November 17, 2009 in Math
Silicon Alley Insider is running a series of posts called “15 _______ questions that will make you feel stupid.” The blank has been filled twice with “Google interview” and most recently with “management consultant interview.” I particularly enjoyed one of the Google questions:
If the probability of observing a car in 30 minutes on a highway is 0.95, what is the probability of observing a car in 10 minutes (assuming constant default probability)?
I have no idea how the word “default” snuck in there – I’m guessing whoever wrote this had a need to relate things back to dangerous CDS! – but the question is a good one. However, the answers posted on the site are absolutely horrendous. One ardent commentator wrote:
“observing a car in 30 minutes on a highway”
If 30 min = 95%, then 100% probability = 30/0.95 => 31.5 min
(ie, the max interval between 2 cars could be 31.5 min)
Probability in 10 min = 10/31.5
You have to wonder if, by his logic, there’s really a 110% chance of seeing a car in 34.7 minutes?
The correct answer is below…
- The probability of observing no cars in 30 minutes is 1-95%, or 5%
- The probability of observing no cars in 10 minutes, p, must agree with the statement p^3 = 5%, since three consecutive carless 10 minute periods will pass with 5% probability.
- Therefore, p = 36.8%
- And the probability of observing a car in 10 minutes is 1-p, or 63.2%.
The FT has posted a lengthy video interview with the brilliant mathematician Benoit Mandelbrot, whose book The (Mis)behavior of Markets first inspired me to enter finance and risk management in particular.
I do find that some of John Auther’s questions mar an otherwise interesting (but extremely high-level) overview of Mandelbrot’s thoughts on finance. Right from the beginning, he introduces Mandelbrot by discussing his early critique of “the theory of efficient markets, which led to the very complicated investment product that crashed disastrously in the last two years, causing the crisis” – a claim I disagree with – and later questions demonstrate some level of discomfort with the subject material. But these are relatively minor quibbles – on the whole it’s a great interview.
The second part of the interview looks into how the efficient markets hypothesis was able to capture the attention of financial economists, and Mandelbrot concludes that it is because it makes the world appear simpler than it actually is – a direct analogue to the manner in which VaR rose to popularity.
In any case, though there’s little in the interview you haven’t heard before, Mandelbrot is always fascinating and the video is well worth your time.
(via Charles Davi)
Speaking of LTCM (and of this Sunday’s Times Magazine, for that matter), here’s an interview that’s going to run with Myron Scholes, who comes off like a bad comedian. The questions are poor and the answers arguably worse. Let’s take a look, shall we?
The second question: “You’re known as the intellectual father of the credit default swap…” Saying this about Scholes is a bit like calling Edison the intellectual father of the personal computer. The Black-Scholes framework introduced modern derivative pricing, yes, but the leap to credit default swaps was independent of their research. Moreover, only the most basic tenets of the BS model are used in pricing CDS. In fact, the math behind a Credit Default Swap is far simpler than any equity option. A high school student could price a CDS; a college student would need to pick up stochastic calculus and multivariable differentiation before he could work through the Black Scholes derivation.
The third question: “In 1997, you shared the Nobel in economics for your Black-Scholes theory. Is the name intended as a riff on the black-hole theory?”
Really?
It’s called Black-Scholes because Fischer Black and Myron Scholes wrote a paper together in 1973, not because Myron wanted to create an intergalactic pun. In fact, he shared the prize with Robert Merton because Merton was the first to expand on the math of pricing options, also in 1973. Merton actually coined the term “Black Scholes model” in his paper; had he not been so modest it would certainly be known today as BSM – which is how many academics refer to it anyway.
The fifth question and answer: Q: “In retrospect, is it fair to say that the idea that banks could manage risk was a total illusion?” A: “What you’re saying is negative. Life is positive too. Every side of a coin has another side.”
Thanks for the insight, Myron.
The seventh question and answer: Q: “Some economists believe that mathematical models like yours lulled banks into a false sense of security, and I am wondering if you have revised your ideas as a consequence. ” A: “I haven’t changed my ideas. A bank needs models to measure risk. The problem, however, is that any one bank can measure its risk, but it also has to know what the risk taken by other banks in the system happens to be at any particular moment.”
A perfect opportunity wasted to point out that models are just the tool. Instead, Scholes here implies that each bank measured its risk correctly but misjudged every other bank. But besides the fact we know this to be false, such a situation wouldn’t necessarily result in the financial destruction we observe today. That would actually be far preferable to the current state of affairs!
Let us repeat the assumptions of the Black-Scholes framework:
- Investors can borrow and lend at the risk free rate
- Stock prices follow geometric brownian motion with constant variance
- There are no transaction costs, taxes or dividends
- You can short sell as easily as you can buy
- Your option is European
If any of those are violated then the model does not reflect reality. 1-4 are always violated; 5 is up to the investor. Scholes should have aknowledged this instead of running on about how his ideas have extreme utility.
The eighth question and answer: Q:” What good is a theory of risk management if it applies to one tree instead of the forest?” A: “Most of the time, your risk management works. With a systemic event such as the recent shocks following the collapse of Lehman Brothers, obviously the risk-management system of any one bank appears, after the fact, to be incomplete. We ended up where banks couldn’t liquidate their risk, and the system tended to freeze up.”
This reminds me of the saying “VaR is like an airbag that works perfectly every day, but breaks as soon as your car crashes.” Most of the time your risk management works? No – most of the time you don’t need your active risk management! Your risk management only has to work a couple times a year and if it doesn’t work those times then it doesn’t work at all. Risk management systems “appear incomplete” because they are incomplete and insufficient.
The sixth question and answer: Q: “The writer Nassim Nicholas Taleb contends that instead of giving advice on managing risk, you ‘should be in a retirement home doing sudoku.’” A: “If someone says to you, ‘Go to an old-folks’ home,’ that’s kind of ridiculous, because a lot of old people are doing terrific things for society. I never tried sudoku. Maybe he spends his time doing sudoku.”
It’s hard to say whether I hate Scholes’ answer or agree with Taleb more strongly.