Stochastic Processes and Financial Mathematics
(part one)
5.2 Hedging in the one-period model
We saw in Section 1.3 that the ‘no arbitrage’ assumption could force some prices to take particular values. It is not immediately obvious if the absence of arbitrage forces a unique value for every price; we will show in this section that it does.
First, let us write down exactly what it is that we need to price.
The function
the value of a unit of stock at time
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Example 5.2.2 A European call option gives its holder the right (but not the obligation) to buy, at time
, a single unit of stock for a fixed price that is agreed at time . As for futures, is known as the strike price.Suppose we hold a European call option at time
. Then, if , we could exercise our right to buy a unit of stock at price , immediately sell the stock for and consequently earn in cash. Alternatively if then our option is worthless.Since
is equal to either to either or , the only interesting case is when . In this case, the contingent claim for our European call option isIn the first case our right to buy is worth exercising; in the second case it is not. A simpler way to write this contingent claim is
In general, given any contract, we can work out its contingent claim. We therefore plan to find a general way of pricing contingent claims. In Section 1.3 we relied on finding specific trading strategies to determine prices (one, from the point of view of the buyer, that gave an upper bound and one, from the point of view of the seller, to give a lower bound). Our first step in this section is to find a general way of constructing trading strategies.
The process of finding a replicating portfolio is known simply as replicating or hedging. The above definition means that, if we hold the portfolio
If a contingent claim
We say that a market is complete if every contingent claim can be replicated. Therefore, if the market is complete, we can price any contingent claim.
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Example 5.2.4 Suppose that
and , and that we are looking at the contingent claimWe can represent this situation as a tree, with a branch for each possible movement of the stock, and the resulting value of our contingent claim written in a square box.
Suppose that we wish to replicate
. That is, we need a portfolio such that : This is a pair of linear equations that we can solve. The solution (which is left for you to check) is , . Hence the price of our contingent claim at time is .
Let us now take an arbitrary contingent claim
By (5.1), if we write
A unique solution exists when the determinant is non-zero, that is when
It is an assumption of the model that
And, in this case, we can solve (5.7) to get
The formula (5.9) is known as the risk-neutral valuation formula. It says that to find the price
of
Note the similarity of (5.9) to (5.4). In fact, (5.4) is a special case of (5.9), namely the case where
To sum up:
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Proposition 5.2.6 Let
be a contingent claim. Then the (unique) replicating portfolio for can be found by solving , which can be written as a pair of linear equations: The general solution is (5.8). The value (and hence, the price) of at time is
For example, we can now both price and hedge the European call option.
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Example 5.2.7 In (5.5) we found the contingent claim of a European call option with strike price
to be . By the first part of Proposition 5.2.6, to find a replicating portfolio we must solve , which is This has the solution (again, left for you to check) . By the second part of Proposition 5.2.6 the value of the European call option at time is