A stock is currently priced at $\$n$. Tomorrow it will either double in value (to $\
In addition, you have access to a casino where you can bet any amount $x$ on the outcome of the stock's move. If your prediction is correct, you win $\$x$ (net gain $\$x$). If your prediction is wrong, you lose your $\$x$ bet.
Design a strategy that guarantees a profit regardless of which way the stock moves. What is the guaranteed profit when $n = 100$?
How to Think About It: This is a pure arbitrage problem -- the asymmetry in the stock's payoff creates an edge that the casino lets you lock in. First, figure out which direction to trade the stock. Buying gives you $+n$ upside and $-n/2$ downside, while selling gives you $+n/2$ upside and $-n$ downside. Buying is strictly better because the upside exceeds the downside. Once you have bought the stock, you hedge the bad scenario (stock halving) at the casino. The question becomes: how much should you bet?
Quick Estimate: If the stock doubles, you make $n$ on the stock. If it halves, you lose $n/2$. The average payoff without hedging is $(n - n/2)/2 = n/4$. Since the casino lets you shift money between states at fair odds, you should be able to lock in something close to $n/4$. Let us check if $n/4$ is achievable exactly.
Approach: Set up equations to find the bet size that equalizes profit across both outcomes.
Formal Solution:
*Step 1 -- Choose the stock direction.*
Buying vs. selling the stock:
Buying has a better payoff profile ($+n$ vs. $-n/2$) compared to selling ($-n$ vs. $+n/2$). The upside of buying ($n$) exceeds the downside ($n/2$) in absolute terms, so buy the stock.
*Step 2 -- Hedge at the casino.*
Bet $\$x$ at the casino that the stock will go down (hedging the bad scenario for our long stock position).
Set the two payoffs equal for a guaranteed outcome:
$n - x = x - \frac{n}{2}$
$\frac{3n}{2} = 2x$
$x = \frac{3n}{4}$
*Step 3 -- Compute the guaranteed profit.*
Substitute $x = 3n/4$ into either scenario:
Both give a guaranteed profit of:
$\text{Guaranteed profit} = \frac{n}{4}$
Answer: Buy the stock and bet $\$3n/4$ at the casino that the stock will go down. This locks in a guaranteed profit of $n/4$. When $n = 100$, the guaranteed profit is $\
The key insight is that the stock's payoff is asymmetric: doubling gives you $+n$ but halving only costs you $n/2$. This asymmetry means there is "free" expected value baked into buying the stock. The casino acts as a mechanism to convert that expected value into a guaranteed payoff by transferring money from the good state to the bad state at fair odds.
This is the same principle behind delta hedging in options. If you own an asset with convex payoff (here, the stock's doubling-vs-halving is effectively convex), you can hedge away the directional risk and lock in the "convexity premium." In real markets, the casino is replaced by options or other derivatives, and the guaranteed profit corresponds to the mispricing between the derivative and the replicating portfolio. The $n/4$ profit here is the arbitrage profit from the market's failure to price the stock's volatility correctly relative to the casino's fair odds.