Editing Rational pricing (section)

==Pricing derivatives==
A [[derivative (finance)|derivative]] is an instrument that allows for buying and selling of the same asset on two markets&nbsp;– the [[spot price|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. See [[Fundamental theorem of arbitrage-free pricing]].

===Futures===<!-- This section is linked from [[Contango]] -->
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 (the "[[Rational pricing#An asset with a known future-price|asset with a known future-price]]", as above); see [[Spot–future parity]]. Thus, for a simple, non-dividend paying asset, the value of the future/forward, <math>F(t)\,</math>, will be found by accumulating the present value <math>S(t)\,</math> at time <math>t\,</math> to maturity <math>T\,</math> by the rate of risk-free return <math>r\,</math>.

:<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|futures contract pricing]].

Any deviation from this equality allows for arbitrage as follows.

* In the case where the forward price is ''higher'':
# The arbitrageur sells the futures contract and buys the underlying today (on the spot market) with borrowed money.
# On the delivery date, the arbitrageur hands over the underlying, and receives the agreed forward price.
# He then repays the lender the borrowed amount plus interest.
# The difference between the two amounts is the arbitrage profit.

* In the case where the forward price is ''lower'':
# The arbitrageur buys the futures contract and sells the underlying today (on the spot market); he invests the proceeds.
# On the delivery date, he cashes in the matured investment, which has appreciated at the risk free rate.
# He then receives the underlying and pays the agreed forward price using the matured investment. [If he was [[short selling|short]] the underlying, he returns it now.]
# The difference between the two amounts is the arbitrage profit.

===Swaps===
Rational pricing underpins the logic of [[Swap (finance)|swap]] valuation. Here, two [[Counterparty|counterparties]] "swap" obligations, effectively exchanging [[cash flow]] streams calculated against a notional [[:wikt:principal|principal]] amount, and the value of the swap is the [[present value]] (PV) of both sets of future cash flows "netted off" against each other. 

To be arbitrage free, the terms of a swap contract are such that, initially, the [[Net present value|''Net'' present value]] of these future cash flows is equal to zero; see {{slink|Swap (finance)#Valuation and Pricing}}.  Once traded, swaps can (must) also be priced using rational pricing. 

The examples below are for [[interest rate swap]]s{{snd}} and are representative of pure rational pricing as these exclude [[credit risk]]{{snd}} although the principle applies to [[:Category:Swaps (finance)|any type of swap]].
<!-- Note that since the [[2007–2012 global financial crisis]], pricing is under a "multi-curve" framework, whereas previously it was off a single, "self discounting", curve; see [[Financial economics#Derivative pricing]] for context. Of course, under both approaches, pricing must be arbitrage free, and the logic below therefore holds under either, although see [[Interest rate swap#Valuation and pricing]] for formulae. (not arbitrage related, and therefore confusing...)-->

====Valuation at initiation====
Consider a fixed-to-floating Interest rate swap where Party A pays a fixed rate ("[[Swap rate]]"), and Party B pays a floating rate. Here, the ''fixed rate'' would be such that the present value of future fixed rate payments by Party A is equal to the present value of the ''expected'' future floating rate payments (i.e. the NPV is zero). Were this not the case, an arbitrageur, C, could:
# Assume the position with the ''lower'' present value of payments, and borrow funds equal to this present value
# Meet the cash flow obligations on the position by using the borrowed funds, and receive the corresponding payments—which have a higher present value
# Use the received payments to repay the debt on the borrowed funds
# Pocket the difference&nbsp;– where the difference between the present value of the loan and the present value of the inflows is the arbitrage profit

====Subsequent valuation====
The Floating leg of an interest rate swap can be "decomposed" into a series of [[forward rate agreement]]s. Here, since the swap has identical payments to the FRA,  arbitrage free pricing must apply as above&nbsp;– i.e. the value of this leg is equal to the value of the corresponding FRAs. Similarly, the "receive-fixed" leg of a swap can be valued by comparison to a [[Bond (finance)|bond]] with the same schedule of payments. (Relatedly, given that their [[underlying]]s have the same cash flows, [[bond option]]s and [[swaption]]s are equatable.)  See further under {{slink|Swap (finance)#Using bond prices}}. The difference between the [[Interest rate cap and floor]] values equate to the swap value, per similar arbitrage arguments.

===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 (finance)|option]] contract, however, exercise is dependent on the price of the underlying, and hence payment is uncertain. Option pricing models therefore include logic that either "locks in" or "infers" this future value; both approaches deliver identical results. Methods that lock-in future cash flows assume ''arbitrage free pricing'', and those that infer expected value assume ''[[Rational pricing#Risk neutral valuation|risk neutral valuation]]''.

To do this, (in their simplest, though widely used form) both approaches assume a "binomial model" for the behavior of the [[underlying instrument]], which allows for only two states&nbsp;– 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]]). Then, given these two states, the "arbitrage free" approach creates a position that has an identical value in either state&nbsp;– the cash flow in one period is therefore known, and arbitrage pricing is applicable. The risk neutral approach infers expected option value from the [[Option time value#Intrinsic value|intrinsic value]]s 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; see [[Black–Scholes equation|The Black–Scholes PDE]]. 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 correspondingly short. The Binomial options model allows for a high number of very short time-steps (if [[Source code|coded]] correctly), while Black–Scholes, in fact, models a [[Continuous-time Markov process|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=====<!-- This section is linked from [[Black–Scholes]] -->
It is possible to create a position consisting of '''Δ''' shares and 1 [[call option|call]] sold, 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 certain value corresponds to the forward price above ([[Rational pricing#An asset with a known future-price|"An asset with a known future price"]]), 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.

# Solve for Δ such that:
#: value of position in one period = Δ × ''S up'' - [[Option time value#Intrinsic value|<math>max</math>]] (''S up''&nbsp;– strike price, 0)  =  Δ × ''S down'' - [[Option time value#Intrinsic value|<math>max</math>]] (''S down''&nbsp;– strike price, 0)
# Solve for the value of the call, using Δ, where:
#: value of position today = value of position in one period ÷ (1 + r) =  Δ × ''S current''&nbsp;– value of call

=====The replicating portfolio=====
{{Main|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 above ([[Rational pricing#Assets with identical cash flows|"Assets with identical cash flows"]]), 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.

# Solve simultaneously for Δ and B such that:
#* Δ × ''S up'' - B × (1 + r) =  [[Option time value#Intrinsic value|<math>\max</math>]] (0, ''S up''&nbsp;– strike price)
#* Δ × ''S down'' - B × (1 + r) =  [[Option time value#Intrinsic value|<math>\max</math>]] (0, ''S down''&nbsp;– strike price)
# Solve for the value of the call, using Δ and B, where:
#* call = Δ × ''S current'' - B

Note that there is no discounting here{{snd}} the interest rate appears only as part of the construction. This approach is therefore used in preference to others where it is not clear whether the risk free rate may be applied as the [[discount window|discount rate]] at each decision point, or whether, instead, a [[Capital asset pricing model#Asset-specific required return|premium over risk free]], differing by state, would be required. The best example of this would be under [[real options analysis]]<ref name="Reilly & Brown">See Ch. 23, Sec. 5, in: Frank Reilly, Keith Brown (2011). "Investment Analysis and Portfolio Management." (10th Edition). South-Western College Pub. {{ISBN|0538482389}}</ref> where managements' actions actually change the risk characteristics of the project in question, and hence the [[Required rate of return]] could differ in the up- and down-states. Here, in the above formulae, we then have: "Δ × ''S up'' - B × (1 + r '''''up''''')..." and "Δ × ''S down'' - B × (1 + r '''''down''''')...". See {{slink|Real options valuation#Technical considerations}}. (Another case where the modelling assumptions may depart from rational pricing is the [[Employee stock option#Valuation|valuation of employee stock options]].)

====Risk neutral valuation====<!-- This section is linked from [[Black–Scholes]] -->
Here the value of the option is calculated using the [[Risk-neutral measure|risk neutrality]] assumption. Under this assumption, the "[[expected value]]" (as opposed to "locked in" value) is [[discounted]]. The expected value is calculated using the [[Option time value#Intrinsic value|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 interest rate|risk-free rate]].

# Solve for p
#: under risk-neutrality, for no arbitrage to be possible in the share, today's price must represent its expected value discounted at the risk free rate (i.e., the share price is a [[Martingale (probability theory)|Martingale]]):
#:<math>
\begin{align}
S &= \frac{p \times S_u + (1-p)\times S_d}{1 + r} \\
&= \frac{p\times u\times S + (1-p)\times d\times S}{1 + r} \\
\Rightarrow p &= \frac{(1+r) - d}{u-d}\\
\end{align}
</math>
# 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:
#:<math>
\begin{align}
C &= \frac{p\times C_u + (1-p) \times C_d}{1+r} \\
&= \frac{p\times \max(S_u - k, 0)  + (1-p) \times\max(S_d -k, 0)}{1+r} \\
\end{align}
</math>

=====The risk neutrality assumption=====
Note that above, the risk neutral formula does not refer to the expected or forecast return of the underlying, nor its [[Volatility (finance)|volatility]]{{snd}} 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" use identical formulae expressed differently. Given this equivalence, it is valid to assume "risk neutrality" when pricing derivatives. A more formal relationship is described via the [[fundamental theorem of arbitrage-free pricing]].