# 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.

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## 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 (short) 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 (short sell) 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.

## 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 securities - i.e. the corresponding risk free rate. Were this not the case, arbitrage would be possible and would bring the price back into line with the price based on the government issued securities; see Bond valuation.

The pricing formula is as below, where each cash flow [itex]C_t\,[itex] is discounted at the rate [itex]r_t\,[itex] which matches that of the corresponding government zero coupon instrument:

Price = [itex] P_0 = \sum_{t=1}^T\frac{C_t}{(1+r_t)^t}[itex]

## 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 arbitrage is not 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, [itex]F(t)\,[itex], will be found by discounting the present value [itex]S(t)\,[itex] at time [itex]t\,[itex] to maturity [itex]T\,[itex] by the rate of risk-free return [itex]r\,[itex].

[itex]F(t) = S(t)\times (1+r)^{(T-t)}\,[itex]

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 follows.

• 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

As above, where the value of an asset in the future is known (or expected), this value can be used to determine the asset's rational price today. In an option contract, however, 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. 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. Here, the value of the share in the up-state is S × u, and in the down-state is S × d - where u and d are multipliers with d < 1 < u and assuming d < 1+r < u; see the binomial options model.

Given these two states, the "arbitrage free" approach creates a position which will have an identical value in either state - the cash flow in one period is therefore known, and arbitrage pricing is applicable. The risk neutral approach infers expected option value from the intrinsic values at the later two nodes.

Although this logic appears far removed from the Black-Scholes formula and the lattice approach in the Binomial options model, it in fact underlies both models. 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 high number of very short time-steps (if coded correctly); Black-Scholes, in fact, models a continuous process.

The examples below 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 calls sold and 1 share owned, such that the positions value will be identical in the S up and S down states, and hence known with certainty (see Delta hedging). This value corresponds to the forward price above, and as above, for no arbitrage to be possible, the present value of the position must be its expected future value discounted at the risk free rate, r. The value of a call is then found by equating the two.

1) Solve for Δ such that:

value of position in one period = S up - Δ × (S up  strike price ) = S down - Δ × (S down  strike price)

2) solve for the value of the call, using Δ, where:

value of position today = value of position in one period ÷ (1 + r) = S current  Δ × value of call

#### The replicating portfolio

It is possible to create a position consisting of Δ shares and \$B borrowed at the risk free rate, which will produce identical cash flows to one option on the underlying share. The position created is known as a "replicating portfolio" since its cash flows replicate those of the option. As shown, in the absence of arbitrage opportunities, since the cash flows produced are identical, the price of the option today must be the same as the value of the position today.

1) Solve simultaneously for Δ and B such that:

i) Δ × S up - B × (1 + r) = (S up  strike price )
ii) Δ × S down - B × (1 + r) = (S down  strike price )

2) solve for the value of the call, using Δ and B, where:

call = Δ × S current - B

### Risk neutral valuation

Here the value of the option is calculated using the 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, with u and d as price multipliers as above. These are then weighted by their respective probabilities: probability p of an up move in the underlying, and probability (1-p) of a down move. The expected value is then discounted at r, the risk free rate.

1) solve for p

for no arbitrage to be possible in the share, todays price must represent its expected value discounted at the risk free rate:
S = [ p × (up value) + (1-p) ×(down value) ] ÷ (1+r) = [ p × S × u + (1-p) × S × d ] ÷ (1+r)
then, p = [(1+r) - d ] ÷ [ u - d ]

2) solve for call value, using p

for no arbitrage to be possible in the call, todays price must represent its expected value discounted at the risk free rate:
Option value = [ p × Option up + (1-p)× Option down] ÷ (1+r)
= [ p × (S up - strike) + (1-p)× (S down - strike) ] ÷ (1+r)

#### The risk neutrality assumption

Note that above, the risk neutral formula does not refer to the volatility of the underlying  p as solved, relates to the risk-neutral measure as opposed to the actual probability distribution of prices. Nevertheless, 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.

## Pricing shares

The Arbitrage pricing theory (APT), a general theory of asset pricing, has become influential in the pricing of shares. APT holds that the expected return of a financial asset can be modelled as a linear function of various macro-economic factors, where sensitivity to changes in each factor is represented by a factor specific beta coefficient. The model derived rate of return will then be used to price the asset correctly - the asset price should equal the expected end of period price discounted at the rate implied by model. If the price diverges, arbitrage should bring it back into line.

Note that under true arbitrage, the investor locks-in a guaranteed payoff, whereas under APT arbitrage the investor locks-in a positive expected payoff, as below. The APT thus assumes "arbitrage in expectations" - i.e that arbitrage by investors will bring asset prices back into line with the returns expected by the model.

If APT holds, then a risky asset can be described as satisfying the following relation:

[itex]E\left(r_j\right) = r_f + b_{j1}F_1 + b_{j2}F_2 + ... + b_{jn}F_n + \epsilon_j[itex]
where
• [itex]E(r_j)[itex] is the risky asset's expected return,
• [itex]r_f[itex] is the risk free rate,
• [itex]F_k[itex] is the macroeconomic factor,
• [itex]b_{jk}[itex] is the sensitivity of the asset to factor [itex]k[itex],
• and [itex]\epsilon_j[itex] is the risky asset's idiosyncratic random shock with mean zero.

Under the APT, an asset is mispriced if its current price diverges from the price predicted by the model. The APT describes the mechanism whereby arbitrage by investors will bring an asset which is mispriced back into line with its expected price. To perform the arbitrage, the investor creates a correctly priced asset (a synthetic asset) being a portfolio consisting of other correctly priced assets. This portfolio will have the same net-exposure to each of the macroeconomic factors as the mispriced asset but a different expected return. The arbitrageur creates the portfolio by identifying [itex]x[itex] correctly priced assets (one per factor plus one) and then weighting the assets such that portfolio beta per factor is the same as for the mispriced asset. Thus, when the investor is long the asset and short the portfolio (or vice versa) he has created a position which has a positive expected return (the difference between asset return and portfolio return) and which has a net-zero exposure to any macroeconomic factor and is therefore risk free. The arbitrageur is thus in a position to make a risk free profit; see the APT article for detail on the arbitrage mechanics.

The Capital asset pricing model (CAPM) is an earlier, (more) influential theory on asset pricing. Although based on different assumptions, the CAPM can, in some ways, be considered a "special case" of the APT; specifically, the CAPM's Securities market line represents a single-factor model of the asset price, where Beta is exposure to changes in value of the Market.

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