theta calculus

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

\Theta

T

and a branch operator.

Theta operator

The

\Theta

operator refers to a time step without activity or passive observation. Whenever

\Theta

occurs in a sequence of event operators the economic state is propagated by the "outside world" in a possibly random manner. The process operator

\Theta

can be taken to the power of

\Delta t

to describe none unit time steps.

\Theta

is mathematically defined as the expected value of the argument function

f

applied to tomorrow's state

X(t+\Delta t)

given a current state

x=X(t)

\Theta^{\Delta t} f(x) := \mathbb{E}\left t)\big)\big| X(t)=x\right

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

T

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

\Delta x

if applied more than once. Applied to a function

f

that depends on the value of the parameter

x

the

T_x

operator is defined as follows:

T_x^{\Delta x} f(x) = f(x+\Delta x)

The exponent

\Delta x

may functionally depend on

x

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

O_1

and

O_2

. 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

C

O_2\end{matrix}

The choice condition

C

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

C_{\max}

C_{\max} = 1_{O_1 > O_2}

= Examples = The ultimate goal of the

\Theta

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

c

Bond deal

A simple bond deal looks like this:

   T_c^{-90} \Theta T_c^{100}\,

Firstly, withdraw 90 units from your cash account

c

. 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:

\left(\Theta T_c^5\right)^{10} T_c^{100}

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

A

and

B

after option maturity, i.e. wait

M

times and decide.

B\end{matrix}

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

a

and

b

. Each player gets a utility from the final outcome.

U = U_a,U_b = 2a-b \,

We play as follows: A decides, fully anticipating B's reaction. B decides, knowing A's action.

T_b^{-1}\end{matrix} \right) U

Evaluating the operator term with initial values of

a=0, b=0

results in

-1,-1

. = Evaluation of strategies = In order to derive any information from

\Theta

-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

f(x)

lign=center\| 大
$x=X_0$	}	valign=top\| ...strategy... $f(x)$

Example

In the following expamle we start with an initial value of

c=0

. As strategy we use the previously defined bond deal with a

\Theta

that charges interest on our dept

\Theta=T_c{}^{rc}

. Then we ask for the final value on cash account

c

lign=center\| 大
${}_{c=0}$	}	valign=top\| $T_c^{-90} T_c^{rc} T_c^{100}$ $c$ =
lign=right\| 大	$T_c^{-90} T_c^{rc}$ $(c+100)$ =
lign=right\| 大	$T_c^{-90}$ $((1+r)c+100)$ =
lign=right\| 大	$((1+r)(c-90)+100)$ = -90(1+r)+100

= Links =

An introduction to Theta-calculus

*The Theta-notation for stochastic processes

<< Previous

Word Browser

Next >>

bdellovibrionaceae
vibrionaceae
circadian clock
invisalign
list of united states army airfields
pasteurellaceae
northern news
willem schermerhorn
prairie public television
jacques lubtchansky
hugh graham, 1st baron atholstan

don zagier
morning glory cloud
elections in the democratic republic of congo
franois benot hoffmann
union of industrial and employers' confederations of europe
peter pook
daniel bovet
neisseriaceae
methylophilaceae
hailstorm over truk lagoon
desulfurellaceae

operation hailstone
metamorphosis (illusion)
xanthomonadaceae
johor bahru landmarks
mr. driller drill spirits
trilateral offices
art criticism
cardiobacteriaceae
timeline of the 2004 u.s. presidential election controversy and irregularities
crucial ministries
types of staging

alteromonadales
cleveland college of art and design
museo archeologico nazionale napoli
les ollires sur eyrieux
arek hersh
hinckley township, medina county, ohio
technosoft
dante gabriel ramrez erazo
broward
caldor
cinema of turkey