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Valuation theory

This theory assumes that everybody is rational. (What is rational?)

Primitive valuables

The value of the null valuable is zero. The value of an amount of cash is the amount. \[ \newcommand\fVal{\mathsf{val}} \newcommand\fEmpty{\mathsf{empty}} \newcommand\fCash{\mathsf{cash}} \newcommand\fNeg{\mathsf{neg}} \newcommand\fBoth{\mathsf{both}} \newcommand\fMax{\mathsf{max}} \newcommand\fMin{\mathsf{min}} \begin{align*} \fVal ~ \fEmpty &= 0 \\ \fVal ~ (\fCash ~ a) &= a. \end{align*} \]

Combinators of valuables

The expression \(\fNeg~x\) represents giving up \(x\). The expression \(\fBoth~x~y\) represents owning both \(x\) and \(y\). The expression \(\fMax~x~y\) represents choosing the one of \(x\) or \(y\) that has the greatest value. The expression \(\fMin~x~y\) represents choosing the one of \(x\) or \(y\) that has the least value. \[ \begin{align*} \fVal ~ (\fNeg~x) &= - \fVal~x \\ \fVal ~ (\fBoth ~ x ~ y) &= \fVal ~ x + \fVal ~ y \\ \fVal ~ (\fMax ~ x ~ y) &= \fMax ~ (\fVal ~ x) ~ (\fVal ~ y). \\ \fVal ~ (\fMin ~ x ~ y) &= \fMin ~ (\fVal ~ x) ~ (\fVal ~ y). \end{align*} \]

If \(x\) and \(y\) are valuables, we will also write \[ \begin{align*} -x &= \fNeg~x \\ x+y &= \fBoth~x~y \\ x-y &= \fBoth~x~(\fNeg~y) \end{align*} \] with \(-\) taking precedence over \(+\) and \(+\) associating to the right: \[ - x + y + z = (-x) + (y + z). \]

Delayed valuable

The value of a valuable that becomes owned after a delay \(d\) is \[ \newcommand\fDelay{\mathsf{delay}} \newcommand\fDiscRate{\textsf{discrate}} \fVal ~ (\fDelay ~ d ~ x) = (\fDiscRate~x)^d \cdot \fVal~x \] where \(\fDiscRate~x\) is the discount rate of the valuable: \[ \fDiscRate~(\fCash~\_) = \frac{1}{1+r} \] where \(r\) is the risk-free interest rate. Both \(d\) and \(r\) must use the same unit of time.

Chance to own a valuable

The value of a chance to own \(x\) with probability \(p\) is \[ \newcommand\fChance{\textsf{chance}} \newcommand\fRiskApt{\textsf{riskapt}} \fVal ~ (\fChance ~ p ~ x) = \fRiskApt~p \cdot \fVal~x. \] where everyone is assumed to have the same risk appetite function \(\fRiskApt\) satisfying this functional equation: \[ \begin{align*} \fRiskApt~0 &= 0 \\ \fRiskApt~1 &= 1 \\ \fRiskApt~(p\cdot q) &= \fRiskApt~p\cdot\fRiskApt~q. \end{align*} \] The function \(\fRiskApt\) will be a power function, that is \[ \fRiskApt~p = p^k \] where \(k\) is a constant. If \(k = 1\) (and thus \(\fRiskApt~p = p\)), the valuation is risk-neutral. If \(k > 1\) (and thus \(\fRiskApt~p < p\)), the valuation is risk-averse.

If you are interested about Cauchy’s functional equation, there is an overview at Math Stack Exchange.

(The function \(\fRiskApt\) is an endomorphism of the group \(([0,1],\cdot)\), if this helps you find things on the Internet.)

Laws of valuables

These laws should be intuitive: \[ \begin{align*} -(x+y) &= -x + -y \\ - \fMax~x~y &= \fMin~(-x)~(-y). \end{align*} \]

Delaying a delay just makes a bigger delay for valuation purposes: \[ \fVal \circ \fDelay~a \circ \fDelay~b = \fVal \circ \fDelay~(a+b). \]

Chance and delay can be commuted for valuation purposes: \[ \fVal \circ \fChance ~ p \circ \fDelay ~ t = \fVal \circ \fDelay ~ t \circ \fChance ~ p. \]

Chance to get a chance just makes a smaller chance for valuation purposes: \[ \fVal \circ \fChance ~ p \circ \fChance ~ q = \fVal \circ \fChance~(p \cdot q). \]

Delay distributes over \(\fBoth\) for valuation purposes: \[ \fVal~(\fDelay~d~(\fBoth~x~y)) = \fVal~(\fBoth~(\fDelay~d~x)~(\fDelay~d~y)). \]

Annuity-like valuables

This is an annuity with \(n\) installments, each having amount \(a\), paid every period \(p\):

\[ \newcommand\fAnnuity{\textsf{annuity}} \begin{align*} \fAnnuity ~ p ~ 0 ~ a &= 0 \\ \fAnnuity ~ p ~ n ~ a &= \fCash ~ a + \fDelay ~ p ~ (\fAnnuity ~ p ~ (n-1) ~ a). \end{align*} \]

Perpetuity with period \(p\) and individual installment amount \(a\): \[ \newcommand\fPerpetuity{\textsf{perpetuity}} \begin{align*} \fPerpetuity ~ p ~ a &= \fCash ~ a + \fDelay ~ p ~ (\fPerpetuity ~ p ~ a) \\ \fVal~(\fPerpetuity~p~a) &= \sum_{k=0}^\infty \frac{a}{(1+r)^{k \cdot p}} = \frac{a}{1-\frac{1}{(1+r)^p}}. \end{align*} \]

Time deposit with period \(p\) unit time, interest rate \(r\) per unit time, and principal amount \(a\): \[ \newcommand\fDeposit{\textsf{deposit}} \newcommand\fInterest{\textsf{interest}} \begin{align*} \fDeposit ~ p ~ r ~ a &= \fCash ~ a + \fDelay ~ p ~ (\fInterest ~ p ~ r ~ a) \\ \fInterest ~ p ~ r ~ a &= r \cdot a + \fDelay ~ p ~ (\fInterest ~ p ~ r ~ ((1+r)\cdot a)). \end{align*} \]

Action valuables

Buying or selling something

\[ \newcommand\fBuy{\textsf{buy}} \newcommand\fSell{\textsf{sell}} \newcommand\fBuyprice{\textsf{buyprice}} \newcommand\fSellprice{\textsf{sellprice}} \begin{align*} \fBuy~x &= - \fCash~(\fBuyprice~x) + x \\ \fSell~x &= \fCash~(\fSellprice~x) - x \end{align*} \]

Buying and holding a stock forever

Buying a stock and holding it forever is equivalent to paying for a perpetuity: \[ \newcommand\fBuyhold{\textsf{buyhold}} \fBuyhold ~ s = -\fCash~ c + \fPerpetuity ~ p ~ d \] where \(c\) is the purchase amount, \(p\) is the dividend period, and \(d\) is the individual dividend amount.

Derivatives

Forward contracts

The buyer of a forward contract pays now to get the underlying later: \[ \newcommand\fForward{\textsf{forward}} \newcommand\fEuopt{\textsf{euopt}} \newcommand\fAmopt{\textsf{amopt}} \newcommand\fCurrentprice{\textsf{currentprice}} \fForward~d~x = - \fCash~(\fCurrentprice~x) + \fDelay~d~x. \]

Option contracts

From the buyer’s point of view, an European option with expiration \(d\) on an underlying valuable \(x\) is \[ \fEuopt ~ d ~ x = \fDelay ~ d ~ (\fMax~\fEmpty~x). \]

From the buyer’s point of view, an American option with expiration \(d\) on an underlying valuable \(x\) is \[ \fAmopt ~ d ~ x = \begin{cases} \fEmpty &\text{if } d < 0 \\ \fMax~x~(\fDelay~\delta~(\fAmopt ~ (d - \delta) ~ x)) &\text{otherwise} \end{cases} \] where \(\delta\) is the hyperreal infinitesimal.

Note

This is a very sketchy draft. Many other stuffs need to be written down here.