Ashwin Rao presented a seminar on improving pricing and hedging of agency mortgage-backed securities. The current industry standard approach uses an option-adjusted spread to account for prepayment risk, but this has challenges. Rao proposed an alternative approach that calibrates and trades based on the market price of prepayment risk rather than a constant option-adjusted spread. This involves modeling sources of prepayment risk other than interest rates, such as forecast errors in prepayment models, and deriving risk-neutral processes for these factors using their market prices of risk.
Slides from Abu Dhabi Prroject Financing Conference (2002) on "Negotiating the Terms & Conditions of the Project Debt and Achieving Financial Close"
The activities of large, internationally active financial institutions have grown increasingly
Complex and diverse in recent years.This increasing complexity has necessarily been accompanied by a process of innovation in how these institutions measure and monitor their exposure to different kinds of risk. One set of risk management techniques that has attracted a great deal of attention over the past several years, both among practitioners and regulators, is "stress testing", which can be loosely defined as the examination of the potential effects on a firm’s financial condition of a set of specified changes in risk factors, corresponding to exceptional but plausible events. A concept of security analysis and portfolio management services has been very famous and old among various institutions. This report represents practices application of portfolio management techniques in the portfolio section. Portfolio management is an integrated and exhaustive of fundamental and technical methods which are used for calculation of annul return and earnings per share for the portfolio. Modern portfolio theory suggests that the traditional approach to portfolio analysis, selection and management may yield less than optimum results. Hence a more scientific approach is required, based on estimates of risk and return of the portfolio and the attitudes of the investor toward a risk-return trade-off stemming from the analysis of the individual Securities.
"You can download this product from SlideTeam.net"
Use our content ready Strategic Portfolio Management PowerPoint Presentation Slides to showcase assets management of various securities in order to meet Investment goals. Investment management strategy PowerPoint complete deck comprises of professional slides such as objectives of portfolio management, types of investments, market scenario overview, investment instruments, securities portfolio, analysis and valuation of equity securities, industry analysis PESTEL, SWOT analysis, discounted cash flow method, financial statement analysis, company cash flow statement, investment in special situations, fixed income and leveraged securities, bond valuation system, reinvestment risk table, type of convertible securities, options analysis, warrants summarization overview, derivative products, put and call options, stock index futures and options, stick indexes comparison table, broaden the investment perspective, international security market highlights, global market trends, mutual funds investment criteria overview, investment in real estate, diversified real estate classification, KPIs and dashboards etc. Download investment portfolio management PPT visuals to analyze risk and return on investment. https://bit.ly/3sauHXW
The presentation has the basics of Mortgage Qualification for home buyers and discusses principles of underwriting that will help Realtors to understand the loan process.
Slides from Abu Dhabi Prroject Financing Conference (2002) on "Negotiating the Terms & Conditions of the Project Debt and Achieving Financial Close"
The activities of large, internationally active financial institutions have grown increasingly
Complex and diverse in recent years.This increasing complexity has necessarily been accompanied by a process of innovation in how these institutions measure and monitor their exposure to different kinds of risk. One set of risk management techniques that has attracted a great deal of attention over the past several years, both among practitioners and regulators, is "stress testing", which can be loosely defined as the examination of the potential effects on a firm’s financial condition of a set of specified changes in risk factors, corresponding to exceptional but plausible events. A concept of security analysis and portfolio management services has been very famous and old among various institutions. This report represents practices application of portfolio management techniques in the portfolio section. Portfolio management is an integrated and exhaustive of fundamental and technical methods which are used for calculation of annul return and earnings per share for the portfolio. Modern portfolio theory suggests that the traditional approach to portfolio analysis, selection and management may yield less than optimum results. Hence a more scientific approach is required, based on estimates of risk and return of the portfolio and the attitudes of the investor toward a risk-return trade-off stemming from the analysis of the individual Securities.
"You can download this product from SlideTeam.net"
Use our content ready Strategic Portfolio Management PowerPoint Presentation Slides to showcase assets management of various securities in order to meet Investment goals. Investment management strategy PowerPoint complete deck comprises of professional slides such as objectives of portfolio management, types of investments, market scenario overview, investment instruments, securities portfolio, analysis and valuation of equity securities, industry analysis PESTEL, SWOT analysis, discounted cash flow method, financial statement analysis, company cash flow statement, investment in special situations, fixed income and leveraged securities, bond valuation system, reinvestment risk table, type of convertible securities, options analysis, warrants summarization overview, derivative products, put and call options, stock index futures and options, stick indexes comparison table, broaden the investment perspective, international security market highlights, global market trends, mutual funds investment criteria overview, investment in real estate, diversified real estate classification, KPIs and dashboards etc. Download investment portfolio management PPT visuals to analyze risk and return on investment. https://bit.ly/3sauHXW
The presentation has the basics of Mortgage Qualification for home buyers and discusses principles of underwriting that will help Realtors to understand the loan process.
Interest rate risk management what regulators want in 2015 7.15.2015Craig Taggart MBA
Areas covered in this section
Why Interest Rate Risk (IRR) should not be ignored
• Forward Rate Agreements (FRA’s) Forwards, Futures
• Swaps, Options
Why Bank Regulators continue to have a poor handle on interest rate risk
• Interest Rate Caps, floors, Collars
• LIBOR and UBS & Barclays rigging rates
• How should Financial Institutions determine which IRR vendor models are appropriate?
IRR Measurement methodologies are institutions
Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution Machine Learning related case solution
Stochastic Control/Reinforcement Learning for Optimal Market MakingAshwin Rao
Optimal Market Making is the problem of dynamically adjusting bid and ask prices/sizes on the Limit Order Book so as to maximize Expected Utility of Gains. This is a stochastic control problem that can be tackled with classical Dynamic Programming techniques or with Reinforcement Learning (using a market-learnt simulator)
Understanding Dynamic Programming through Bellman OperatorsAshwin Rao
Policy Iteration and Value Iteration algorithms are best understood by viewing them from the lens of Bellman Policy Operator and Bellman Optimality Operator
A.I. for Dynamic Decisioning under Uncertainty (for real-world problems in Re...Ashwin Rao
Slides from the Research Seminar talk I gave at Nvidia. The topic was: A.I. for Dynamic Decisioning under Uncertainty (for Real-World problems in Retail and in Financial Trading)
Overview of Stochastic Calculus FoundationsAshwin Rao
This is a quick refresher/overview of Stochastic Calculus Foundations. This assumes you have done a Stochastic Calculus course previously and now want to review/revise the material to prepare for a course that lists Stochastic Calculus as a pre-req. In these 11 slides, I list the key content you must be familiar with within Stochastic Calculus.
Risk-Aversion, Risk-Premium and Utility TheoryAshwin Rao
This lecture helps understand the concepts of Risk-Aversion and Risk-Premium viewed from the lens of Utility Theory. These are foundational economic concepts used widely in Financial applications - Portfolio problems and Pricing problems, to name a couple.
To make Reinforcement Learning Algorithms work in the real-world, one has to get around (what Sutton calls) the "deadly triad": the combination of bootstrapping, function approximation and off-policy evaluation. The first step here is to understand Value Function Vector Space/Geometry and then make one's way into Gradient TD Algorithms (a big breakthrough to overcome the "deadly triad").
Stanford CME 241 - Reinforcement Learning for Stochastic Control Problems in ...Ashwin Rao
I am pleased to introduce a new and exciting course, as part of ICME at Stanford University. I will be teaching CME 241 (Reinforcement Learning for Stochastic Control Problems in Finance) in Winter 2019.
HJB Equation and Merton's Portfolio ProblemAshwin Rao
Deriving the solution to Merton's Portfolio Problem (Optimal Asset Allocation and Consumption) using the elegant formulation of Hamilton-Jacobi-Bellman equation.
A Quick and Terse Introduction to Efficient Frontier MathematicsAshwin Rao
A Quick and Terse Introduction to Efficient Frontier Mathematics. Only a basic background in Linear Algebra, Probability and Optimization is expected to cover this material and gain a reasonable understanding of this topic within one hour.
Recursive Formulation of Gradient in a Dense Feed-Forward Deep Neural NetworkAshwin Rao
Recursive Formulation of Gradient in a Dense Feed-Forward Deep Neural Network. Derived for a fairly general setting where the supervisory variable has a conditional probability density modeled as an arbitrary Generalized Linear Model's "normal-form" probability density, and whose output layer activation function is the GLM canonical link function.
USDA Loans in California: A Comprehensive Overview.pptxmarketing367770
USDA Loans in California: A Comprehensive Overview
If you're dreaming of owning a home in California's rural or suburban areas, a USDA loan might be the perfect solution. The U.S. Department of Agriculture (USDA) offers these loans to help low-to-moderate-income individuals and families achieve homeownership.
Key Features of USDA Loans:
Zero Down Payment: USDA loans require no down payment, making homeownership more accessible.
Competitive Interest Rates: These loans often come with lower interest rates compared to conventional loans.
Flexible Credit Requirements: USDA loans have more lenient credit score requirements, helping those with less-than-perfect credit.
Guaranteed Loan Program: The USDA guarantees a portion of the loan, reducing risk for lenders and expanding borrowing options.
Eligibility Criteria:
Location: The property must be located in a USDA-designated rural or suburban area. Many areas in California qualify.
Income Limits: Applicants must meet income guidelines, which vary by region and household size.
Primary Residence: The home must be used as the borrower's primary residence.
Application Process:
Find a USDA-Approved Lender: Not all lenders offer USDA loans, so it's essential to choose one approved by the USDA.
Pre-Qualification: Determine your eligibility and the amount you can borrow.
Property Search: Look for properties in eligible rural or suburban areas.
Loan Application: Submit your application, including financial and personal information.
Processing and Approval: The lender and USDA will review your application. If approved, you can proceed to closing.
USDA loans are an excellent option for those looking to buy a home in California's rural and suburban areas. With no down payment and flexible requirements, these loans make homeownership more attainable for many families. Explore your eligibility today and take the first step toward owning your dream home.
how to sell pi coins at high rate quickly.DOT TECH
Where can I sell my pi coins at a high rate.
Pi is not launched yet on any exchange. But one can easily sell his or her pi coins to investors who want to hold pi till mainnet launch.
This means crypto whales want to hold pi. And you can get a good rate for selling pi to them. I will leave the telegram contact of my personal pi vendor below.
A vendor is someone who buys from a miner and resell it to a holder or crypto whale.
Here is the telegram contact of my vendor:
@Pi_vendor_247
how to swap pi coins to foreign currency withdrawable.DOT TECH
As of my last update, Pi is still in the testing phase and is not tradable on any exchanges.
However, Pi Network has announced plans to launch its Testnet and Mainnet in the future, which may include listing Pi on exchanges.
The current method for selling pi coins involves exchanging them with a pi vendor who purchases pi coins for investment reasons.
If you want to sell your pi coins, reach out to a pi vendor and sell them to anyone looking to sell pi coins from any country around the globe.
Below is the contact information for my personal pi vendor.
Telegram: @Pi_vendor_247
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
US Economic Outlook - Being Decided - M Capital Group August 2021.pdfpchutichetpong
The U.S. economy is continuing its impressive recovery from the COVID-19 pandemic and not slowing down despite re-occurring bumps. The U.S. savings rate reached its highest ever recorded level at 34% in April 2020 and Americans seem ready to spend. The sectors that had been hurt the most by the pandemic specifically reduced consumer spending, like retail, leisure, hospitality, and travel, are now experiencing massive growth in revenue and job openings.
Could this growth lead to a “Roaring Twenties”? As quickly as the U.S. economy contracted, experiencing a 9.1% drop in economic output relative to the business cycle in Q2 2020, the largest in recorded history, it has rebounded beyond expectations. This surprising growth seems to be fueled by the U.S. government’s aggressive fiscal and monetary policies, and an increase in consumer spending as mobility restrictions are lifted. Unemployment rates between June 2020 and June 2021 decreased by 5.2%, while the demand for labor is increasing, coupled with increasing wages to incentivize Americans to rejoin the labor force. Schools and businesses are expected to fully reopen soon. In parallel, vaccination rates across the country and the world continue to rise, with full vaccination rates of 50% and 14.8% respectively.
However, it is not completely smooth sailing from here. According to M Capital Group, the main risks that threaten the continued growth of the U.S. economy are inflation, unsettled trade relations, and another wave of Covid-19 mutations that could shut down the world again. Have we learned from the past year of COVID-19 and adapted our economy accordingly?
“In order for the U.S. economy to continue growing, whether there is another wave or not, the U.S. needs to focus on diversifying supply chains, supporting business investment, and maintaining consumer spending,” says Grace Feeley, a research analyst at M Capital Group.
While the economic indicators are positive, the risks are coming closer to manifesting and threatening such growth. The new variants spreading throughout the world, Delta, Lambda, and Gamma, are vaccine-resistant and muddy the predictions made about the economy and health of the country. These variants bring back the feeling of uncertainty that has wreaked havoc not only on the stock market but the mindset of people around the world. MCG provides unique insight on how to mitigate these risks to possibly ensure a bright economic future.
how to sell pi coins effectively (from 50 - 100k pi)DOT TECH
Anywhere in the world, including Africa, America, and Europe, you can sell Pi Network Coins online and receive cash through online payment options.
Pi has not yet been launched on any exchange because we are currently using the confined Mainnet. The planned launch date for Pi is June 28, 2026.
Reselling to investors who want to hold until the mainnet launch in 2026 is currently the sole way to sell.
Consequently, right now. All you need to do is select the right pi network provider.
Who is a pi merchant?
An individual who buys coins from miners on the pi network and resells them to investors hoping to hang onto them until the mainnet is launched is known as a pi merchant.
debuts.
I'll provide you the Telegram username
@Pi_vendor_247
how can I sell pi coins after successfully completing KYCDOT TECH
Pi coins is not launched yet in any exchange 💱 this means it's not swappable, the current pi displaying on coin market cap is the iou version of pi. And you can learn all about that on my previous post.
RIGHT NOW THE ONLY WAY you can sell pi coins is through verified pi merchants. A pi merchant is someone who buys pi coins and resell them to exchanges and crypto whales. Looking forward to hold massive quantities of pi coins before the mainnet launch.
This is because pi network is not doing any pre-sale or ico offerings, the only way to get my coins is from buying from miners. So a merchant facilitates the transactions between the miners and these exchanges holding pi.
I and my friends has sold more than 6000 pi coins successfully with this method. I will be happy to share the contact of my personal pi merchant. The one i trade with, if you have your own merchant you can trade with them. For those who are new.
Message: @Pi_vendor_247 on telegram.
I wouldn't advise you selling all percentage of the pi coins. Leave at least a before so its a win win during open mainnet. Have a nice day pioneers ♥️
#kyc #mainnet #picoins #pi #sellpi #piwallet
#pinetwork
How to get verified on Coinbase Account?_.docxBuy bitget
t's important to note that buying verified Coinbase accounts is not recommended and may violate Coinbase's terms of service. Instead of searching to "buy verified Coinbase accounts," follow the proper steps to verify your own account to ensure compliance and security.
where can I find a legit pi merchant onlineDOT TECH
Yes. This is very easy what you need is a recommendation from someone who has successfully traded pi coins before with a merchant.
Who is a pi merchant?
A pi merchant is someone who buys pi network coins and resell them to Investors looking forward to hold thousands of pi coins before the open mainnet.
I will leave the telegram contact of my personal pi merchant to trade with
@Pi_vendor_247
how to sell pi coins in all Africa Countries.DOT TECH
Yes. You can sell your pi network for other cryptocurrencies like Bitcoin, usdt , Ethereum and other currencies And this is done easily with the help from a pi merchant.
What is a pi merchant ?
Since pi is not launched yet in any exchange. The only way you can sell right now is through merchants.
A verified Pi merchant is someone who buys pi network coins from miners and resell them to investors looking forward to hold massive quantities of pi coins before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
@Pi_vendor_247
Introduction to Indian Financial System ()Avanish Goel
The financial system of a country is an important tool for economic development of the country, as it helps in creation of wealth by linking savings with investments.
It facilitates the flow of funds form the households (savers) to business firms (investors) to aid in wealth creation and development of both the parties
Turin Startup Ecosystem 2024 - Ricerca sulle Startup e il Sistema dell'Innov...Quotidiano Piemontese
Turin Startup Ecosystem 2024
Una ricerca de il Club degli Investitori, in collaborazione con ToTeM Torino Tech Map e con il supporto della ESCP Business School e di Growth Capital
Turin Startup Ecosystem 2024 - Ricerca sulle Startup e il Sistema dell'Innov...
Towards Improved Pricing and Hedging of Agency Mortgage-backed Securities
1. Towards Improved Pricing and Hedging of
Agency Mortgage-Backed Securities (MBS)
Ashwin Rao
Presentation for AFTLab Seminar at Stanford University
May 31, 2018
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 1 / 54
2. Overview
1 Mortgage Basics
2 Agency Securitization
3 Industry-Standard approach to Pricing & Hedging
Economic and Behavioral Intuition of Prepayment Models
Challenges with Trading based on “OAS”
4 Proposal: Pricing/Hedging with “risk-neutral prepayments”
Mathematical Formalism for Industry-Standard approach
Modeling of Prepayment Risks other than Interest Rate Risk
Trading based on calibrated Price of Prepayment Risk
Are we Happy? Can we do better?
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 2 / 54
3. Mortgage Basics: Borrower’s Perspective
A family wants to buy a house worth $500K
Can make down-payment of $100K ⇒ Loan to Value (LTV) of 80%
Borrower has a good FICO score of 750
Monthly family income is $6000, Current monthly debt is only $100
So qualifies for a good mortgage rate (30 year, Fixed Rate) of 4%
Monthly payment (Interest + Principal) amounts to about $1900
Debt to Income (DTI) is 1900+100
6000 = 33%
Psychology less about debt mgmt, more about cash flows (DTI)
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 3 / 54
5. Mortgage Basics: Lender’s Perspective
The lender requires documentation on income, other assets/debts
The lender is long a “Bond” receiving monthly Principal & Interest
Lender’s biggest risk is the borrower defaulting on monthly payments
Think of this as the Borrower’s (American) Put option on the House,
with Strike = Remaining Principal (Default ⇒ House taken away)
Lender will offer a rate commensurate with LTV, DTI, FICO
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 5 / 54
6. Lender also exposed to Voluntary Prepayment Risk
Prepayment: Payment of Principal (full or partial) before schedule
Borrower Defaulting is refered to as Involuntary Prepayment
Voluntary Prepayment mainly due to Home Sale or Refinancing
Think of this as the Borrower’s (American) Call option on the Loan,
with Strike = Remaining Principal
Like any callable bond, Call Option is In-the-money when Price > Par
In-the-money when offered rate is below rate on borrower’s mortgage
Refinancing: Borrowers often exercise deep in-the-money (suboptimal)
HomeSale: Often happens when call option is out-of-the-money
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 6 / 54
7. Agency Securitization of a Pool of Mortgages
“Agency” refers to the Government-sponsored entities (GSE)
Pool of Conforming loans (conservative loan size, LTV, DTI, docs)
Agency receives collective Principal & Interest from the pool
In Exchange for cash extended to banks to originate mortgages
Agency issues a Mortgage-Backed Security (backed by mortgage pool)
Passthrough MBS simply forwards mortgage cash flows to investors
Structured MBS: Pro Rata, Sequential, Planned Amortization Classes
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 7 / 54
8. Agency MBS securitized from a Conforming Pool of Loans
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 8 / 54
9. Agency MBS are essentially credit-risk-free
Conforming loans ⇒ risk of borrower delinquencies/default is low
Agency guarantees investors principal even when a borrower defaults
Defaulted loan is taken out of pool, investor receives full principal
So Agency MBS investor has no credit-risk (government backing!)
Agency receives a guarantee fee for this credit protection
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 9 / 54
10. Investor Perspective of Prepayment Risk for Passthroughs
Prepayments ⇒ Call option exercise on a fraction of the Pool
So, Prepayments will move Price towards Par
Price for a Premium (Price > Par) reduces when Prepayments rise
Price for a Discount (Price < Par) increases when Prepayments rise
Price for the Par Passthrough is insensitive to Prepayments
Prepayments for Premiums (Price > Par) are mainly Refinancings
Prepayments for Discounts (Price < Par) are mainly HomeSales
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 10 / 54
11. Investor Perspective of Hedging Passthroughs
Duration Definition: − 1
P
∂P
∂r
Duration viewed as cashflows-PV-weighted average time of cash flows
Duration is lower when Price is higher
Duration reduces when Prepayments rise
Short Prepayment Option ⇒ Convexity and Vega are negative
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 11 / 54
13. Industry-Standard approach to Pricing & Hedging
The key components of a typical Agency MBS Pricing/Hedging System:
Stochastic interest rate model calibrated to liquid rate/vol instruments
Monte Carlo simulation of interest rate model
Econometric model of prepayments
Model for mortgage rates (as a function of interest rates and vol)
Cash flow generator for various MBS Structures
Pricing/Sensitivities based on notion of “Option-Adjusted Spread”
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 13 / 54
14. Interest Rate Model
Calibration to liquid rate/vol instruments
Multi-factor model needed to capture non-parallel curve moves
Capturing Vol Skew/Smile very important for MBS
So need to also calibrate to OTM swaptions’ market vol
Model Choices: Local Vol, Stochastic Vol, CEV, Jump Diffusion
Short rate or HJM models
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 14 / 54
15. Monte-Carlo Simulation
Two reasons for needing to do non-Markovian Pricing of MBS:
Burnout and Media effects
Many CMOs have cash flows that depend on history of cash flows
Sequence of draws of Brownian increments gives a path of states
This provides discount factors at each time step
For cash flows, need analytic/grid mapping from State to Rates/Vols
Variance reduction:
Antithetic variables (E[dzt] = 0 for all t)
Ensure E[(dzt)2
] = dt, E[(dzs)(dzt)] = 0 using Orthonormalization
Control Variates, Quasi Random Numbers, Brownian Bridge
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 15 / 54
16. Prepayment Modeling
Econometric forecast of fraction of pool terminating each month
Explanatory variables built from historical data involving:
Loan Characteristics (Loan Size, Coupon, Term ...)
Borrower Information (Income, FICO, Other Debts ...)
Collateral Information (Single/Multi-Family, RE Taxes, ZIP ...)
Economic Conditions (Interest Rates, Unemployment, Home Prices ...)
4 sub-models, one for each of the following types of prepayments:
HomeSale (typically job-related or buying more expensive house)
Refinancing (to get a lower rate, or for cash-out refinance)
Default (⇒ loan paid off and removed from pool)
Curtailment (partial prepayment to curtail the life of the loan)
Most prepayments for agency loans are home sales and refinancings
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 16 / 54
17. Economic and Behavioral Intuition of HomeSales Model
HomeSales volume varies significantly by different seasons of the year
Ramp up in home sales as a function of loan age
Higher home prices leads to homeowners “trading up”
Higher credit borrowers face less frictions
“Lock-in” effect: Borrowers with low mortgage rate (relative to
offered rate) reluctant to move as they like to hold on to their low rate
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 17 / 54
19. Economic and Behavioral Intuition of Refinancing Model
Incentive is typically the % improvement in monthly payments
Mortgage Rate difference (“moneyness”) is used as an approximation
Lower credit quality diminishes the ability to refinance
Credit quality indicators are SATO, FICO, LTV, DTI, Home Prices
Higher home prices lead to cash-out refinancing
Prepayment Model is a function of history of rates (non-Markovian)
Burnout Effect (Heterogeneity in borrower refinancing efficiency)
Media Effect (Spike in commercials when mortgage rates reach lows)
Topic for another day: Loan-level Deep Learning models
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 19 / 54
22. Regression model for mortgage rates
Prepayment model typically based on mortgage rate “moneyness”
Current Coupon (CC) model and Primary-Secondary Spread Model
Historical CC regressed against historical rates and vol
Instead, one can regress pricing-generated CC against rates and vol
But then this CC model is input to pricing
This fixed point is resolved by iterating to convergence
Alternative approach: Price Moneyness and Backward Induction
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 22 / 54
23. Pricing/Sensitivities based on “Option-Adjusted Spread”
If T is maturity in months and N is number of Monte Carlo paths,
Price =
1
N
N
i=1
T
t=1
(Principali,t + Interesti,t) · e− t
1(ri,u+OAS)
OAS is a constant spread across time steps and MC paths
Risk-Premium for all risk factors other than interest rates
Pricing/Hedging with this OAS-based (industry-standard) approach:
For MBS with market prices, assess rich/cheap based on implied OAS
For illiquid MBS, compute Price from trader-defined(!) OAS
Price Sensitivities (duration, convexity, vega) with constant OAS
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 23 / 54
25. Significant challenges in Trading based on OAS
Implicit assumption of constancy of OAS, but market doesn’t agree
Passthrough MBS exhibit an “OAS smile”
Empirical Duration performs better than Model Duration
Ill-formed theories on “OAS Directionality”
IO/PO from same collateral have vastly different OAS
IO Duration + PO Duration = Underlying Pool Duration
Unclear what OAS to set to price illiquid MBS
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 25 / 54
27. Towards an improved approach for Pricing/Hedging
First establish mathematical formalism for Industry-Standard approach
Calibrate Market Price of Prepayment Risk (modulo Interest Rates)
Trade based on constancy of calibrated Price of Risk (instead of OAS)
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 27 / 54
28. Notation for Continuous-Cashflow Pricing
dr(t) = αr (r, t) · dt + σr (r, t) · dzr (t)
V (t) is Value at time t of (uncertain) cash flows after time t
B(t) is Balance = Remaining Principal at time t
P(t) is Price = “Normalized” Value = Value
Balance = V (t)
B(t)
c(r, t) is Coupon (interest paid per unit Balance per unit time)
π(r, t) is Principal paid per unit Balance per unit time
π(r, t) = sched(t) + sale(r, t) + refi(r, t) + curtail(r, t) + defaults(r, t)
Cashflow over time dt is (c(r, t) + π(r, t)) · B(t) · dt
dB(t) = −π(r, t) · B(t) · dt
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 28 / 54
29. Derivation of Continuous-Cashflow Pricing PDE
First assume MBS is a pure interest rate derivative, i.e., assume
Prepayments are guaranteed to obey our model π(r, t) function.
Follow a derivation similar to zero-coupon-bond-pricing PDE.
Create portfolio of two MBS whose dV doesn’t have dz term
So portfolio Value grows at risk-free rate r
The two MBS have the same Market Price of Interest Rate Risk λr
∂P
∂t
+ α
(Q)
r
∂P
∂r
+
σ2
2
∂2P
∂r2
+ c + π = (π + r)P
dr(t) = α
(Q)
r dt + σr dz
(Q)
r (t) where α
(Q)
r = αr − λr σr
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 29 / 54
30. Pricing PDE for Industry-Standard (OAS-based) approach
Now relax the assumption that MBS is a pure interest rate derivative
Actual prepayments depend on stochastic (risk) factors other than r
Each of them command a Market Price of Risk (i.e., a risk premium)
Our Model Pricing is devoid of the risk premium of these other factors
For Pricing to match Market, we need to adjust by this risk premium
∂P
∂t
+ α
(Q)
r
∂P
∂r
+
σ2
2
∂2P
∂r2
+ c + π = (π + r + s)P
where s is our “good-old” Option-Adjusted Spread (OAS)
P(0) = EQ[
T
0
(c + π) · e− u
0 (π+r+s)du
dt]
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 30 / 54
31. Modeling Prepayment Risk Factors modulo Interest Rates
To unravel s, model risk factors for prepayments other than r
We won’t model economic factors like employment, refinance frictions
Instead we model processes for forecast-errors of prepayment model
refi (x, r, t) = refi(r, t) · x(t)
sale (y, r, t) = sale(r, t) · y(t)
dx(t) = βx (1 − x)dt + σx dzx (t) with x(0) = 1
dy(t) = βy (1 − y)dt + σy dzy (t) with y(0) = 1
Analogous to rate risk, shifting to risk-neutral measure, we have:
dx(t) = α
(Q)
x dt + σx dz
(Q)
x (t) where α
(Q)
x = βx (1 − x) − λx σx
dy(t) = α
(Q)
y dt + σy dz
(Q)
y (t) where α
(Q)
y = βy (1 − y) − λy σy
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 31 / 54
32. Calibrating the Market Price of Risk λx and λy
∂P
∂t
+ α
(Q)
r
∂P
∂r
+
σ2
r
2
∂2P
∂r2
+ α
(Q)
x
∂P
∂x
+
σ2
x
2
∂2P
∂x2
+ α
(Q)
y
∂P
∂y
+
σ2
y
2
∂2P
∂y2
+c + π = (π + r + rs)P
π (x, y, r, t) = sched(t) + refi(r, t) · x(t) + sales(r, t) · y(t) + . . .
rs is the spread due to any residual risk such as liquidity or credit
σx , σy , βx , βy estimated from errors in prepayment forecasts
λx , λy calibrated to liquid MBS market prices (shifting x, y drifts)
P(0) = EQ[
T
0
(c + π ) · e− u
0 (π +r+rs)du
dt]
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 32 / 54
34. Pricing and Hedging with “risk-neutral prepayments”
π(Q)(r, t) = π (EQ[x], EQ[y], r, t) called “risk-neutral prepayments”
λx , λy calibration ⇒ EQ[x] > 1, EQ[y] < 1,
which dictates the alteration of π(r, t) to π(Q)(r, t)
Trading with “risk-neutral prepayments” and residual spread rs
For MBS with market prices, assess rich/cheap based on implied rs
For illiquid MBS, Price using a prudent rs (easier than a OAS input)
Price Sensitivities using constant rs (instead of constant OAS)
Effectively, we trade based on constancy of λx , λy and rs
(versus constancy of OAS)
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 34 / 54
37. Practical considerations in designing the Pricer
I used TBA prices for calibration (liquid IOs/POs could also be used)
I calibrated Market Price of Risk of uncertain initial values x(0) and
y(0), and Market Price of Risk of dzx , dzy
I set rs = Agency-Swap spread plus appropriate liquidity spread
(we can do better by building a proper model for residual risk)
Small ∂2P
∂x2 , ∂2P
∂y2 ⇒ we can avoid stochasticity of x, y in Pricer
So calibrated λx , λy essentially give us EQ[x(t)], EQ[y(t)]
This gives us an altered prepayment function π(Q)(r, t)
Practically, use same old Pricer, simply alter π(r, t) to π(Q)(r, t)
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 37 / 54
38. Are we Happy? Can we do better?
“OAS Smile” captured (but sometimes small residual smile remains)
Model Durations match Empirical (“OAS Directionality” captured)
Model prices for IOs/POs are fairly close to their market prices
IOs: rs-based Duration more negative than OAS-based Duration
Superior hedge performance relative to OAS-based Duration
Further work (will help in crisis periods) - make rs a function of:
General credit risk
Agency-specific credit risk
Supply/Demand-based liquidity risk
Modeling x, y as a jump-diffusion might also be a good idea
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 38 / 54
41. Summary and Finishing Comments
“Risk-Neutral Prepayments” have been discussed since the mid-90s.
But practitioners shy away claiming “Pricing Theory is not for MBS”!
I was first influenced to implement this by Alex Levin in 2006
My motivations: A) Improved Rich/Cheap, & B) Hedge Performance
My recommendation from a Trading perspective:
Develop a loan-level econometric prepayment model
Capitalize on recent advances in Deep Learning techniques
Adjust this model with Market Price of Risk calibration to MBS Prices
Further work on residual Liquidity/Credit Risk (for times of crises!)
Can we extend this idea to non-Agency (default-risky) MBS?
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 41 / 54
42. Appendix 1: Derivation of Pricing PDE
dr(t) = α(r, t) · dt + σ(r, t) · dz(t)
V (t) is Value at time t of (uncertain) cash flows after time t
B(t) is Balance = Remaining Principal at time t
P(t) is Price = “Normalized” Value = Value
Balance = V (t)
B(t)
c(r, t) is Coupon (interest paid per unit Balance per unit time)
π(r, t) is Principal paid per unit Balance per unit time
π(r, t) = sched(t) + sale(r, t) + refi(r, t) + curtail(r, t) + defaults(r, t)
Cashflow over time dt is (c(r, t) + π(r, t)) · B(t) · dt
dB(t) = −π(r, t) · B(t) · dt
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 42 / 54
43. Appendix 1: Pricing PDE (assuming only IR risk)
Assume MBS is a pure interest rate derivative, i.e., assume
Prepayments are guaranteed to obey our model π(r, t) function.
Consider two different MBS with notation subscripted with “1” and “2”.
dV1 = B1dP1 + P1dB1
Ito’s Lemma for dP1 and substituting dB1 = −π1B1dt gives:
dV1 = B1(
∂P1
∂t
dt +
∂P1
∂r
dr +
1
2
σ2 ∂2P1
∂r2
dt) − P1π1B1dt
= B1(
∂P1
∂t
+ α
∂P1
∂r
− πP1 +
1
2
σ2 ∂2P1
∂r2
)dt + B1σ
∂P1
∂r
dz
dV1
V1
= (
1
P1
∂P1
∂t
+
α
P1
∂P1
∂r
− π1 +
σ2
2P1
∂2P1
∂r2
)dt + (
σ
P1
∂P1
∂r
)dz (1)
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 43 / 54
44. Appendix 1: Pricing PDE (assuming only IR risk)
Denote the Ito drift and dispersion of dV1
V1
as α1 and σ1.
dV1
V1
= α1dt + σ1dz (2)
α1 =
1
P1
∂P1
∂t
+
α
P1
∂P1
∂r
− π1 +
σ2
2P1
∂2P1
∂r2
(3)
σ1 =
σ
P1
∂P1
∂r
(4)
Consider a portfolio W = V1 + V2 = P1B1 + P2B2 with values of B1 and
B2 such that we can eliminate the dz term in the expression for dW
B1
∂P1
∂r
= −B2
∂P2
∂r
(5)
Since we have eliminated the dz term, portfolio W together with the
cashflow it generates should grow at the risk-free rate r. In other words,
dV1 + dV2 + B1(c1 + π1)dt + B2(c2 + π2)dt = r(V1 + V2)dt
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 44 / 54
46. Appendix 1: Pricing PDE (assuming only IR risk)
Using equations 3 and 4, the above equation can be expressed as:
α1 + c1+π1
P1
− r
σ1
=
α2 + c2+π2
P2
− r
σ2
(6)
Note the numerator of LHS of Eq 6 is the expected excess return per
unit time of investing in MBS 1 (expected growth rate of process V1
together with its P & I cash flows, less r).
The denominator is the standard deviation of the return per unit time.
Their ratio is the familiar Market Price of Interest-Rate Risk λr (which
is the same for every security exposed to only interest-rate risk).
This is what equation 6 is telling us, which can be re-expressed as:
α1 + c1+π1
P1
− r
σ1
=
α2 + c2+π2
P2
− r
σ2
= λr
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 46 / 54
47. Appendix 1: Pricing PDE (assuming only IR risk)
Substituting in the above equation for α1 from equation 3 and for σ1 from
equation 4, we get:
1
P1
∂P1
∂t + α
P1
∂P1
∂r − π1 + σ2
2P1
∂2P1
∂r2 + c1+π1
P1
− r
σ
P1
∂P1
∂r
= λr
Reorganize and drop the subscript 1 in P1, c1, π1 to arrive at the PDE:
∂P
∂t
+ (α − λr σ)
∂P
∂r
+
σ2
2
∂2P
∂r2
+ c + π = (π + r)P
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 47 / 54
48. Appendix 1: Pricing PDE (assuming only IR risk)
If we shift to the risk-neutral measure (denoted as Q), V1 together with
the cash flows it generates should grow at risk-free rate r. Hence,
dV1
V1
= (r −
c1 + π1
P1
)dt + σ1dz(Q)
= (α1 − λr σ1)dt + σ1dz(Q)
Since, dz(Q) = λr dt + dz, the Ito process for the short-rate r in the
risk-neutral measure can be written as:
dr = (α − λr σ) · dt + σ · dz(Q)
= α(Q)
· dt + σ · dz(Q)
where α(Q) = α − λr σ is the risk-neutral drift for r.
So, the PDE can also be expressed in terms of the risk-neutral drift of r:
∂P
∂t
+ α(Q) ∂P
∂r
+
σ2
2
∂2P
∂r2
+ c + π = (π + r)P
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 48 / 54
49. Appendix 2: Martingale-based Pricing
Define the money-market process M(t) as:
dM(t) = r(t) · M(t) · dt, M(0) = 1
Consider a process θ(t) derived from V (t) and the cash flows, as follows:
dθ(t) = dV (t) + (c(r, t) + π(r, t))B(t)dt
Assume MBS is a pure interest rate derivative, i.e., assume
Prepayments are guaranteed to obey our model π(r, t) function.
Then, θ(t)
M(t) is a martingale in risk-neutral measure Q. So, for any T > 0,
θ(0)
M(0)
= EQ[
θ(T)
M(T)
]
Since θ(0) = V (0), M(0) = 1, and without loss of generality, B(0) = 1,
P(0) =
θ(0)
M(0)
= EQ[
V (T) +
T
0 (c(r, t) + π(r, t)) · B(t) · dt
M(T)
]
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 49 / 54
50. Appendix 2: Martingale-based Pricing
If we set T to MBS maturity, V (T) = B(T) = 0. Then,
P(0) = EQ[
T
0
(c(r, t) + π(r, t)) · B(t)
M(t)
dt]
= EQ[
T
0
(c(r, t) + π(r, t)) · B(t) · e− u
0 r(u)du
dt]
= EQ[
T
0
(c(r, t) + π(r, t)) · e− u
0 (r(u)+π(r,u))du
dt]
(7)
Conceptualize as cash flows discounted at rate r + π in Q-measure.
We refer to this as the Expected Discounted Cashflow (EDC) formula.
Note: This formula holds only when assuming there is no risk factor other
than interest rates. Otherwise, we require further (OAS) discounting.
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 50 / 54
51. Appendix 3: Closed-Form Approx. for Price/Sensitivities
Closed-form Solution for Expected Discounted Cashflow (EDC)
formula when π(r, t) = π0(t) + rπ1 and c(r, t) = c0(t) + r · c1(t)
Closed-form Solution for Pricing PDE when r follows an affine process
and π(r, t) = π0(t) + rπ1 and c(r, t) = c0(t) + r · c1(t)
Closed-form Solution for EDC Price formula when π is linear in r with
hard max and min
The above gives bad greeks due to piecewise prepayment function.
But this serves quite useful as a control variate.
We can do analytical partial derivatives of EDC Price formula w.r.t:
a time-parallel shift to r(t) (Duration approximation)
coupon c
a time-parallel shift to π(r, t) (Prepayment sensitivity)
a time-parallel shift to refi multiplier x(t) or sale multiplier y(t)
residual spread rs
These closed-forms useful to reason about pricing/sensitivities
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 51 / 54
52. Appendix 4: Markovian Pricing (Alex Levin APD paper)
Model Burnout by creating a few (2-3) cohorts in the pool, each of
which is homogeneous in terms of refinancing efficiency
Each cohort’s prepayment incentive is Markovian
For each cohort, at every grid node, probabilistically discount sum of:
Interest cash flow c
Principal cash flow π
(1 − π) · P
This gives a backward induction for P for each cohort
MBS Price = sum of cohort Prices
Alternatively, we can solve Pricing PDE with Crank-Nicholson
Making π a function of P (instead of r) yields a clean model
Caveat: ARMs and some CMOs have non-Markovian cash flows.
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 52 / 54
53. Appendix 5: Sign of Price of Risk
For risk factor f , the vol of dV
V w.r.t f is σf
P
∂P
∂f .
So the risk-premium for the MBS due to the risk factor f is:
λf σf
P
∂P
∂f
We calibrate λx and λy to liquid passthrough market prices
Premiums are mainly exposed to refinancing risk and ∂P
∂x is negative
OAS smile ⇒ λx (Price of Refinancing-Forecast Risk) is negative
Discounts are mainly exposed to home sales risk and ∂P
∂y is positive
OAS smile ⇒ λy (Price of HomeSales-Forecast Risk) is positive
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 53 / 54
54. Appendix 6: Direction of risk-neutral adjustment of drift
For risk factor f , it’s drift is subtracted by λf σf to make the process for f
risk-neutral. Risk-premium s due to the risk factor f is given by:
s =
λf σf
P
∂P
∂f
So, the drift of f is subtracted by:
sP
(∂P
∂f )
s > 0 ⇒ drift is adjusted in a direction that worsens price
So, adjusted Refinancing multiplier EQ[x(t)] > 1
And adjusted HomeSales multiplier EQ[y(t)] < 1
Ashwin Rao (AFTLab Seminar @ Stanford) Agency MBS Pricing May 31, 2018 54 / 54