Rational pricing
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.
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.
- An asset with a known price in the future, must today trade at that price discounted at the 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 <math> C_t <math> is discounted at the rate <math> r_t <math> which matches that of the corresponding government zero coupon instrument.
- Price = <math> P_0 = \sum_{t=1}^T\frac{C_t}{(1+r_t)^t}<math>
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 / reference rate, and the spot price will be related such that no arbitrage is possible.
Futures
In a futures contract, 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.
- <math>F(t) = S(t)\times (1+r)^{(T-t)}<math>
This relationship may be modified for storage costs, dividends, dividend yields, and convenience yields; see futures contract pricing.
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.
Options
In an Option contract, exercise is dependent on the price of the underlying, and hence payment is uncertain. Option pricing models therefore include logic which either "locks in" or "infers" the value in one period's time. As above, where the value of an asset in the future is known (or expected), this can be used to determine the asset's rational price today. Methods which "lock-in" future cash flows assume ?arbitrage free pricing?; those which infer expected value assume ?risk neutral valuation?. Both assumptions deliver identical results.
Both approaches assume a ?Binomial model? for the behavior of the underlying instrument, which allows for only two states - up or down. If S is the current price, then in the next period the price will either be S up or S down; see the binomial options model. In the arbitrage free approach, given these two states, it is possible to create a position which will have an identical value in either state - the cash flow in one period is therefore known. The risk neutral approach infers expected option value from the intrinsic values at the later two nodes.
The assumption of binomial behaviour in the underlying price is defensible as the number of time steps between today (valuation) and exercise increases, and the period per time-step is increasingly short. The Binomial options model allows for a very high number of time-steps (if coded correctly); Black-Scholes, in fact, models a continuous process.
The examples that follow have shares as the underlying, but may be generalised to other instruments. The value of a put option can be derived as below, or may be found from the value of the call using put-call parity.
Arbitrage Free Pricing
Here, the future payoff is "locked in" using either "delta hedging" or the "replicating portfolio" approach. As above, this payoff is then discounted, and the result is used in the valuation of the option today.
Delta hedging
It is possible to create a position consisting of Δ risk neutrality assumption. Under this assumption, the ?expected value? (as opposed to "locked in" value) is discounted. The expected value is calculated using the intrinsic values from the later two nodes (?Option up? and ?Option down?) weighted by their respective probabilities ? ?probability? p of an up move in the underlying, and ?probability? (1-p) of a down move. The value of the share if it went up is S × u, the value if it went down is S × d. Here, u and d are multipliers with d < 1 < u. The expected value is then discounted at r, the volatility of the underlying. In fact, p as solved, relates to the risk-neutral measure as opposed to the actual probability distribution of prices. Still, both Arbitrage free pricing and Risk neutral valuation deliver identical results; in fact it can be shown that ?Delta hedging? and ?Risk neutral valuation? are identical formulae expressed differently. Given this equivalence, it is valid to assume ?risk neutrality? when pricing derivatives.
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