Introduction

We want to buy oil in 6 months time. We can buy it now for $30/barrel or in 6 months time for eg. $36 with probability p, or $18 with probably 1-p. (Assuming 0 interest rates, storage costs etc.)

A cap contract compensates the holder for any increase in price, eg. it would pay off $6 if the price rose to $36, or $0 if the price fell. This reduces risk – the pain of the worst case scenario is lessened. Is an example of a financial derivative, or contigent claim. Where do caps come from? – insurance companies (risk transfer) – oil company (reduces risk for both parties).

How do we value the cap? One (incorrect) way would be to take the expected value = .

Replicating strategies

Is it possible to reproduce this contract with some strategy of buying and selling existing assets? How about: buy 1/3 barrel now for $10 (using $6 borrowed money) then sell the oil in 6 months time – has the same payoffs as the cap, so the cap should cost the same as implementing this strategy, $4.

More generally, if a contract payoff can be replicated by a trading strategy with existing assets, then the contract has the same cost as the strategy.

Hidden assumption: no arbitrage – no strategy can have zero cost and non-negative payoffs, unless the payoffs are all 0. Why should be believe arbitrage is impossible? – exploiting arbitrage opportunities tends to diminish them.

More general single period models

• Have assets (stocks, bonds, oil, etc.).
• Current prices given by vector .
• Asset 0 is cash, .

• possible outcomes, , with .
• The possible values of the asset prices at time , are given by a matrix,
• , where is the interest rate

A strategy is a vector of length giving the amount of each asset to hold.

• cost to set up a strategy =
• possible eventual values =
• Any component of may be negative (short selling)
• arbitrage strategy is one where and , or and

Assumption: If prices are consistent then no arbitrage strategy should exist.

Theorem: No arbitrage strategies there exists a vector with , all . is called an equivalent martingale measure. Can be rewritten as , ie. according to the probabilities every asset is a fair bet.

Proof: suppose exists, then for any strategy we have . If then , unless , hence .

How do you value a new asset?

Suppose the payoff the new asset is a ramdon variable with possible value scenarios given by a vector of length . Find a replicating strategy , ie. . Then the value of the new asset is .

Possible to calculate the value without explicitly finding :

ie. take the expected discounted value with q-probabilities, called risk neutral valuation.

But what if isn’t unique?

Theorem: A vector can be replicated has the same value for all EMMs . In particular, if there is a unique EMM then all payoff vectors can be replicated. Markets with a unique EMM are called complete.

A model with few assets and many scenarios will be incomplete – a problem for realistic models. What to do? Find a better model!