Is there arbitrage?

Question

An economist writes a 1-period expectation model for valuing options. The model assumes that the stock starts at S and moves to $2S$ or $\frac{1}{2}S$ in 1 year's time with equal probability. Strike is equal to $K$

Assume rates are zero.

I found that the value of the call option using the economist's model is $$S-\frac{1}{2}K$$ and using the Binomial 1-period pricing model it is $$\frac{2}{3}S-\frac{1}{3}K$$

So we have 2 models

• 1-period binomial model
• 1-period expectation model

Is there arbitrage between the two models? If so, how can we capture it?

Answer 1

First of all, economists think in terms of risk premia. There's a concept in the economic branch of asset pricing, called "stochastic discount factor", which differentiates economists from mathematicians. That being said, the question asks you to find the value of the option today. As an economist, you will attach 1/2 of the probability to each outcome; hence,

Call Price Today = $\frac{1}{2}$ (Payoff from Call Price Up) + $\frac{1}{2}$ (Payoff from Call Price Down)

Bear in mind that the tree you construct for the stock is equivalent to the tree constructed for the calls. Then, denote the payoff in the usual way, i.e. $max(S(1)-K, 0)$, which leads to the following:

Payoff from Call Price Up = $max (2S-K, 0)$

Payoff from Call Price Down = $max (0.5S-K, 0)$

Now, make the assumption that the strike price, $K$, is such that $$S(down) < K < S(up)$$ Therefore, the call that goes down will be OTM, namely will have a payoff of zero; while the other one, which is ITM, will be worth $2S - K$. Therefore, the economist will price the option as:

$p(Call) = \frac{1}{2} * (2S - K) + \frac{1}{2} * (0)$

For a mathematician using risk-neutral pricing, things are slightly different. There's no need to know the actual probability as it is possible to construct synthetic ones. However, note that economics and mathematics are highly interlaced here. Risk-neutrality lies on important economic intuition too, which relies around risk aversion. Hence, in the binomial tree, you simply calculate the risk-neutral probability as you did and find that:

$p(Call) = \frac{1}{3} (2S-K) + \frac{2}{3} (0) $

Here comes the tricky part of the question. I think what is meant is that the 1-period expectation model presents a mis-pricing, which makes the price of the call overvalued. If you were an arbitrageur, you would know that because you calculated what the price of the call is using risk-neutrality. Hence, you would sell the overvalued call and buy the underlying. This strategy is known as "naked call strategy".

I hope this helps!