Paradox? law of large numbers vs option theory

Question

Flip a fair coin: if you get heads you win $120$, if you get tails you get $0$. How much would you pay to play this game? $60$ right ?

Now let's consider the following situation: You have an asset S that has value $S_0= 100$ at $t=0$. At time $t=1$, $S$ changes value to $S_1=200$ with probability $\frac{1}{2}$ or $S_1=40$ with probability $\frac{1}{2}$.

How much would you pay for an option that pays $max(S_1-80, 0)$? If you work out the math, you will find that the options value is $45$.

But if you take a step back, this situation is identical to the first one (flipping the coin), there is a $50-50$ chance of winning $120$ or $0$.

So how much should someone pay to play any of these two games, $45$ or $60$? According to option theory, if you pay $50$ you will lose money but that not really the case, especially if we repeat this game a big number of times. How to reconcile these two situations? does it really make sense to price options the way we do in practice?

If the price is $50$, it seems that everyone is happy, the option dealer who can sell an option worth $45$ at $50$ and make $5$ as a profit and the gambler who on average is winning an extra $10$ per game. How can this be?

Answer 1

In a perfect financial market, arbitrage opportunities should not be possible:

There must not exist a trading strategy $\phi=(\phi_{0},\phi_{1})$ investing money into the bank account and the stock such that

1. $\phi_{0}+100\phi_{1}=0$ ("initial value of $0$")
2. $\phi_{0}+P_{1}\phi_{1}\ge 0$ ("never lose in $t=1$")
3. $\phi_{0}+P_{1}\phi_{1}> 0$ with probability $>0$

because that would imply that we have can make a riskless profit - which does not make much sense mathematically.

These conditions are satisfied if and only if there is a risk-neutral probability measure (also called martingale measure) $Q$ under which the (discounted, which is irrelevant here) stock prices are all martingales. We then have to price derivatives under Q - otherwise again, there is arbitrage.

Here, $Q$ would be given by $Q(P_{1}=200)=0.375$ and thus the value of the option is $$V(0)=E_{Q}[(P_{1}-80)^{+}]=0.375\cdot 120=45$$

Now why is $P$ suddenly irrelevant for pricing? That is because the present day value of the stock already includes all information we need for pricing, and all individual risk preferences of the market participants.

The stock simply wouldn't valuate at $P_{0}=100$ if everyone would be prepared to buy it and risk losing $60$ units with a probability of $\frac{1}{2}$. It would instead immediately go up as a consequence of smart traders setting up arbitrage pottfolios.

For more details, see

https://quant.stackexchange.com/questions/103/how-does-the-risk-neutral-pricing-framework-work

Answer 2

The answer given by MF14 is very good but maybe I can add something.

Consider a world in which the coin lottery, the stock and the option exist. Assume that the agents are risk neutral in the sense that they are indifferent between earning $60$ and playing a game with expected value $60$. Also, the risk-free rate is $0$. Then, under no arbitrage conditions:

• The lotery price will be $L_0=E[L_1]=60$
• The stock price will be $S_0= E[S_1]=120$
• The call option price will be $C_0=E_Q[S_1-80|S_1>80]=120 \cdot Q(S_1=200)=60$

Here the physical (real world) probability measure $P$ coincides with the risk-neutral probability measure $Q$.

Consider now a world in which the coin lottery, the stock and the option exist, but now the stock price is $S_0=100$. This implies that agents are risk averse (concave utility functions) as they are pricing the stock below its expected return. The risk-neutral probability measure implied by $S_0=100$ is such that $Q(S_1=200)=0.375$. Then we have:

• The stock price is $S_0=E_Q[S_1]$=100
• The option price is $C_0=E_Q[S_1-80|S_1>80]=120 \cdot Q(S_1=200)=45$
• The lotery price should be $L_0=45$ as it is equivalent to the option.

Essentially, when you write that the lotery price is $L_0=60$ and the stock price is $S_0=100$ you are defining implicitly two different preference profiles for the agents (and two different risk-neutral probability measures). These prices cannot exist in the same world.

Finally, as you noted, in the risk-averse world we can make profit on average if we buy any of the assets at the prices given. That is just the risk premium.