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I'm trying to calculate the today's price of a European put option, which strikes at price $K$, so that the payoff is $(K-S_1)^+$, in an arbitrage-free, one-period market.

As of now, I can only think of $Price=\frac{1}{1+R} E[X]$, here with $X=(K-S_1)^+$, and $E[X]=\sum\limits_{i|Z_i<K} q_i(K-Z_i)$, where $q_i$ is the probability that the stock will have price $Z_i$ at point of time 1: $q_i=P(S_1=Z_i)$. But that would mean I just plug the information into a formula and stop. I feel I'm not finished yet, did I oversee anything?

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Well, you're correct, except that you need to price under the risk-free probability measure rather than the real-world probability measure. – Chris Taylor Feb 24 '12 at 19:00
up vote 6 down vote accepted

I think the most enlightening way to look at this problem is from a no-arbitrage point of view. Let's say that the possible prices of the stock after one period are $(1+u)S$ and $(1-d)S$ with probability $p$ and $1-p$. Further, let the payoffs in the case of a positive and negative move be $P$ and $Q$, i.e.

$$P = [K-(1+u)S]^+$$ $$Q = [K-(1-d)S]^+$$

I'm not actually going to use these expressions for the rest of the answer. The method I present will be generic for any contract, but I wanted to make it clear that this is where the fact that the contract is a put option enters the calculation.

The idea is that we will construct a risk-free (i.e. non-random) portfolio by combining the option and the stock. Since the portfolio is non-random, its value after one period is determined - it must earn the risk-free rate of interest. If it was worth more, then we could earn a non-risky profit by borrowing as much money as possible and buying the portfolio. If it was worth less, then we would sell it short and use the proceeds to buy a risk-free bond, again booking a risk-free profit. This is all that we will need to determine the price of the option.

Our risk-free portfolio consists of one option contract (valued at $V$) and a short position in $\Delta$ units of the stock:

$$\Pi = V - \Delta S$$

After one period, the value of the portfolio is known with certainty. We want the portfolio to be risk-free, i.e. to have the same value in the case that the stock moves up or down in price, so we must have

$$P - \Delta(1+u)S = Q - \Delta(1-d)S$$

Now we can solve for $\Delta$:

$$\Delta = \frac{P-Q}{(u+d)S}$$

Now, since the value of our portfolio is non-random, we must earn the risk-free rate of interest:

$$\Pi' = (1+r)\Pi$$

which means

$$Q - \Delta(1-d)S = (1+r)(V-\Delta S)$$

and you can now solve for the value of the portfolio:

$$(1+r)V = Q - \Delta(1-d)S + \Delta(1+r)S$$

Notice that this is independent of $p$! So the probabilities of the stock moving up or down don't matter - what is important is that there is a unique price for the option, guaranteed by the assumption that no arbitrages exist.

It's helpful to take things a bit further than this, to make the connection between what I've done here ("no-arbitrage pricing theory") and pricing with expectation. If we substitute our expression for $\Delta$ into the equation for the value of the option contract we get

$$\begin{align} (1+r)V & = Q - \frac{(P-Q)(1-d)}{u+d} + \frac{(P-Q)(1+r)}{u+d} \\ & = \frac{Q(u+d)}{u+d} - \frac{(P-Q)(1-d)}{u+d} + \frac{(P-Q)(1+r)}{u+d} \\ & = \frac{d+r}{u+d}P + \frac{u-r}{u+d}Q \end{align}$$

This looks somewhat suggestive. It looks even more suggestive when we notice that

$$\frac{d+r}{u+d} + \frac{u-r}{u+d} = \frac{u+d}{u+d} = 1$$

and additionally, both of the coefficients are positive1. This means that we can interpret this expression for the value for the option as a probability-weighted sum of the two possible outcomes, i.e. as an expectation. However, we're not taking the expectation under the real-world probability $p$. Instead we have introduced a new pseudo-probability

$$q = \frac{r+d}{u+d}$$

which is referred to as risk-neutral probability. This is the key insight in derivative pricing: that pricing using the assumption of no arbitrage is equivalent to taking expectations, as long as you use the risk-neutral probability measure. If you understand this, then you can understand all derivative pricing!

1 Or rather, they're positive as long as $u>r$ and $r>-d$, which both make sense in context: the first says that the potential upswing in the stock is greater than what we'd earn by holding a risk-free asset, and the second says that the risk-free rate can't be too negative (in fact we'd expect $r>0$ in general, so that you earn interest on your assets rather than having to pay a tax on them).

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@MDCCXXIXFar out. Ramanujan lives! – TheBirdistheWord Feb 24 '12 at 22:32
That was awesome! :) – Marie. P. Feb 26 '12 at 19:29
Glad I could help. – Chris Taylor Feb 27 '12 at 9:06

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