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Theta calculus is a mathematical calculus for the
description of stochastic processes, financial derivatives and
possible strategies in game theory. It uses operators to
describe elementary activities or events. All kinds of contracts, strategies and multiperiod games can be captured in terms of
quantitative implications by a vocabulary of three words:
Each elementary activity is represented by
an operator, that can be interpreted in an operator sequence as a
chronologically ordered list of events.
Elementary operators
The basic idea of theta calculus is a notation of all
strategies, games and portfolios in terms of three operators referring
to the activities: waiting, transacting and deciding. These
activities are written by the mathematical symbols
<math>\Theta<math>, <math>T<math> and a branch operator.
Theta operator
The <math>\Theta<math> operator refers to a time step without
activity or passive observation. Whenever <math>\Theta<math> occurs
in a sequence of event operators the economic state is propagated
by the "outside world" in a possibly random manner.
The process operator <math>\Theta<math> can be taken to the power of
<math>\Delta t<math> to describe none unit time steps. <math>\Theta<math> is mathematically defined as the expected value of the argument function <math>f<math> applied to tomorrow's state <math>X(t+\Delta t)<math> given a current state <math>x=X(t)<math>.
<math>
\Theta^{\Delta t} f(x) :=
\mathbb{E}\left[f\big(X(t+\Delta t)\big)\big| X(t)=x\right]
<math>
The operator always corresponds to a Markovian
process, or one that can be turned into Markovian form.
For the valuation of derivative securities the expectated value should be computed with respect to the risk-neutral measure.
Transaction operator
The transaction operator <math>T<math> increases a process paramater
by one unit. It is used for the transfer of goods or assets between
acounting variables and to apply deterministic impacts on any process
variable.
Mathematically, the operator replaces every instance of the index
variable with the variable plus the one, or an operator exponent <math>\Delta x<math> if
applied more than once. Applied to a function <math>f<math> that depends on the value of the parameter <math>x<math> the
<math>T_x<math> operator is defined as follows:
<math>
T_x^{\Delta x} f(x) = f(x+\Delta x)
<math>
The exponent <math>\Delta x<math> may functionally depend on
<math>x<math>.
Decision operator
An option is defined by the alternatives among which can be selected
and by the entity that does decide. Feasible choices are specified by
the two portfolios <math>O_1<math> and <math>O_2<math>.
Depending on the choice, one
of the optional substrategies determines the remaining sequence after
the decision. The deciding entity is characterized by her
choice condition <math>C<math>.
<math>
C O_1 + (1-C) O_2
=:
\begin{matrix}{}_{{}^{C}}\\ \Big\langle\\{}^{{}_{1-C}}\end{matrix}
\begin{matrix}O_1\\[1ex] O_2\end{matrix}
<math>
The choice condition <math>C<math> is an operator.
In the most common cases the function itself
carries the information on which choice is preferred. We can choose the more valuable scenario the with the choice condition <math>C_{\max}<math>.
<math>
C_{\max} = 1_{O_1 > O_2}
<math>
Examples
The ultimate goal of the <math>\Theta<math> notation is the ability
to describe complex contracts, game rules and strategies. We can now
derive mathematical terms that fully represent our strategy, including
all outstanding events, all embedded options, the sensitivity to
random events and the room for further activity.
Financial Products
The calculus can be used to describe financial investments and
Derivative securities. We will generally assume, that your
account balance is <math>c<math>
Bond deal
A simple bond deal looks like this:
<math>
T_c^{-90} \Theta T_c^{100}\,
<math>
Firstly, withdraw 90 units from your cash account <math>c<math>. Then,
wait one period (maturity is one). Finally add 100 units onto your account.
Coupon bond
A coupon bond with maturity 10 and coupon rate 5 looks like that:
<math>
\left(\Theta T_c^5\right)^{10} T_c^{100}
<math>
Read: Wait and receive 5. Repeat 10 times. Then receive 100.
Option
An option is a contract with the right to choose between predefined
alternative investments <math>A<math> and <math>B<math> after option
maturity, i.e. wait <math>M<math> times and decide.
<math>
\Theta^M
\Big\langle
\begin{matrix}A\\[1ex] B\end{matrix}
<math>
Games
We will investigate a game, known as the prisoners dilemma in game theory, with
two players A and B, each facing a choice between complying
(+1) and defecting (-1). The selected action of the players are
represented by variables <math>a<math> and <math>b<math>. Each player gets a utility from
the final outcome.
<math>
U = [U_a,U_b] = [2b-a, 2a-b]\,
<math>
We play as follows: A decides, fully anticipating B's reaction.
B decides, knowing A's action.
<math>
\left(
\begin{matrix}\\ \Big\langle\\{}^{{}_{\max_1}}\end{matrix}
\begin{matrix}T_a^{+1}\\[1ex] T_a^{-1}\end{matrix}
\right)
\left(
\begin{matrix}\\ \Big\langle\\{}^{{}_{\max_2}}\end{matrix}
\begin{matrix}T_b^{+1}\\[1ex] T_b^{-1}\end{matrix}
\right)
U
<math>
Evaluating the operator term with initial values of <math>a=0, b=0<math> results in <math>[-1,-1]<math>.
Evaluation of strategies
In order to derive any information from <math>\Theta<math>-calculus terms
one must determine the initial state of the process variables, run through
the defined strategy and ask a question about the final state.
Initialize...Strategy....Question?
The evaluation operator 大 is used as an operator to insert initial values. It is applied from the left hand side an is always the first operator in a sequence. It simply replaces every remaining occurance of a state variable with its initial value.
After running through the strategy we can ask for the expected value of any
function of state variables <math>f(x)<math>
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| ...strategy... <math>f(x)<math>
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Example
In the following expamle we start with an initial value of <math>c=0<math>.
As strategy we use the previously
defined bond deal with a <math>\Theta<math> that charges interest
on our dept <math>\Theta=T_c{}^{rc}<math>. Then we ask for the final value on cash account <math>c<math>.
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| <math>T_c^{-90} T_c^{rc} T_c^{100}<math> <math>c<math> =
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| 大
| <math>T_c^{-90} T_c^{rc}<math> <math>(c+100)<math> =
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| 大
| <math>T_c^{-90}<math> <math>((1+r)c+100)<math> =
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| 大
| <math>((1+r)(c-90)+100)<math> = -90(1+r)+100
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