INTERNATIONAL JOURNAL OF COMPUTER SCIENCE & INFORMATION
SYSTEM
Volume10 Issue03, March-2025, pg. 1-9
E-ISSN: 2536-7919
P-ISSN: 2536-7900
SJIF 2019: 4.58 2020: 5.046 2021: 5.328
2025, IJCSIS, https://scientiamreearch.org
pg. 1
Published Date: -
01-03-2025
DEVELOPING A NEW ENCRYPTION TECHNOLOGY USING
RECONFIGURABLE MULTI-VALUED LOGIC OPERATORS
Michael A. Smith
Department of Computer Science, University of Wisconsin, Madison, WI 53706,
United States
Abstract: This paper presents a real option approach to the valuation of the default risk of residential
mortgages. Traditional models of mortgage valuation often overlook the strategic decision-making of
borrowers, who may choose to default or refinance based on changing market conditions. We propose
a dynamic framework that incorporates the borrower's flexibility to respond to fluctuations in house
prices and interest rates. By modeling the mortgage default decision as a real option, akin to an
American put option, we capture the optimal default boundary where it becomes advantageous for
the borrower to default. Our results show that default risk is highly sensitive to changes in house
prices, with declining house values increasing the likelihood of default. Additionally, lower interest
rates reduce default risk, but this effect is tempered by falling home values. The real option approach
provides a more realistic and accurate valuation of mortgage default risk, offering valuable insights for
investors and financial institutions managing mortgage-backed securities. This research highlights the
importance of borrower flexibility in mortgage risk assessment and suggests that real options can
significantly enhance mortgage pricing models.
Keywords: Reconfigurable Logic, Multi-Valued Logic, Encryption Technology, Cryptography, Logic
Operators, Secure Communication, Digital Security, Encryption Algorithms, Information Protection,
Advanced Cryptographic Systems.
INTRODUCTION
The default risk associated with residential mortgages is a critical factor in determining their value,
especially given the substantial role of mortgage-backed securities in global financial markets. Traditional
approaches to mortgage valuation often assume fixed parameters, such as interest rates, home values,
and borrower behaviors. However, these assumptions fail to account for the inherent flexibility of
borrowers, who may choose to default or refinance based on evolving market conditions. A real option
approach offers a dynamic framework for valuing default risk by incorporating the strategic decision-
making of borrowers, which can significantly affect the value of residential mortgages.
INTERNATIONAL JOURNAL OF COMPUTER SCIENCE & INFORMATION
SYSTEM
Volume10 Issue03, March-2025, pg. 1-9
E-ISSN: 2536-7919
P-ISSN: 2536-7900
SJIF 2019: 4.58 2020: 5.046 2021: 5.328
2025, IJCSIS, https://scientiamreearch.org
pg. 2
Published Date: -
01-03-2025
This article explores the application of a real option approach to the valuation of the default risk of
residential mortgages. We propose a model that captures the borrower's decision-making flexibility in
response to changing market conditions, particularly house prices and interest rates. This paper
contributes to the understanding of mortgage default risk by integrating the concept of real options, a
method that has been extensively used in financial decision-making but is underutilized in the context of
mortgage risk.
Literature Review
Several studies have focused on modeling default risk in residential mortgages. The traditional methods,
including reduced-form models and structural models, rely heavily on static assumptions about market
factors and borrower behavior. For instance, the Merton model (1974) uses the framework of option
pricing to value the default risk of corporate debt, applying similar concepts to mortgage-backed
securities. However, these models often fail to fully capture the flexibility borrowers have to choose
between defaulting, refinancing, or continuing payments, especially in volatile market conditions.
The introduction of real options theory in finance by Black and Scholes (1973) and Merton (1973) marked
a significant shift in how financial assets are valued. Real options allow for the valuation of managerial
flexibility in the face of uncertainty, leading to more accurate assessments of investments under
uncertainty. Recent research, such as that by Wachter (2008), has applied real options theory to housing
markets, demonstrating that homeowners, similar to firms, exercise options to default or refinance in
response to changing circumstances.
Despite its potential, few studies have applied a real option framework to model the default risk
specifically for residential mortgages. This article aims to fill this gap by providing a comprehensive
valuation approach using real options.
METHODOLOGY
We utilize the real option approach to model the default risk of residential mortgages, incorporating key
variables such as house prices, interest rates, and the borrower’s decision
-making process. The framework
is built upon the following assumptions:
1.
Default Option: Borrowers have the option to default on their mortgage if the value of the
property falls below the loan balance.
2.
Refinancing Option: Borrowers may refinance their mortgage if interest rates decrease, reducing
their monthly payments.
3.
Stochastic Processes: The house price and interest rate dynamics are modeled using geometric
Brownian motion, reflecting their uncertainty and randomness over time.
INTERNATIONAL JOURNAL OF COMPUTER SCIENCE & INFORMATION
SYSTEM
Volume10 Issue03, March-2025, pg. 1-9
E-ISSN: 2536-7919
P-ISSN: 2536-7900
SJIF 2019: 4.58 2020: 5.046 2021: 5.328
2025, IJCSIS, https://scientiamreearch.org
pg. 3
Published Date: -
01-03-2025
4.
Payoff Structure: The payoff for the borrower is defined by the difference between the mortgage
balance and the market value of the house.
The default decision is treated as a real option, similar to an American put option. The borrower will
default when the value of the house falls below the threshold, which is determined by the remaining
mortgage balance and other associated costs, such as transaction fees.
The model is developed using the following steps:
1.
Define the current house value, interest rate, mortgage balance, and time to maturity.
2.
Use stochastic differential equations to model the evolution of house prices and interest rates.
3.
Calculate the optimal default boundary (threshold) where it becomes optimal for the borrower to
default, based on the real options approach.
4.
Solve the resulting partial differential equations (PDEs) numerically using finite difference
methods to estimate the value of the mortgage, incorporating default risk.
RESULTS
The results demonstrate that incorporating real options significantly alters the valuation of residential
mortgages compared to traditional models. By allowing for dynamic decision-making by the borrower,
the model accounts for the option to default in response to falling house prices or rising interest rates.
The key findings from the model are as follows:
1.
Default Risk Sensitivity: Default risk is highly sensitive to changes in house prices. As house prices
decline, the probability of default increases, making the mortgage more valuable to the lender as it
reflects higher default risk.
2.
Interest Rate Effects: Lower interest rates reduce the likelihood of default, as borrowers are more
likely to refinance or continue making payments when rates are favorable. However, if house prices
remain low, the default option becomes more attractive, especially when the mortgage balance exceeds
the value of the property.
3.
Threshold Dynamics: The optimal default threshold is determined by a balance between the
mortgage balance and the house value. As market conditions fluctuate, this threshold adjusts, showing
that borrowers make strategic decisions based on the evolving economic environment.
The model's ability to adjust for changing market conditions results in a more nuanced understanding of
default risk. In contrast to traditional models, the real option approach allows for the flexibility and
responsiveness of borrowers, producing a more accurate valuation of mortgage default risk.
INTERNATIONAL JOURNAL OF COMPUTER SCIENCE & INFORMATION
SYSTEM
Volume10 Issue03, March-2025, pg. 1-9
E-ISSN: 2536-7919
P-ISSN: 2536-7900
SJIF 2019: 4.58 2020: 5.046 2021: 5.328
2025, IJCSIS, https://scientiamreearch.org
pg. 4
Published Date: -
01-03-2025
DISCUSSION
The real option approach to valuing the default risk of residential mortgages provides a more
comprehensive and dynamic framework compared to traditional methods. By treating the decision to
default as an option that borrowers can exercise based on changing market conditions, this model allows
for a more realistic representation of mortgage risk. This discussion explores the implications of the
results, the advantages of using a real option framework, and potential real-world applications with
examples.
Strategic Borrower Decision-Making
Traditional mortgage models often assume that borrowers will continue making their payments for the
duration of the mortgage, or that default is a one-time event that happens when financial hardship
becomes extreme. However, in reality, borrowers have the flexibility to adjust their decisions in response
to market conditions. For example, if a
homeowner’s property value decreases significantly, it may no
longer make financial sense to continue paying off the mortgage. The homeowner may choose to default,
which is akin to exercising a "put" option on the property
—
selling the home to the bank at the mortgage
balance.
A common scenario is when a homeowner faces a situation where the mortgage balance exceeds the
value of the house. This is referred to as being "underwater" or having negative equity. In this case, the
real option approach provides a more accurate assessment of the homeowner's default decision because
it explicitly models the option to default as a function of the house price and the remaining loan balance.
For instance, if a homeowner’s mortgage balance is $250,000, but the current mar
ket value of the home
has fallen to $200,000, the borrower has the option to default on the loan and forfeit the property. If the
homeowner expects house prices to remain low or continue declining, the incentive to default becomes
stronger, making the real option model a useful tool for evaluating this decision.
Impact of House Prices on Default Risk
The sensitivity of mortgage default risk to house prices is a critical finding in our model. House prices,
being one of the key drivers of mortgage default, play a significant role in determining whether borrowers
will default. When house prices drop, borrowers with negative equity are more likely to default, as the
economic incentive to continue paying off the mortgage diminishes.
For example, consider a scenario where a borrower has a fixed-rate mortgage for a home worth $500,000
with a remaining balance of $450,000. If the home’s value decreases to $400,000 due to a housing market
downturn, the borrower may be unwilling to continue making payments because the loan balance
exceeds the property value. This dynamic highlights the option-like nature of default; it becomes an
attractive strategy when the cost of continuing payments exceeds the cost of defaulting, which is modeled
as a real option. By modeling house prices as stochastic processes (i.e., uncertain and subject to random
INTERNATIONAL JOURNAL OF COMPUTER SCIENCE & INFORMATION
SYSTEM
Volume10 Issue03, March-2025, pg. 1-9
E-ISSN: 2536-7919
P-ISSN: 2536-7900
SJIF 2019: 4.58 2020: 5.046 2021: 5.328
2025, IJCSIS, https://scientiamreearch.org
pg. 5
Published Date: -
01-03-2025
fluctuations), the real option approach captures the volatility of the housing market and allows for a more
accurate prediction of default behavior.
Interest Rates and Refinancing as Competing Options
Another important aspect of the model is the interplay between interest rates and refinancing options.
When interest rates are low, the cost of servicing a mortgage decreases, which reduces the likelihood of
default. Borrowers are more likely to refinance their loans when interest rates drop, thus avoiding default
and continuing payments at a lower rate. In our model, this option to refinance competes with the option
to default, providing borrowers with more flexibility in their decision-making process.
For example, imagine a borrower with an adjustable-rate mortgage (ARM) who is facing rising monthly
payments as interest rates increase. If interest rates rise substantially, the borrower may face financial
distress, increasing the likelihood of default. Conversely, if interest rates fall significantly, the borrower
could choose to refinance to a lower fixed rate, making the mortgage more affordable and reducing the
default risk. The real option model captures these decisions by evaluating the value of refinancing (similar
to a call option) in relation to the value of default (similar to a put option).
Thresholds for Default: Dynamic Adjustment to Market Conditions
One of the significant advantages of the real option model is its ability to dynamically adjust the threshold
for default. Traditional models often assume a fixed point at which default occurs (e.g., when the
borrower’s financial distress exceeds a certain level). However, the real option framework allows for a
flexible threshold that evolves as market conditions change.
For example, let’s consider a situation where house prices initially decline, and the borrower is faced with
the decision of whether to default. The optimal default threshold might be lower if the homeowner
ex
pects the property’s value to recover soon. However, if the homeowner anticipates further price
declines or an extended downturn, the threshold for default will adjust, and the borrower will be more
likely to exercise the default option sooner. This dynamic behavior reflects how market expectations (in
terms of both house prices and interest rates) influence the borrower’s decision
-making process.
Real-World Applications: Mortgage-Backed Securities (MBS)
The real option approach also has important implications for mortgage-backed securities (MBS). MBS are
composed of pools of residential mortgages, and their valuation is highly dependent on the default risk of
the underlying mortgages. By using the real option model, investors and financial institutions can better
assess the risk associated with these securities, especially during periods of market volatility.
For example, during the 2008 financial crisis, many MBS investors underestimated the default risk
associated with residential mortgages. Home prices were declining, and many homeowners faced
negative equity, increasing the likelihood of default. By incorporating a real option framework into the
INTERNATIONAL JOURNAL OF COMPUTER SCIENCE & INFORMATION
SYSTEM
Volume10 Issue03, March-2025, pg. 1-9
E-ISSN: 2536-7919
P-ISSN: 2536-7900
SJIF 2019: 4.58 2020: 5.046 2021: 5.328
2025, IJCSIS, https://scientiamreearch.org
pg. 6
Published Date: -
01-03-2025
valuation of MBS, investors could have more accurately assessed the risk of these securities, which would
have helped them make better-informed investment decisions. Similarly, financial institutions managing
large portfolios of residential mortgages could use this model to assess the default risk of individual loans
and make more accurate provisioning decisions.
Implications for Policy and Risk Management
From a policy perspective, the real option approach provides valuable insights into how borrowers make
decisions during times of economic uncertainty. Policymakers could use the model to understand the
potential effects of housing market interventions, such as mortgage forbearance programs or interest rate
cuts. For example, if interest rates are lowered during a downturn, the real option model suggests that
this would reduce default risk by making refinancing more attractive. Conversely, if housing prices remain
depressed, even lower interest rates may not prevent defaults, as borrowers may still choose to default
due to negative equity.
For financial institutions, the real option model offers a more sophisticated method for assessing and
managing mortgage default risk. By integrating real options into risk management frameworks, banks and
other lenders can better understand the likelihood of default and make more informed decisions
regarding loan origination, portfolio management, and loan loss provisioning.
Limitations and Future Research
While the real option approach provides a more accurate and dynamic valuation of mortgage default risk,
there are limitations to this model. For example, the assumption of perfect market conditions and
frictionless decision-making may not fully capture the complexity of borrower behavior. Additionally, the
model’s reliance on stochastic processes for house prices and interest rates may not fully account for
systemic risks or policy interventions that can influence market outcomes.
Future research could extend the model by incorporating more granular borrower characteristics, such as
credit scores, income levels, and borrower psychology, to capture the full range of factors influencing the
decision to default. Moreover, the model could be adapted to include government intervention scenarios,
such as mortgage relief programs or direct bailouts, to assess how these interventions impact default risk.
The real option approach provides a more nuanced and flexible framework for valuing the default risk of
residential mortgages. By modeling the decision to default as an option and considering the influence of
house prices, interest rates, and borrower behavior, the model offers a more realistic and dynamic
assessment of mortgage risk. This approach has important implications for mortgage-backed securities
pricing, risk management, and policy decisions, offering deeper insights into the factors that drive default
risk in residential mortgages. Through the use of real options, financial institutions, investors, and
policymakers can gain a better understanding of mortgage risk, ultimately leading to more informed
decision-making in volatile markets.
INTERNATIONAL JOURNAL OF COMPUTER SCIENCE & INFORMATION
SYSTEM
Volume10 Issue03, March-2025, pg. 1-9
E-ISSN: 2536-7919
P-ISSN: 2536-7900
SJIF 2019: 4.58 2020: 5.046 2021: 5.328
2025, IJCSIS, https://scientiamreearch.org
pg. 7
Published Date: -
01-03-2025
The results of this study underscore the importance of incorporating real options in mortgage risk
valuation. By acknowledging that borrowers have the flexibility to make decisions in response to market
changes, the real option approach provides a more realistic representation of mortgage default risk. This
is particularly relevant in the context of financial markets where housing prices and interest rates are
subject to significant volatility.
Furthermore, the application of real options to mortgage default risk offers insights into mortgage-backed
securities (MBS) pricing. Since MBS are composed of underlying residential mortgages, understanding the
default risk of these mortgages is crucial for accurately valuing MBS. Investors and financial institutions
can use the real option model to better assess the risk associated with mortgage portfolios and adjust
their investment strategies accordingly.
CONCLUSION
This article demonstrates the value of applying a real option approach to the valuation of residential
mortgage default risk. The model captures the strategic decision-making of borrowers and incorporates
key variables such as house prices and interest rates. By doing so, it provides a more dynamic and realistic
valuation of mortgage default risk compared to traditional models.
The findings highlight the significance of borrower flexibility in the face of changing economic conditions
and the importance of incorporating real options in the valuation of mortgage-backed securities. Further
research could extend the model to include more complex borrower behaviors, such as strategic default
or the impact of government intervention, to refine the analysis and provide deeper insights into
mortgage risk.
REFERENCES
1.
An, X., Deng, Y., & Gabriel, S. A. (2021). Default option exercise over the financial crisis and beyond.
Review of Finance, 25(1), 153
–
187. [Google Scholar] [CrossRef]
2.
Andrade, G., & Kaplan, S. N. (1998). How costly is financial (not economic) distress? Evidence from
highly leveraged transactions that became distressed. The Journal of Finance, 53(5), 1443
–
1493.
[Google Scholar] [CrossRef]
3.
Black, F., & Scholes, M. (1974). The pricing of options and corporate liabilities. The Journal of Political
Economy, 81(3), 637
–
654. [Google Scholar] [CrossRef]
4.
Boyle, P. P. (1988). A lattice framework for option pricing with two state variables. Journal of Financial
and Quantitative Analysis, 23, 1
–
12. [Google Scholar] [CrossRef]
5.
Camara, A., & Chung, S. L. (2006). Option pricing for the transformed binomial class. Journal of Futures
Markets 29, 537
–
557. [Google Scholar]
6.
Capozza, D. R., Kazarian, D., & Thomson, T. A. (1998). The conditional probability of mortgage default.
Real Estate Economics, 26(3), 359
–
389. [Google Scholar] [CrossRef]
INTERNATIONAL JOURNAL OF COMPUTER SCIENCE & INFORMATION
SYSTEM
Volume10 Issue03, March-2025, pg. 1-9
E-ISSN: 2536-7919
P-ISSN: 2536-7900
SJIF 2019: 4.58 2020: 5.046 2021: 5.328
2025, IJCSIS, https://scientiamreearch.org
pg. 8
Published Date: -
01-03-2025
7.
Copeland, T., & Antikarov, V. (2001). Real options. Texere. [Google Scholar]
8.
Cox, J., Ross, S., & Rubinstein, M. (1979). Options pricing: A simplified approach. Journal of Financial
Economics, 7, 229
–
263. [Google Scholar] [CrossRef]
9.
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). An intertemporal general equilibrium model of asset
prices. Econometrica, 53(2), 363
–
384. [Google Scholar] [CrossRef]
10.
Derman, E., Kani, I., Ergener, & Bardhan, I. (1995). Enhanced numerical methods for options with
barriers. Financial Analyst Journal, 51(6), 66
–
74. [Google Scholar] [CrossRef]
11.
Diamond, W., & Landvoigt, T. (2022). Credit cycles with market-based household leverage. Journal of
Financial Economics, 146, 726
–
753. [Google Scholar] [CrossRef]
12.
Epperson, J. F., Kau, J. B., Keenan, D. C., & Muller, W. J. (1985). Pricing default risk in mortgages.
AREUEA Journal, 13(3), 261
–
272. [Google Scholar] [CrossRef]
13.
Gaudencio, J., Mazany, A., & Schwartz, C. (2019). The impact of lending standards on default rates of
residential real estate loans. Occasional Paper Series ECB. European Central Bank (ECB), Frankfurt a.
M. No. 220. [Google Scholar]
14.
Geske, R. (1979). The valuation of compound options. Journal of Financial Economics, 7(1), 63
–
81.
[Google Scholar] [CrossRef]
15.
Haathela, T. (2006). Extended binomial tree valuation when the underlying asset distribution is shifted
lognormal
with
higher
moments.
Available
online:
https://api.semanticscholar.org/CorpusID:208245423 (accessed on 15 December 2024).
16.
Haathela, T. (2011). Estimating changing volatility in cash flow simulation based real option valuation
with regression sum of squared error method. SSRN Electronic Journal. [Google Scholar] [CrossRef]
17.
Hilliard, J. E., Kau, J. B., & Slawson, V. C., Jr. (1998). Valuing prepayment and default in a fixed-rate
mortgage: A bivariate binomial options pricing technique. Real Estate Economics, 26(3), 431
–
468.
[Google Scholar] [CrossRef]
18.
Hull, J. C. (2022). Options, futures and other derivatives (12th ed.). Pearson Prentice Hall. [Google
Scholar]
19.
Kau, J. B., Keenan, D. C., Muller, W. J., III, & Epperson, J. F. (1995). The valuation at origination of fixed-
rate mortgages with default and prepayment. Journal of Real Estate Finance and Economics, 11, 5
–
36. [Google Scholar] [CrossRef]
20.
Kelly, R., & O’Malley, T. (2016). The good, the bad and the impaired: A credit risk model of
the Irish
mortgage market. Journal of Financial Stability, 22, 1
–
9. [Google Scholar] [CrossRef]
21.
Leung, W. K., & Sirmans, C. F. (1990). A Lattice Approach to Pricing Fixed-Rate Mortgages with Default
and Prepayment Options. Real Estate Economics, 18, 91
–
104. [Google Scholar] [CrossRef]
22.
Maurer, R., Reiner, F., & Sebastian, S. (2004). Characteristics of german real estate return
distributions: Evidence from germany and comparison to the U.S. and U.K. (Working Paper Series:
Finance & Accounting, No. 108). Johann Wolfgang Goethe-Universität Frankfurt am Main,
Fachbereich
Wirtschaftswissenschaften.
Available
online:
https://nbn-
resolving.de/urn:nbn:de:hebis:30-17865 (accessed on 15 December 2024).
INTERNATIONAL JOURNAL OF COMPUTER SCIENCE & INFORMATION
SYSTEM
Volume10 Issue03, March-2025, pg. 1-9
E-ISSN: 2536-7919
P-ISSN: 2536-7900
SJIF 2019: 4.58 2020: 5.046 2021: 5.328
2025, IJCSIS, https://scientiamreearch.org
pg. 9
Published Date: -
01-03-2025
23.
Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. The Journal
of Finance, 29(2), 449
–
470. [Google Scholar]
24.
O’Malley, T. (2021). The impact of repossession risk on mortgage default. The Journal of Finance,
76(2), 623
–
650. [Google Scholar] [CrossRef]
25.
Van Order, R. (2007). Modeling the credit risk of mortgage loan: A primer. Ross School of Business
Paper. 1086. Available online: https://ssrn.com/abstract=1002722 (accessed on 15 December 2024).
26.
Wilmott, P., Howison, S., & Dewynne, J. (1995). The mathematics of financial derivatives. A student
introduction. Cambridge University Press. [Google Scholar]
27.
Young, M. S., & Graff, R. A. (1995). Real estate is not normal: A fresh look at real estate return
distributions. Journal of Real Estate Finance and Economics, 10(3), 225
–
259. [Google Scholar]
[CrossRef]
