# Valuation theory

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

## Primitive valuables


## 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


## Chance to own a valuable


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*}


## Action valuables


### 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


### 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.