=Paper=
{{Paper
|id=Vol-2239/article_12
|storemode=property
|title=OntoREA© Accounting and Finance Model: Including Option Contracts in the MDD-Context
|pdfUrl=https://ceur-ws.org/Vol-2239/article_12.pdf
|volume=Vol-2239
|authors=Christian Fischer-Pauzenberger,Matthias Ondra,Walter Schwaiger
|dblpUrl=https://dblp.org/rec/conf/vmbo/Fischer-Pauzenberger18
}}
==OntoREA© Accounting and Finance Model: Including Option Contracts in the MDD-Context==
https://ceur-ws.org/Vol-2239/article_12.pdf
OntoREA© Accounting and Finance Model:
Including Option Contracts in the MDD-Context
Christian Fischer-Pauzenberger1[0000-0002-9071-9523], Matthias Ondra, Walter S.A.
Schwaiger1[0000-0003-1868-7018]
1
Technische Universität Wien, Institute of Management Science, Theresianumgasse 27, 1040
Wien, Austria
{christian.fischer-pauzenberger, matthias.ondra,
walter.schwaiger}@tuwien.ac.at
Abstract. OntoREA© is a specification of the Accounting and Finance domain
in the OntoUML language [1] which includes derivative instruments as well.
The authors use a forward contract as running example to demonstrate the va-
lidity of the OntoREA© Accounting and Finance model within the design sci-
ence resource methodology (DSRM) [2]. They claim but they do not proof that
the model not only holds for forward but also for option contracts. In this article
the missing proof is given by refining the relational MySQL model. In the mod-
el driven development (MDD) context [3] this model constitutes the platform
specific model (PSM) which is derived from the platform independent model
(PIM) in form of the OntoREA© model. The refined PSM model adequately
represents European option contracts and it should be especially useful for
business analysts as well as for educational purposes.
Keywords: OntoREA© Accounting and Finance Model, Design Science Re-
search Methodology DSRM, Model Driven Development MDD, Conceptual
Modeling, Derivative Instruments, Dynamic Hedge Portfolio
1 Introduction
The no-arbitrage pricing theory also holds for options. For options a dynamic
hedge portfolio exactly replicates (duplicates) the option value over time. In a dynam-
ic replication policy fractions of the stock are bought and they are partially financed
by a loan liability. Over time the investment and financing has to be continuously
adjusted according to the revealing stock prices. At the expiration date the hedge
portfolio gives the same result as the initial buying of an option contract.[1]
In contrast to options the hedge portfolio composition of a forward contract does
not change over time and that’s why it is called a static hedge portfolio. The static
hedge portfolio representation was used in [1] to demonstrate the suitability of the
OntoREA© Accounting and Finance model as conceptual platform independent mod-
el (PIM) for the development of a platform specific MySQL relational database model
(PSM) within the model-driven software development context (MDD).
The hedge portfolio representation of derivative instruments is one of the core fea-
tures of the OntoREA© model and it is expressed in the upper left part of Figure 1 in
2
form of the Collective class Derivative Instrument and its MemberOf relationship to
the Kind class Economic Resource. In simple terms the meta-physical stereotypes of
the OntoUML language have the following meaning (for a full description of the con-
ceptual and meta-physical OntoUML details see [4]): A derivative instrument is rep-
resented as a rigid and identity-providing portfolio collective that consists of two eco-
nomic resources that are themselves rigid and identity providing kinds.
«Phase» {disjoint, complete}
Asset «Collective»
Derivative
Instrument
«Phase»
Liability
«MemberOf»
2 1..*
«Phase»
Off Balance «Kind» «Kind» participation
«Kind»
Economic Economic Event * «Formal» 2 Economic Agent
Resource
1..* 1..* 1..* 1..* 2
{disjoint, complete}
out-/inflow
{disjoint, complete} «Formal»
in-/outflow
«Phase» «Formal» 1
Asset
«SubKind» «SubKind»
Debit Event Credit Event
«Phase» 1
duality
Liability «Material»
1..* 1..*
1..* 1..*
value constraint
«Phase»
Equity «Mediation»
«Relator» «Mediation»
1 Balanced Duality 1
«Phase»
Claim participation
«Relator»
«Mediation» Balanced
0..1 «Formal»
1 Reciprocity
commited (Economic 1
inflow Contract) «Mediation»
«Formal» 1..* 1..*
«SubKind» present value «SubKind»
Debit constraint Credit
reciprocity
0..1 Commitment Commitment
1..* «Material» 1..*
commited
outflow
«Formal» 0..1
{disjoint, complete}
«Kind»
Commitment fulfillment
0..1 «Formal»
Figure 1: OntoREA© Accounting and Finance Model - Conceptual PIM Level
The hedge portfolio [5] representation of derivative instruments was originated by
the Nobel laureates Black/Scholes [6] and Merton [7] who developed to the no-
arbitrage pricing theory. This representation holds true for unconditional derivatives
(e.g. forward contracts) as well as for conditional derivatives (e.g. option contracts).
The main difference between un- and conditional derivates lies in the dynamic behav-
ior of the hedge portfolio composition. Unconditional derivatives include the obliga-
tion for the buyer of the contract to buy the underlying asset in the future. Due to this
obligation the hedge portfolio composition does not change over time. Conditional
derivatives include the right for the buyer of the contract to buy the asset in the future.
3
As the probability of executing the option is changing over time, the hedge portfolio
composition changes as well.
The research objective of this article is to provide the proof that the conceptual On-
toREA© PIM model incorporates not only the static but also the dynamic hedge port-
folio representation of derivative instruments. For this purpose a refined MySQL
database PSM model is constructed that includes the peculiarities of dynamic hedging
portfolios as well, so that un- and conditional derivative instruments are covered in
the proposed PSM model.
This paper is organized as follows: In the next section the no-arbitrage pricing the-
ory and its hedge portfolio foundation are presented. In following section the refined
relational database model (PSM) for un- and conditional derivative instruments is
presented and its applicability to conditional option contracts is demonstrated with a
stock call running example. The last section concludes the paper and shows directions
for further research.
2 No-Arbitrage Pricing: Hedge Portfolio Representation
The no-arbitrage pricing theory was – as already mentioned – developed by the
Nobel laureates Black/Scholes [6] and Merton [7]. They show that there is only one
price for the derivative instruments, i.e. the no-arbitrage price that does not allow
arbitrage possibilities. They derive the no-arbitrage price for European stock call op-
tions. European stock calls have the peculiarity that the right to buy refers to a stock
asset, which is the underlying of the contract, and that the right can be exercised by
the buyer of the contract only at expiration date (European style). The no-arbitrage
price for the European stock call is given by the Black/Scholes formula:
𝑣𝑎𝑙𝑢𝑒 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒
(𝑎𝑠𝑠𝑒𝑡/𝑙𝑒𝑓𝑡 𝑙𝑒𝑔) (𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦/𝑟𝑖𝑔ℎ𝑡 𝑙𝑒𝑔)
⏞
𝐹𝑎𝑖𝑟 𝑉𝑎𝑙𝑢𝑒 𝐶𝑎𝑙𝑙,𝑡 = ⏞ 𝑁(𝑑1,𝑡 ) −𝑋0,𝑇 ∙ 𝑁(𝑑2,𝑡 ) ∙ 𝑒 (−𝑙𝑛(1+𝑅0,𝑇 ) ∙ 𝑇𝑡,𝑇)
+𝑃𝐴,𝑡 ∙ ⏟ (1)
𝑎𝑠𝑠𝑒𝑡 𝑤𝑒𝑖𝑔ℎ𝑡
As can be seen by the annotations in equation 1, the no-arbitrage price, which is
called fair value, corresponds according to the hedge portfolio of two parts: The value
of the asset (asset value) on the left side (left leg) and the present value of the liability
(loan liability) on the right side.
The asset weight , i.e. N(d1,t) gives the fraction of the underlying stock that is hold
in the hedge portfolio. It is calculated by evaluating the standard normal distribution
function N() at the value of d1,t. The d1,t-value (for further details see [6]) is a function
of the stock Price PA,t and the time to maturity Tt,T. Consequently this value changes
over the life cycle of the call option. The asset weight is a probabilistic term that ex-
presses the probability of a stock option execution. It ranges between zero and 100 %.
The present value of the loan liability is calculated by weighting the exercise price
X0,T with the weighting factor N(d2,t) and discounting the resulting product by multi-
plying it with the discount factor exp(-ln(1+R0,T)*Tt,T). The discount factor is calculat-
4
ed in form of a continuous compounding by inserting the interest rate R0,T over the
whole life time of the option, i.e. from 0 to T, and the time to maturity Tt,T into the
Euler exponential.
Finally, by using the t variable for the pricing date, the Black/Scholes formula is
generically defined so that it can be applied for the initial (i.e. t = 0) and the subse-
quent (i.e. t > 0) pricing.
Table 1: European Stock Call (running example) – Specification and Pricing
Contracting date: 01.01.
Expiration date: 31.12.
Exercise price: 100
Initial stock price: 100
Volatility: 20%
Initial interest rate: 5%
Asset weight N(d1): 63.68%
Liability weight N(d2): 55.96%
Stock Asset: 63.68
Loan Liability: 53.23
Fair value: = A – L 10.45 (A)
Table 1 contains the specification of a European stock call and its initial pricing at
the beginning of the year (01.01.) according to the Black/Scholes formula, which is
evaluated at the contracting date, i.e. t = 0. The fair value of the stock option, i.e. its
no-arbitrage price, amounts to 10.45 and it is calculated by subtracting the present
value of the loan liability (53.23) from the value of the stock asset (63.68).
Due to its generic definition the Black/Scholes formula can be used for pricing
purpose not only at the call’s contracting date but also at subsequent dates. The big
advantage of the formula is that it not only gives the call price, but also the composi-
tion of the hedge portfolio at all pricing dates.
The composition information is of special importance in the case of a dynamic call
replication policy, where the call is not bought initially but instead it is synthetically
created by implementing and rebalancing the dynamic hedge portfolio over time. In
this case the asset weights N(d1,t) are of special importance. They indicate the frac-
tions of the underlying stock assets in the hedge portfolio.
For demonstrative purposes a pricing after each quarter is assumed. Furthermore it
is assumed that the stock price from initially 100 does not change after the first and
second quarter and then increases to 120. In this constellation – as can be seen in the
first column of Table 2 – the asset weights start decreasing from 63.68 % (01.01.:
100) to 61.91 % (31.03.: 100) and to 59.77 % (30.06.: 100) and consequently increase
to 97.72 % (30.09.: 120). The initial decrease at the stable price of 100 indicates a
decreasing execution probability and consequently a smaller stock position is hold in
the hedge portfolio. The stock price increase increases the execution probability and
consequently the stock position in the hedge portfolio. The changing asset weights
over the call’s life time demonstrate what is meant by saying that the composition of
the option’s hedge portfolio is changing over time. This changing composition in the
dynamic hedge portfolio is contrasted to the stable composition in the static hedge
portfolio of stock forwards where at each point in time exactly one unit of the under-
lying stock is held in the hedge portfolio.
5
3 Hedge Portfolio: Switching from PIM- to PSM-Level
After having a deeper understanding of the hedge portfolio in the Black/Scholes
formula, the transformation of its conceptualization in the OntoREA© Accounting
and Finance model – as the Collective class Derivative Instrument with a MemberOf
relationship to the Kind class Economic Resource – into a MySQL database model
can be addressed. In the MDD context this transformation corresponds to the switch
from an abstract conceptual PIM model into a specific database PSM model. Associ-
ated with this concretization step is an informational extension that is accomplished
by adding additional attributes and tables in the PSM model in order to capture the
more detailed contents at the PSM model level.
Financial_Security_Pricing_Transactional Financial_Security_Master
«column» «column»
*pfK ISIN: VARCHAR(50) 0..* «FK» 1 *PK ISIN: VARCHAR(50)
*PK timestamp: DATETIME Description: VARCHAR(50)
* Price: DOUBLE Marketplace: VARCHAR(50)
1
«FK»
0..*
Asset «FK» Derivative_Instrument_Master
«column» 0..1 «column»
*pfK Derivative_Instrument_Dynamic_ID: INT 1 *PK Derivative_Instrument_Master_ID: INT
FK ISIN: VARCHAR(50)
Derivative_Instrument_Transactional Type_Of_Stock_Derivative: ENUM
* Exercise_Or_Forward_Price: DOUBLE
Liability «column» * Contracting_Date: DATETIME
«FK»
*PK Derivative_Instrument_Transactional_ID: INT * Expiration_Date: DATETIME
«column» 0..1 1 *FK Derivative_Instrument_Master_ID: INT * Contracting_Security_Price: DOUBLE
1..* «FK» 1
*pfK Derivative_Instrument_Dynamic_ID: INT *FK Event_ID: INT Contract_Size: DOUBLE
* timestamp: DATETIME Number_Of_Contracts: DOUBLE
* Fair_Value: DOUBLE * Volatility: DOUBLE
0..* * Interest_Rate: DOUBLE
Off_Balance 1 1 1 Mark_To_Model: ENUM
«FK» «FK»
«column»
*pfK Derivative_Instrument_Dynamic_ID: INT 0..1
«FK» «FK»
1 1
Economic_Resource_Risky_Income Economic_Resource_Fixed_Income
«column» «column»
*pfK Resource_ID: INT *pfK Resource_ID: INT
FK Derivative_Instrument_Transactional_ID: INT FK Derivative_Instrument_Transactional_ID: INT
Nd1t: DOUBLE * Present_Value: DOUBLE
* Value: DOUBLE Asset_Or_Liability: ENUM
Asset_Or_Liability: ENUM
0..* 0..*
«FK» «FK»
1 1 1
Resource Event Agent
«column» «column» «column»
*PK Resource_ID: INT *pfK Event_ID: INT *PK Agent_ID: INT
FK Agent_ID: INT 0..* «FK» 1
1 1
«FK»
Figure 2: MySQL Relational Database Model – Database PSM Level
Figure 2 contains the refined database PSM model that concretizes the OntoREA©
conceptualization of the derivative instruments’ hedge portfolio representation. It
covers not only unconditional but also conditional derivative instruments. Compared
to the development of the MySQL database (PSM) model related to the static hedge
portfolio representation in [1], the dynamic hedge portfolio peculiarities for the stock
options are now explicitly incorporated for the:
1. Collective class Derivative Instrument,
2. MemberOf relationship between Collective class Derivative Instrument and Kind
class Economic Resource and
3. Formal relationship in-/outflow (out-/inflow) between Kind class Economic Re-
source and SubKind class Debit Event (Credit Event).
6
Ad 1) The Collective class Derivative Instrument is transformed via the four tables
in the right upper corner of Figure 2. The splitting into four tables allows a clear dis-
tinction of information that is stable over time (master information) and information
that changes (transactional information).
• The table Derivative_Instrument_Master contains the stable information which
specifies the derivative instruments. Its attribute Type_Of_Stock_Derivative is of
ENUM type so that un- and conditional derivatives are covered in the MySQL
database model. It can be seen that the table also contains all information from
Table 1 which are used to specify the stock call.
• The table Derivative_Instrument_Transactional contains the pricing information
which is associated to the initial and subsequent pricing dates measured with the
Attribut timestamp. In the case of a dynamic replication policy the hedge portfo-
lio composition adjustments are connected with capital market transactions.
That’s why the table has the Transactional suffix.
• The two tables Financial_Security_Pricing_Master and Financial_Security_Pri-
cing_Transactional are included in order to allow the separate specification of the
derivative’s underlying asset (i.e. financial security) which is fully defined by its
international security identification number (ISIN). The usage of two tables is
due to the intended separation of master and transactional information.
Ad 2) The MemberOf relationship is transformed via the two tables in the lower
left part of Figure 2 according to the financial categorization of financial instruments
into risky income and fixed income resources. Both tables have a foreign key to the
table Derivative_Instrument_Transactional. The missing NOT NULL CONSTRAINT
(*) indicates that not all of these resource types have to be part of a hedge portfolio.
The inclusion of the two tables promotes the understanding of the different financial
natures of the two hedge portfolio constituents, i.e. the stock asset (risky income) and
the loan liability (fixed income).
Ad 3) The Formal relationship in-/outflows (out-/inflows) between the Kind class
Economic Resource and the SubKind class Debit Event (Credit Event) is transformed
via the relationship between the table Derivative_Instrument_Transactional and the
reflexive table Event at the bottom of Figure 2. This relationship is needed for trigger-
ing events that are – as can be seen in Figure 1 – connected to classes Debit Event and
Credit Event. At the pricing dates (timestamp) events are triggered in two cases, i.e.
either if the Mark_To_Model mode is active (see table Derivative_Instrument_Mas-
ter) or if a dynamic replication policy is in place. By triggering the events the changes
in the hedge portfolio compositions are recognized via balanced double-entries in the
accounting system.
After having presented the construction of the MySQL database PIM its working is
addressed. The storage of information starts at the contracting date (initial pricing
date). At that date the contracts for the derivative instruments are specified and the
corresponding information is stored in the master tables. As time goes by – that is at
runtime – additional information is revealed. Of special importance for the calcula-
tions of the hedge portfolio compositions are the revealing asset prices (see table Fi-
nancial_Security_Pricing_Transactional). Furthermore the timestap information in the
table Derivative_Instrument_Transactional is important as it allows the calculation of
7
the remaining time to maturity (Tt,T) together with the attribute Expiration_Date in the
table Derivative_Instrument_Master. With this information the hedge portfolio com-
positions are determined and the corresponding information processing activities are
triggered.
Table 2: European Stock Call – Specification and Subsequent Pricing
PSM Relational Schema
Dynamic Hedging Attribute Table
Contracting date: 01.01. Contracting_Date Derivative_Information_Master
Initial Pricing/Static Data
Expiration date: 31.12. Expiration_Date Derivative_Information_Master
Exercise price: 100 Exercise_Or_Forward_Price Derivative_Information_Master
Initial stock price: 100 Contracting_Security_Price Derivative_Information_Master
Volatility: 20% Volatility Derivative_Information_Master
Initial interest rate: 5% Interest_Rate Derivative_Information_Master
Asset weight N(d1): 63,68% Nd1t Economic_Resource_Risky_Income
Liability weight N(d2): 55,96% - -
Stock Asset: 63,68 Value Economic_Resource_Risky_Income
Loan Liability: 53,23 Present_Value Economic_Resource_Fixed_Income
Fair value: = A – L 10.45 (A) Fair_Value Derivative_Instrument_Transactional
Pricing date #1: 31.03. timestamp Financial_Security_Pricing_Transactional
Actual stock price: 100 Price Financial_Security_Pricing_Transactional
Actual time to maturity: 9 months - -
Asset weight N(d1): 61,91% Nd1t Economic_Resource_Risky_Income
Stock Asset: 61,91 Value Economic_Resource_Risky_Income
Loan Liability: 52,13 Present_Value Economic_Resource_Fixed_Income
Fair value: = A – L 9.78 (A) Fair_Value Derivative_Instrument_Transactional
Pricing date #2: 30.06. timestamp Financial_Security_Pricing_Transactional
Subsequent Pricing/Dynamic Data
Actual stock price: 100 Price Financial_Security_Pricing_Transactional
Actual time to maturity: 6 months - -
Asset weight N(d1): 59,77% Nd1t Economic_Resource_Risky_Income
Stock Asset: 59,77 Value Economic_Resource_Risky_Income
Loan Liability: 50,65 Present_Value Economic_Resource_Fixed_Income
Fair value: = A – L 9.12 (A) Fair_Value Derivative_Instrument_Transactional
Pricing date #3: 30.09. timestamp Financial_Security_Pricing_Transactional
Actual stock price: 120 Price Financial_Security_Pricing_Transactional
Actual time to maturity: 3 months - -
Asset weight N(d1): 97,72% Nd1t Economic_Resource_Risky_Income
Stock Asset: 117,26 Value Economic_Resource_Risky_Income
Loan Liability: -96,82 Present_Value Economic_Resource_Fixed_Income
Fair value: = A – L 20.44 (A) Fair_Value Derivative_Instrument_Transactional
Pricing date #4: 31.12. timestamp Financial_Security_Pricing_Transactional
Actual stock price: 120 Price Financial_Security_Pricing_Transactional
Actual time to maturity: 0 months - -
Stock Asset: 117,26 Value Economic_Resource_Risky_Income
Loan Liability: 98,03 Present_Value Economic_Resource_Fixed_Income
Fair value: = A - L 19.23 (A) Fair_Value Derivative_Instrument_Transactional
In Table 2 the calculations for the stock call running example can be seen in the
first column. It is interesting to note the last fair value at the end of the year (31.12.)
amounting to 19.23 is close to the intrinsic value of the stock call amounting to 20
which is calculated as difference between the stock asset value of 120 and the exer-
cise price of 100. By executing the subsequent pricings more often the difference can
theoretically be brought to zero. For completeness it should be mentioned that the
resulting fair value is connected to a self-financing policy. According to this policy
the changing asset fractions are either used for redemption of the loan if they decrease
or financed by increasing the loan if the increase.
Finally, the middle and the right column show with which attribute of which table
this information is captured in the MySQL database PIM model.
8
4 Conclusion
The main contribution of this article is the development of a more advanced
MySQL database PSM model compared to [1] that not only covers static hedge port-
folio representations for unconditional derivatives (e.g. stock forward) but also dy-
namic hedge portfolio representations for conditional derivatives (e.g. stock call). The
applicability of the proposed database PSM model was demonstrated for a European
stock call running example. According to the put/call parity analogue results hold for
European stock puts, so that the PSM model covers call and put options. Next to the
applicability demonstration the proposed PSM model fulfills all peculiarities that are
related to the dynamic hedge portfolio representation specified in the OntoREA©
PIM model. In this sense the research objective is reached by delivering the missing
proof in [1] in form of a refined database PSM that covers unconditional as well as
conditional derivatives.
For further research two considerations seem worthwhile. Firstly, the transfor-
mation of the database PSM model into an R/Shiny prototype like in [1] in order to
complete also the 2-step in the forward engineering approach in the MDD context.
Secondly, the explicit policy level specification seems to be a worthwhile endeavor.
This proposed model can be useful for business analysts in the finance domain as
well as for teaching purposes by explaining derivative instruments in form of concep-
tual PIM and database specific PSM models.
5 References
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