On financial valuation

1Notes

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

It is straightforward to write a Haskell program that implements this theory. Indeed a similar thing has been done in 2000 [1].

2Primitive valuables

The value of the null valuable is zero.

The value of an amount of cash is the amount itself.

\[ \newcommand\fVal{\mathsf{val}} \newcommand\fEmpty{\mathsf{empty}} \newcommand\fCash{\mathsf{cash}} \newcommand\fNeg{\mathsf{neg}} \newcommand\fBoth{\mathsf{both}} \newcommand\fMax{\mathsf{max}} \newcommand\fMin{\mathsf{min}} \begin{align*} \fVal ~ \fEmpty &= 0 \\ \fVal ~ (\fCash ~ a) &= a. \end{align*} \]

3Basic 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 unary \(-\) taking precedence over binary \(+\), and \(+\) associating to the right: \[ - x + y + z = (-x) + (y + z). \]

4The value of a 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.

5The value of a 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 equation¹: \[ \begin{align*} \fRiskApt~0 &= 0 \\ \fRiskApt~1 &= 1 \\ \fRiskApt~(p\cdot q) &= \fRiskApt~p\cdot\fRiskApt~q. \end{align*} \] The function \(\fRiskApt\) will be a power function², 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.)

6Laws 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: \[ \fVal~(\fDelay~d~(\fBoth~x~y)) = \fVal~(\fBoth~(\fDelay~d~x)~(\fDelay~d~y)). \]

7The value of annuities and perpetuities

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 \(a\): \[ \newcommand\fDeposit{\textsf{deposit}} \newcommand\fInterest{\textsf{interest}} \begin{align*} \fDeposit ~ p ~ r ~ a &= \fCash ~ a + \fDelay ~ p ~ (\fInterest ~ p ~ r ~ a) \\ \fInterest ~ p ~ r ~ a &= r \cdot a + \fDelay ~ p ~ (\fInterest ~ p ~ r ~ ((1+r)\cdot a)). \end{align*} \]

8The value of actions

8.1Buying or selling something

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

8.2Buying 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.

9The value of forward contracts?

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

10Option 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.

How do we value them?

11Bibliography

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1. http://en.wikipedia.org/wiki/Functional_equation↩

2. http://math.stackexchange.com/questions/43964/if-fxy-fxfy-then-show-that-fx-xt-for-some-t↩