the most common definition of volatility, and are essential for creating useful and relevant estimators.
Of particular interest is the section on forecasting volatility. Here Sinclair explains the merits and shortcomings of GARCH type models from a practical point of view. The volatility cone is introduced, and although some might find it too heuristic and ad hoc, it does provide a relatively simple and systematic way to bring implied vol and realized vol into comparison.
The chapter on vol surface dynamics is quite weak, which is possibly a price to pay by not including too much theory. However the chapter on hedging is definitely worth a read, as it is not so much a crash course on Greeks calculation, but rather an in-depth investigation on why, what and when to hedge when trying to trade volatility.
http://www.amazon.com/Volatility-Trading-CD-ROM-Wiley/dp/0470181990
Wednesday, February 20, 2013
Those damn dice...
To recap, a dice rolling game contrasts with coin tossing in:
- Coin tossing follows a binomial distribution (which converges to Gaussian); die rolling follows a uniform distribution.
- Tossing m coins gives an expectation value of m/2; rolling an m-sided die gives an expectation value of (m+1)/2.
Since the probability distribution itself for dice game is simpler, the game you would encounter in an interview is usually more complex to compensate for it. Let's start with some easy ones.
DR01
Player A rolls one die for 4 times, aiming to get one 6; player B rolls two dice for 24 times, aiming to get (6,6). Who has a better chance of winning?
DR01 - Answer:
The probabilities are 1-(5/6)^4 vs. 1-(35/36)^24
DR02
When rolling 2 dice, what is P(both are 6 | at least one 6)?
DR02 - Answer:
Bayesian: (1 * 1/36) / (1 - 25/36) = 1/11
DR03
You have two dice, one is 10-sided and the other is 20-sided. What is P(points on 10-sided die > points on 20-sided die)?
DR03 - Answer:
(1 + ... + 9) / 200 = 9/40
See here for more interview questions/brainteasers
- Coin tossing follows a binomial distribution (which converges to Gaussian); die rolling follows a uniform distribution.
- Tossing m coins gives an expectation value of m/2; rolling an m-sided die gives an expectation value of (m+1)/2.
Since the probability distribution itself for dice game is simpler, the game you would encounter in an interview is usually more complex to compensate for it. Let's start with some easy ones.
DR01
Player A rolls one die for 4 times, aiming to get one 6; player B rolls two dice for 24 times, aiming to get (6,6). Who has a better chance of winning?
DR01 - Answer:
The probabilities are 1-(5/6)^4 vs. 1-(35/36)^24
DR02
When rolling 2 dice, what is P(both are 6 | at least one 6)?
DR02 - Answer:
Bayesian: (1 * 1/36) / (1 - 25/36) = 1/11
DR03
You have two dice, one is 10-sided and the other is 20-sided. What is P(points on 10-sided die > points on 20-sided die)?
DR03 - Answer:
(1 + ... + 9) / 200 = 9/40
See here for more interview questions/brainteasers
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