Definition of computational price, pricing process and pricing algorithm

Let T be a set and (E, ξ) a measurable space. A stochastic process X indexed by T with values in (E, ξ) is a collection of random variables X = (X

t

)

t ∈ T

from a fixed probability space (Ω, 𝔽, ℙ)

²⁷²

to (E, ξ),

269 Tavella, Domingo A. Quantitative Methods in Derivative Pricing. An Introduction to Computational Finance. Wiley Finance, 2002.

270 In this chapter section, the font size is increased to 13, in order to facilitate reading. Also, some paragraphs are in brown color when colloquial explanations are given to complement the formal theory exposition.

271 For my probability theory exposition I used the following references:

Spieksma, Flora and van Zanten, Harry (adaptator). “An Introduction to Stochastic Processes in Continuous Time…”. Lecture notes‒ Stochastic processes - Fundamentals course. University of Leiden.

Spring, 2016. (http://www.math.leidenuniv.nl/~spieksma/SPspring08.html)

Lalley, Steven and Mykland Per. “Brownian motion”. Lecture notes‒Statistics 313: Stochastic Processes II course. The University of Chicago. Spring, 2013. (http://galton.uchicago.edu/~lalley/Courses/313/) Lowther, George. ‘Filtrations and Adapted Processes’. Almost Sure blog. November 8, 2009. At:

https://almostsure.wordpress.com/2009/11/08/filtrations-and-adapted-processes/

272 A probability space consists of three elements:

1. A sample spaceΩ which is the set of all possible outcomes, i.e. the universe of the random process possible results.

2. A set of events

𝔽

where each event is a set of possible outcomes, including the empty set (zero outcomes), the universal set (all possible outcomes) and every subset of possible outcomes/results.

𝔽

is an σ-algebra.

ℙ is the assignment of probabilities to the events; i.e. a function from events (domain) to probabilities (image). ℙ is a probability measure that assigns a value between 0 and 1 to each 𝓕.

89

T is a set of all the observations instants of the process, that is, T represent a time index. (E, ξ) represents the state space of the process. For every fixed observation instant t ∈ T, the stochastic process gives us an (E, ξ)-valued random element X

t

on (Ω, F, P). Simply put, a stochastic process is a phenomenon that can be thought of as evolving in time in a random manner.

A standard Brownian process (or a standard Wiener process) {S

t

}

t ≥ 0+

is a process with the following properties:

(1) W

0

= 0.

(2) With probability 1, the function t →W

t

is continuous in t.

(3) The process {S

t

}

t ≥ 0

has stationary, independent increments.

(4) The increment S

t+h

−S

h

has the NORMAL(0,t) distribution.

{S

t

}

t ≥ 0+

means “a family of random variables S

t

indexed by nonnegative real numbers t, defined on a common probability space (Ω,F,P), t ∈ T”. If S

T

is a stochastic Brownian process it means that the increment S

t+h

− S

t

is statistically independent of

‘what happened up to time t’. Simply put S

T

is irregular and erratic.

A collection (𝓕

t

)

t∈T

of sub-σ-algebras is called a filtration if 𝓕

h

⊂ 𝓕

t

for all h ≤ t. A stochastic process X defined on (Ω, 𝔽, ℙ) and indexed by T is called adapted to the filtration if for every t ∈ T, the random variable X

t

is 𝓕

t

-measurable. Then (Ω, 𝔽, (𝓕

t

)

t∈T

, ℙ) is a filtered probability space.

A stochastic process X is adapted if X

t

is an 𝓕

t

-measurable random variable for each time t > 0. Conversely, the filtration generated by any process X is the smallest filtration with respect to which it is adapted. This is given by 𝓕

t X

= σ(X

h

: h ≤ t), and is referred to as the natural filtration of X.

We can think of a filtration as a flow of information, a concept that it is necessary to represent the information available at each time. The σ-algebra F

t

contains the events that can happen ‘up to time t’. One reason for using filtrations is to define ’adapted’

processes. This is just saying that the value X

t

is observable by time t.

An (𝓕

t

)-adapted, real-valued process M = (M

n

)

n ≥ 0

is called a martingale with respect to the filtration (𝓕

t

) if

E [ M

t

] < ∞ ⩝ t ∈ T and E [ M

t

| 𝓕

s

] = M

s

⩝ t ≤ T.

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic

process) for which, at a particular time in the realized sequence, the expectation of

the next value in the sequence is equal to the present observed value even given

knowledge of all prior observed values. A martingale is a stochastic process whose

expected value at each step equals its previous realization/observed value. This is

the most important process in the general theory of stochastic processes. Its defining

characteristic is the so-called martingale property: the best prediction for the next

90

realization is the current value of the process

²⁷³

. Intuitively, a martingale is a process that is ‘constant on average’. Given all information up to time s, the best guess for the value of the process at time t ≥ s is simply the current value M

s

.

Now, let 𝔽 be an abstract stochastic economy and (E, ξ) be the measurable space ℝ

⁺

. Let S be a Brownian strictly positive discrete-time stochastic process S = (S

t

)

t ∈ {0,T}

. S is adapted to the filtration 𝓕 = { 𝓕

0

, 𝓕

T

}.

Let ℙ* be a probability measure equivalent to ℙ, such that the relative (discounted) price process S* is defined as

S

0

*= S

0

S

T

* = (1+r)

^-1

S

T

, where r is the risk-free interest rate.

A price is defined as a realisation (an instance) of the price process (S

t

)

t ∈ T

. This means that a price is a random variable and has an expected value, and that the stochastic price process S* is a martingale.

――――――――

Recall that since the price process S* follows a P*-martingale, then the discounted price process can be conceptualized as a fair game model in a risk-neutral economy, that is, in a stochastic economy in which the probabilities of future price fluctuations are determined by the martingale measure P*.

It is because of this that P* is labeled as a risk-neutral probability and 𝔽 a risk-neutral economy.

Following with our interpretations (Ω, (𝓕t) t∈[0,T]) is the set of all market scenarios endowed with a filtration (𝓕t) t∈[0,T] representing the flow of information with time. Recall as well that a sequence Y1,Y2,Y3,…,Yn is said to be a martingale with respect to another sequence X1,X2,X3,…,Xn...for all n if E(Yn+1 | X1,…,Xn) = Yn.. A filtration is a growing sequence of sigma algebras 𝓕1 ⊆ 𝓕2 ⊆ 𝓕3 … ⊆ 𝓕n

(𝓕0 is trivial). So whenever we write E(Yn | X1,…,Xn) we can alternatively write it as E(Yn+1 | 𝓕n), where 𝓕n is a sigma algebra that makes the variables X1, X2…,Xn measurable. A 𝓕n is simply an increasing sequence of sigma algebras. That is we are conditioning on growing amounts of information. Since the variables X1, X2…,Xn are, in fact, prices (realisations of a price process), this accounts for the economists’ observation that prices accumulate and incorporate all the relevant information in the economy²⁷⁴. Formally put:

Let X = {Xt : t ∈ ℝ⁺} be a real-valued stochastic process: a family of real random variables representing prices all defined on the same probability space Ω representing market scenarios.

Define 𝓕t = “information available by observing the process up to time t”

= what we learn by observing Xs for 0 ≤ s ≤ t ²⁷⁵

We can go even further. Given a set of benchmark assets (St) t ≥ 0, let φ be a trading strategy. Then the gain of the trading strategy (φt) t ≥ 0 is defined via the stochastic integral ∫ φdS with respect to the price process.

As Biagini &Cont (2006) ²⁷⁶ point out, the stochastic models of financial markets represent the evolution of the prices of financial products as stochastic processes defined on a filtered probability space (Ω, (𝓕t)t ≥ 0, ℙ), where it is usually assumed that an “objective" probability measure ℙ describing the random evolution of market prices, is given. However, in financial markets, and even more so in

273 Petrov, Krassimir. ‘A Primer on Martingales’. Petrov Financial. 1998. Available at:

http://www.petrovfinancial.com/?page_id=880 274 ‘Martingale and filtration’. Stack Exchange. At:

https://math.stackexchange.com/questions/13605/martingale-and-filtration

275 Pollard, David “Brownian motion (BM)”. Lecture notes‒ Statistics 251b/551b - Stochastic processes. Yale University. Spring, 2004. (http://www.stat.yale.edu/~pollard/Courses/251.spring04/) 276 Biagini, Sara and Cont, Rama. “Model-free representation of pricing rules as conditional expectations”. December, 2006.

91 the context of derivative pricing, there is no consensus on the underlying model ℙ: Moreover, market consensus is expressed in terms of prices of the underlying assets and their derivatives traded in the market, not in terms of probabilities. Herein comes the significance of an adequate definition of arbitrage under which a “consensus” is correctly specified.

The reason for introducing the martingale measure is dual. It simplifies the evaluation or arbitrage prices (especially in the case of derivative securities), and describes the arbitrage-free property/condition of a given pricing model for primary securities in terms of the behaviour of relative prices. Forcing a price process or price model to rely on a specific choice of ℙ would be a mistake, unless a complete loss of generality is not a concern. Summarizing, the concept of an arbitrage price of a derivative security does not depend on the choice of a particular probability measure ℙ, in a particular price model for primary securities. This means that the arbitrage price depends on the actual selection of ℙ, but is invariant with respect of which ℙ is selected among the class of the mutually equivalent ℙ measures available for selection for the implementation of the price model, and the selection of a particular ℙ is, to some extent at least, subjective. Put it in colloquial language, all investors agree on the fact that the price of primary securities will fluctuate randomly in the future, but they may have different assessments of the corresponding subjective probabilities, however²⁷⁷. This is the most fundamental reason behind the plethora of different pricing algorithms we encounter in the financial markets.

Finally, a definition for pricing algorithm is needed. Intuitively, it has to be related to the pricing process S* we already specified. We define pricing algorithm the following way.

Given Ω a space of market scenarios ω, ω ∈ ℝⁿ, ℙ an abstract economy,

ℙ * a risk neutral measure, 𝒥 a priceable asset and

𝒟(ℙ ,ω, 𝒥) the demand function for asset 𝒥,

a pricing algorithm is an objective optimizing function 𝑓 | 𝑓(ω, 𝒥, 𝒟) → p, where p is the price and p ∈ ℝ.

In simple terms, a pricing algorithm is a well-defined procedure implementing a price process for a particular priceable asset in face of its known demand.

I call for attention to three points here. First, Ω, the space of market scenarios is n-dimensional, while the price p is 1-dimensional, i.e. a scalar. Second, the requirement of a known demand for pricing algorithms. If the demand function is unknown, then we would have ended with an auction algorithm, instead of with a pricing one. Third, I haven’t defined what a priceable asset is, which is the purpose of the following Chapter section.

In document Pricing algorithms and computational price theory : the building blocks of computational finance and IT business applications (sivua 103-106)