# 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.