If there are no arbitrage opportunities in the capital markets, then there exists a (not generally unique) valuation operator, L.— title: A Basic Unified Model for Derivative Instruments author: Keith A. Lewis institute: KALX, LLC fleqn: true classoption: fleqn abstract: Value, hedge, and manage the risk of any instruments …
The Black-Scholes/Merton theory of option valuation has a serious flaw: it assumes continous time trading is possible. Only a finite number of trades can be executed based on information available at the time of the trade.
(Ross 1978) extended the B-S/M theory to value any collection of instruments and identified the condition for models to be arbitrage-free.
If there are no arbitrage opportunities in the capital markets, then there exists a (not generally unique) valuation operator, L.
Ross used the Hahn-Banach theorem to show arbitrage is a geometric notion that has nothing to do with probability. The proposed model puts Ross’ theory on a firm mathematical basis by identifying his valuation operator as a deflator: a set of measures indexed by trading times. Not only are these not unique, as Ross pointed out, the restriction to a finite number of trades underscores the main unsolved problem of Mathematical Finance: when and how much should you trade given available information? This short note does not propose a solution to this, only a framework for further research.
When observation does not agree with theory in physics, it is time to come up with a new theory. Lord Kelvin claimed toward the end of the 19th century
There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.
The Rayleigh-Jeans law for black-body electromagnetic radiation fit experimental observation for low temperatures, but was drastically wrong as the temperature increased. Experiment did not agree with observation.
Für die Richtung des dabeieinzuschlagenden Gedankenganges giebt der Hinblick auf dieUnhaltbarkeit der früher gemachten Voraussetzung einen Finger-zeig. – Max Planck
The untenability of the assumption made earlier provides an indication of the direction of the line of thought to be taken.
Max Planck showed it was possible to fit the observed data by assuming photons could only be emitted in integer multiples of a minimal value. Planck’s main contribution to science was not his eponymous constant that fit the data, it was getting his contemporaries to focus on understanding why that worked. The result was the development of quantum mechanics.
Stock prices also trade in integer multiples of a minimal value. This does not necessarialy imply results from quantum mechanics can be applied to mathematical finance.
There is something drastically wrong with the Black-Scholes/Merton theory due to their assumption of continuous time trading. Merton provided a closed form formula for the value of barrier options in (Merton 1973) using the reflection principal for Brownian motion. A preposterous mathematical artifact of using Ito’s theory is that it implies the value of a barrier option does not change if it knocks in (or out) the second, third, or even the millionth time it is hit.
This untenable conclusion can be avoided by accepting the fact that every hedging strategy involves only a finite number of trades. There are no perfect hedges, something every derivative trader knows after the first week on the job. They just want to know what initial hedge to put on, and when and how much to adjust the initial hedge over time to replicate the derivative contract obligations. Half a century on from B-S/M, the mathematical finance community has not been able to come up with a mutually agreeable answer to this fundamental problem.
Another pernicious influence of the B-S/M model that (Ross 1978) pointed out was arbitrage-free models have nothing to do with probability, they only involved geometry. We eliminate the use of the Hahn-Banach theorem and show his linear valuation operator corresponds to a collection of positive measures indexed by trading time.
Market instruments are not perfectly liquid; there is a bid-ask spread involved in transactions that tends to widen as the amount traded increases. Stock prices are discrete; they trade in integer multiples of their smallest trading unit. The price of an instrument can also depend on the credit worthiness of the counterparties involved in a trade. The Basic Unified Model does not address these salient market realities.
The Basic Unified Model does provide a simple and mathematically rigorous model that can be applied to all instruments. Instruments have prices and cash flows associated with holding them. A trading strategy involves only a finite number of trades based on available information. Trades result in amounts being debited or credited to a trading account and accumulate to a position in each instrument. Nominal prices for instruments can be used to approximate the value (mark-to-market) of the position.
An elementary mathematical consequence of defining price, cash flow, value, and account is that trading strategies create synthetic instruments. The value of the strategy corresponds to the price of an instrument and the amounts in the trading account correspond to cash flows. This is the skeleton key to valuing, hedging, and managing the risk of derivatives. A derivative is a contract. The buyer pays the seller to provide cash flows at times specified in the contract. The Basic Unified Model does not solve the difficult problems involved with valuing, hedging, and managing the risk faced by seller. It only provides a mathematically rigorous notation for investigating this problem.
Let \Omega be a set of possible outcomes. Partial information is modeled by a partition of \Omega. A collection of subsets of \Omega, \{A_j\}, is a partition if they are pairwise disjoint and their union is \Omega. Full information is knowing \omega\in\Omega. Partial information is knowing only to which atom \omega belongs. No information is modeled by the singleton partition \{\Omega\}.
An algebras of sets is a collection of sets closed under union and complement.
Exercise. Show algebras of sets are closed under intersection.
Hint: Let A' = \Omega\setminus A = \{\omega\in\Omega\mid \omega\not\in A\} be the set complement of A in \Omega. Use De Morgan’s Law (A\cap B)' = A'\cup B'.
We can identify a set A with its indicator function 1_A\colon\Omega\to\boldsymbol{R} defined by 1_A(\omega) = 1 if \omega\in A and 1_A(\omega) = 0 if \omega\not\in A. Note 1_{A\cup B} = 1_A + 1_B - 1_{A\cap B}, 1_{A'} = 1 - 1_A, and 1_{A\cap B} = 1_A 1_B,
Exercise. Prove De Morgan’s Law.
Hint: Start from 1_{A'\cup B'} = 1_{A'} + 1_{B'} - 1_{A'\cap B'}.
Calculations on algebras of sets is algebra.
If {\mathcal{A}} is a finite algebra of sets on \Omega then [\omega] = \cap\{A\in{\mathcal{A}}\mid\omega\in A\} is the atom of {\mathcal{A}} containing {\omega\in\Omega}.
Exercise. If B\subseteq[\omega] and B\in{\mathcal{A}} then B = \emptyset or B = [\omega].
Exercise. The atoms of {\mathcal{A}}, [{\mathcal{A}}], form a partition of \Omega.
This shows we can identify a finite algebra of sets with its atoms.
A function X\colon\Omega\to\boldsymbol{R} is {\mathcal{A}}-measurable for an algebra of sets {\mathcal{A}} if \{\omega\in\Omega\mid X(\omega) \le x\} belongs to {\mathcal{A}} for all x\in\boldsymbol{R}.
Exercise. If {\mathcal{A}} is finite, show X is {\mathcal{A}}-measurable if and if and only if it is constant on atoms of {\mathcal{A}}.
This shows X\colon[{\mathcal{A}}]\to\boldsymbol{R} is a function. We jettison the word ‘measurable’ and say X is known given {\mathcal{A}}.
The Unified Model does not involve probability, however as an aid to those schooled in the classical theory we will reconnoiter some elementary facts.
A probability measure is a positive measure with mass 1. If P is a probability measure on \Omega then any function X\colon\Omega\to\boldsymbol{R} is a random variable. The expected value of X is E[X] = \int_\Omega X\,dP.
The conditional probability of B given A is P(B\mid A) = P(B\cap A)/P(A) for B,A\subseteq\Omega.
Exercise. Show B\mapsto P(B\mid A) is a probability measure on A.
This can be generalized to the conditional expectation of a random variable given an algebra of sets. We say Y = E[X\mid{\mathcal{A}}] if Y is known given {\mathcal{A}} and \int_A Y\,dP = \int_A X\,dP for all A\in{\mathcal{A}}.
If X is a random variable and P is a measure we can define the measure XP by (XP)(A) = \int_A X\,dP.
Exercise. Show XP is a measure.
Exercise Show if X\ge 0 and E[X] = 1 then XP is a probability measure.
Hint: Show XP is positive and (XP)(\Omega) = 1.
Exercise. Show Y = E[X|{\mathcal{A}}] if and only if Y(P|_{\mathcal{A}}) = (XP)|_{\mathcal{A}}.
Hint: If P is a measure on \Omega then P|_{\mathcal{A}} is the restriction of the measure to {\mathcal{A}}.
Exercise. Show if A is an atom of {\mathcal{A}} and P(A)\not=0 then E[X\mid{\mathcal{A}}](A) = \int_A X\,dP/P(A).
Conditional expectation is the average over each atom.
Every instrument has a price X_t and a cash flow C_t at any trading time t\in T. Instruments are assumed to be perfectly liquid: they can be bought or sold at the given price in any amount. Cash flows are associated with owning an instrument and are almost always 0; stocks have dividends, bonds have coupons, currencies have no cash flows, commodities have storage costs. European options have exactly one cash flow at expiration.
Let T be a totally ordered set of trading times, I the set of all market instruments, \Omega the sample space of possible outcomes, and {\mathcal{A}}_t an algebra of sets modeling information available at time t\in T. Prices and cash flows are bounded {\mathcal{A}}_t-measurable functions {X_t, C_t\colon\Omega\to\boldsymbol{R}^I}, {t\in T}.
We write X\in B({\mathcal{A}}) if X\colon\Omega\to\boldsymbol{R} is a bounded {\mathcal{A}}-measurable function. Note if {\mathcal{A}} is finite then its atoms are a partition of \Omega and being {\mathcal{A}}-measurable is equivalent to X being constant on atoms. In this case X\colon{\mathcal{A}}\to\boldsymbol{R} is standard mathematical notation for a function when {\mathcal{A}} is identifed with its atoms.
Traders buy and sell shares based on information available at each trading time. They trade a finite number of times and eventually close out all postions.
A trading strategy is a finite collection of strictly increasing stopping times (τ_j) and trades (\Gamma_j) where \Gamma_j:{\mathcal{A}}_{τ_j}\to \boldsymbol{R}^{I} indicating the number of shares to trade in each instrument at time τ_j. If \tau is a stopping time then {\mathcal{A}}_\tau = \{A\in{\mathcal{A}}_t\mid t\in T, t\le\tau\}.
Trades accumulate to a position \Delta_t = \sum_{τ_j < t}\Gamma_j = \sum_{s < t}\Gamma_s where \Gamma_s(\omega) = \Gamma_j(\omega) when s = τ_j(\omega), \omega\in\Omega. A trade at time t is not included in the position at time t; it takes some time for trades to settle.
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The value, or mark-to-market is V_t = (\Delta_t + \Gamma_t)\cdot X_t. It is the amount that would result from closing out the entire position at current market prices, assuming that is possible. The trading account is A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t; cash flows propostional to existing positions are credited and the cost of the trade at time t is debited.
Arbitrage exists if there is a trading stragegy (\Gamma_t)_{t\in T} with A_0 < 0, A_t \ge 0, t > 0, and \sum_{t} \Gamma_t = 0.
Theorem. (Funamental Theorem of Asset Pricing) There is no arbitrage if and only if there exist positive measures (D_t)_{t\in T} on \Omega with X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|_{{\mathcal{A}}_t}
Lemma. For any arbitrage free model and any trading strategy V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{{\mathcal{A}}_t}
Lemma. If X_t D_t = M_t - \sum_{s\le t} C_s D_s where M_t = M_u|_{{\mathcal{A}}_t}, t \le u then there is no arbitrage.
For those paying attention… What about bid-ask spread? Use limit orders.