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Rational pricing is the assumption in financial economics that asset prices (and hence asset pricing models) will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.1 Arbitrage mechanics
Arbitrage is the practice of taking advantage of a state of imbalance between two (or possibly more) markets. Where this mismatch can be exploited (i.e. after transaction costs, storage costs, transport costs, dividends etc.) the arbitrageur "locks in" a risk free profit without investing any of his own money. Arbitrage is possible when one of three conditions is not met:
- The same asset must trade at the same price on all markets ("the law of one price").
- Where this is not true, the arbitrageur will: 1) buy the asset on the market where it has the lower price, and simultaneously sell it on the second market at the higher price 2) deliver the asset to the buyer and receive that higher price 3) pay the seller on the cheaper market with the proceeds and pocket the difference.
- Two assets with identical cash flows must trade at the same price.
- Where this is not true, the arbitrageur will: 1) sell the asset with the higher price and simultaneously buy the asset with the lower price 2) fund his purchase of the cheaper asset with the proceeds from the sale of the expensive asset and pocket the difference 3) deliver on his obligations to the buyer of the expensive asset, using the cash flows from the cheaper asset.
- (Note that this condition can be viewed as an application of the above, where the two assets in question are the asset to be delivered and the risk free asset.)
- (a) where the discounted future price is higher than today's price:
- (1) The arbitrageur agrees to deliver the asset on the future date (i.e. sells forward) and simultaneously buys it today with borrowed money. 2) On the delivery date, the arbitrageur hands over the underlying, and receives the agreed price. 3) He then repays the lender the borrowed amount plus interest. 4) The difference between the agreed price and the amount owed is the arbitrage profit.
- (b) where the discounted future price is lower than today's price:
- (1) The arbitrageur agrees to pay for the asset on the future date (i.e. buys forward) and simultaneously sells the underlying today; he invests the proceeds. 2) On the delivery date, he cashes in the matured investment, which has appreciated at the risk free rate. 3) He then takes delivery of the underlying and pays the agreed price using the matured investment. 4) The difference between the maturity value and the agreed price is the arbitrage profit.
2 Fixed income securities
Fixed income securities have known cash flows (by definition). Further, each cash flow of a fixed income security can readily be matched by trading in some multiple of a risk free government issue Zero-coupon bond with the corresponding maturity. Hence, the price of any fixed income security, must today equal the sum of each of its cash flows discounted at the same rate as the corresponding government security - i.e. the corresponding risk free rate. Were this not the case, arbitrage would be possible; see Bond valuation.
The pricing formula is as below, where each cash flow ' is discounted at the rate ' which matches that of the corresponding government zero coupon instrument.
- Price =
3 Pricing derivative securities
A derivative is an instrument which allows for buying and selling of the same asset on two markets – the spot market and the derivatives market. Mathematical finance assumes that any imbalance between the two markets will be arbitraged away. Thus, in a correctly priced derivative contract, the derivative price, the strike price (or reference rate), and the spot price will be related such that no arbitrage is possible.
4 Futures
In a futures contractA futures contract is a form of forward contract, a contract to buy or sell an asset of any kind at a pre-agreed future point in time, that has been standardised for a wide range of uses. It is traded on a futures exchange. Futures may also differ from fo, for no arbitrage to be possible, the price paid on delivery (the forward price) must be the same as the cost (including interest) of buying and storing the asset. In other words, the rational forward price represents the expected future value of the underlying discounted at the risk free rate. Thus, for a simple, non-dividend paying asset, the value of the future/forward, F(t), will be found by discounting the present value S(t) at time t to maturity T by the rate of risk-free return r.
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This relationship may be modified for storage costs, dividends, dividend yields, and convenience yields; see futures contract pricingA futures contract is a form of forward contract, a contract to buy or sell an asset of any kind at a pre-agreed future point in time, that has been standardised for a wide range of uses. It is traded on a futures exchange. Futures may also differ from fo.
Any deviation from this equality allows for arbitrage as below.
- In the case where the forward price is higher: 1) The arbitrageur sells the futures contract and buys the underlying today (on the spot market) with borrowed money. 2) On the delivery date, the arbitrageur hands over the underlying, and receives the agreed forward price. 3) He then repays the lender the borrowed amount plus interest. 4) The difference between the two amounts is the arbitrage profit.
- In the case where the forward price is lower: 1) The arbitrageur buys the futures contract and sells the underlying today (on the spot market); he invests the proceeds. 2) On the delivery date, he cashes in the matured investment, which has appreciated at the risk free rate. 3) He then receives the underlying and pays the agreed forward price using the matured investment. [If he was short the underlying, he returns it now.] 4) The difference between the two amounts is the arbitrage profit.
