# The dice game and derivatives trading

I happened to a interview question:

Give a equal dice, you will gain the money which is the number you roll, then how much will you pay for the game.

Naturely, the answer is 3.5. But the interview said, the dice game is not the derivatives, you have nothing to hedge it, then you are a speculator. So the answer is not 3.5.

What did he mean?

• Meaning as a risk-averse agent, you are not willing to pay the expected value for a risky gamble. As the single dice throw is not correlated to any other instrument you could trade, there is not way for you to hedge its risk. – LocalVolatility Nov 21 '17 at 11:57
• @LocalVolatility but we still pay the risk-neutral price of a derivative regardless whether we hedge it? – A.Oreo Nov 21 '17 at 13:06
• Beware, risk-neutral theory is a consequence of hedging: the risk-neutral price is originally derived through hedging arguments. – Daneel Olivaw Nov 21 '17 at 13:57
• @DaneelOlivaw that makes sense, so the price should determined higher than 3.5? Or 3.5 can give nothing reference? – A.Oreo Nov 21 '17 at 14:05
• I am not sure what he meant TBH. The comment/answer given by LocalVolatility and dm63 are a way of interpreting this, although given the interviewer says "[...] you are a speculator." maybe the price should be higher than 3.5, instead of lower (I have read in some places a definition of a speculator as a risk-prone investor, i.e. opposite to risk-averse). On the other hand, when I read "[...] you have nothing to hedge it [...]", I think insurance and in insurance pricing is done by expectation under the real-world measure - although assuming we are pooling many independent risks together. – Daneel Olivaw Nov 21 '17 at 14:13

The interviewer meant that he's smart. Quoting Senior VP of People operations at Google,

On the hiring side, we found that brainteasers are a complete waste of time. How many golf balls can you fit into an airplane? How many gas stations in Manhattan? A complete waste of time. They don’t predict anything. They serve primarily to make the interviewer feel smart.

Putting that aside, one possible approach would be to invoke Von Neumann-Morgernstern expected utility to construct a certainty equivalent value for the gamble based upon your level of risk aversion.

Utility functions are used to define a total order over possible outcomes and hence can represent complete, transitive preferences: outcome $X$ is preferred to $Y$ if and only if the utility function assigns $X$ higher utility. Expected utility extends classic utility theory to stochastic outcomes by defining the overall utility $U$ of a stochastic outcome $X$ as the expectation of a bernoulli utility function $u$ whose curvature $-\frac{u''}{u'}$ formalizes a notion of risk aversion.

$$U(X) = \mathbb{E}[u(X)]$$

(Small note: the curvature of $u$ here is extremely important, representing risk aversion, while the curvature of $U$ is irrelevant: any monotonic, increasing transformation of an overall utility function $U$ represents the same preferences.)

A nice Bernoulli utility function $u$ to use is power utility. In a special case this is simply log utility: $u(x) = \log(x)$.

Let $w$ be a scalar representing your wealth. Let $Z$ be payoff from the dice roll (i.e. 1 dollar if dice rolls 1 etc...) Let $c$ be the certainty equivalent of the gamble. The certainty equivalent gives you the same expected utility as your gamble hence $c$ solves the equation: $$u(w + c) = \mathrm{E}[ u(w + Z) ]$$

With log utility:

$$\log(w + c) = \frac{1}{6}\sum_{i=1}^6 \log(w + i)$$

If we have log utility and a wealth of one million dollars ($w = 1,000,000$), then I compute the certainty equivalent of the gamble as $c = 3.49999854$. So it's not 3.5 dollars, but really, it's basically the same unless you pump up your risk aversion or scale up the gamble. (And that wealth is probably dramatically too low if you take into account the present value of all future wages.)

This analysis of course doesn't take into account the value of the time wasted talking about this gamble. A few dollar bet is almost certainly too small to be worth meaningful analysis.

• The last sentence is very true. I would answer the question with questions - what's the size of my bet, and do I have just one bet to make? Or can I make many? – will Nov 22 '17 at 19:56

He meant that you are taking risk, so you might only pay less than 3.5 for it. For example , supposing your whole net worth is usd 35,000 and the game is played in units of usd 10,000. Would you pay 35,000 for the game where you could get back only 10,000? That would crush your total wealth.

• but we still pay the risk-neutral price of a derivative regardless whether we hedge it? Even you have the probability to become a negative net. – A.Oreo Nov 21 '17 at 13:09
• Or you might pay more than 3.5. Suppose that of you have a mobster debt and if you don't pay it, you will go sleep with the fishes. It is contrived, but playing a game with a negative expectation could still put you ahead (which is just demonstrates again that it depends on your utility function) – Bram Nov 22 '17 at 18:14