L03 Volatility and Mortgage-Backed Securities Risk

Managing Volatility Risk

Historical Volatility & Historical Correlation

Volatility

• Can define as std of returns

• Helpful to standardise on an annualised basis

• If volatility is constant, we can derive longer-period volatility from (in)famous 'square-root rule'

• Important for derivatives pricing, hedging, and risk measurement

Historical Vol Forecasts

• One forecasting rule is the historical MA

• This gives unbiased estimate
• If data period is daily, mean return will be low and we can set this to zero

• This usually reduces variance
• Change from to n in denominator makes little difference

• These approaches give each observation the same weight, then no weight, in forecast

• Easy to use

• Problems

  • If true vol is constant, any differences are due only to sampling error
  • Can’t allow for changes in true vol
  • More distant events in sample period have same weight as more recent ones

Some illustrative H vols

• Figure gives two alternative H estimators, using and
• For large , vol estimate is smoother and less responsive
• When shock occurs, both jump and remain high
• For high , plateau is lower but longer lasting

EWMA

• Can ameliorate some of these problems using an exponential weight

• Weight attached to observation decays over time

• Can also be rewritten as updating rule

• Higher means weight declines slowly
• For daily data, RiskMetrics uses
• Figure shows how low- rises most after shock, but decays faster

EWMA Forecasting

• Can use EWMA to forecast:

for any

• EWMA forecast is same as current value

• But flat vol forecast not very appealing

  • Ignores other information; not plausible

GARCH models

• EWMA models also take to be constant

• This is implausible and not appealing

• Popular alternative is a GARCH

• This fits nicely with some stylised features of returns

  • Returns show vol clustering
  • Returns show leptokurtosis – heavy tails

• GARCH can accommodate both of these

• Basic GARCH model

for and

• Errors are often normal, in which case return are conditionally normal
• Returns can be also

GARCH

• Most popular is GARCH

• High implies that vol is persistent and takes a long time to change

• High means that vol is spikey and quick to react

• Often is over 0.7 and less than 0.25

Properties of GARCH

• GARCH depends on same variables as EWMA, but has three parameters not 1

• EWMA is special case with , and

• GARCH with positive intercept allows vol to be mean-reverting

  • This is appealing
  • Long run vol tends to revert to

Forecasting with GARCH

• Letting , can show that period ahead forecast is

• Since , this means vol forecast converges to
• Can also apply to vol term structure

Estimating GARCH models

• Run data through a preliminary filter (e.g., ARMA) to remove serial correlation

• ARMA model should be as parsimonious as possible

• Select particular candidate GARCH specification (e.g., based on PACs)

• Estimate model using ML

• Check model adequacy by testing whether standardised innovations are idd and follow assumed distribution

• Pass or fail model

• If model fails, try another GARCH

Other GARCH models

• IGARCH is applicable when returns not stationary

• EWMA is special case where

• Components GARCH

  • Lets vol converge to a long-term vol that changes over time

• Factor GARCH

  • This links returns from n assets to one or more underlying factors (e.g., as with CAPM)
  • This leads to

Covariances and correlations

• Covarance between and is

• Correlation is

• Correlation only good measure of dependency is returns are elliptical
  Only defined if vols exist – need to check for this

• Estimate correlation

  • Equal-weighted covariance/correlation
  • EWMA covariances/correlation
  • GARCH covariances/correlation

Implied Volatility & Implied Correlation

Implied Volatility

• Black-Scholes model

where

• Implied volatility

• Quote ISD vs premium

  • ISD is more intuitive (similar to quotting bonds with yield instead of price)
  • reflect the market's view
  • ISD is a risk-neutral volatility

• These are volatilities generated from option prices

• Given that other variables (price, etc.) are observable, can infer implied vol from option price (e.g., Black-Scholes)

• Implied vol is a forward-looking estimator

  • Takes account of all information, not just historical backward-looking information

• Implied vols generally regarded as better than historical ones

• Implied vols are dependent on option-pricing model

  • Possible problems due to holes in Black-Scholes, volatility smiles/smirks, etc.

• Also, implied vols only exist for assets on which options have been written

VIX

• VIX is a popular measure of the implied volatility of S&P 500 index options
  
• Properties about VIX
  • average value: on the order of 21%; daily change: about 2.4%
  • assuming normal distribution: daily movements should be within
  • the actual distribution is far from normal

Implied Volatility Surface

• If B-S were correct, the ISDs should be constant across strike prices and maturities

• If we plot ISDs agaist strike prices and maturities we obtain the implied volatility surface
  

  • volatility smile (ISDs vs. strike prices)
  • term structure of volatility
  • spot & forward volatility

Prediction of the Volatility Surface

• Sticky strike

  • there is no structural change in the volatility curve
  • peice movement is largely temporary

• Sticky moneyness

  • permanent shift in the volatility curve

Implied Correlation

• These are directly analogous to implied volatilities
• But can only use if relevant options exist
• Need two or more options, or spread options
• Have same pros and cons as implied vols, and also be unstable
• Exchange option involves the surrender of an asset (B) in exchange for acquiring another (A). The payoff on such a call is

• The valuation formula (Margrabe Model)

  • replace by the price of asset B ()
  • replace the risk-free rate by the yield on asset B ()
  • the volatility is

• Rainbow Option: exposed to two or more sources of uncertainty

  • e.g. An option on a basket

Currency Implied Correlation: A Triplet of Currency Options

• Exchange rates are expressed relative to a base currency (usually USD)

• The cross rate is the exchange rate between two currencies other than the reference currency

• Example: let be the dollar price of GBP and be the doller/euro rate. Then the euro/pound rate is

the volatility of the cross rate is

Portfolio Average Correlation

• Expressing portfolio ISD by the ISD of its components through the average correlation

• the average correlation is a summary measure of diversification benefits across the portfolio

• all else equal, an increasing correlation in creases the total portfolio risk

• A dispersion trade takes a short position in index volatility, which is offset by long position in the volatility of the index components

Covariance Matrices

Forecasting covariance matrices

• Estimated cov or corr matrices must be positive definite or positive semi-definite

  • This imposes constraints on how we can estimate them
  • Can’t estimate parameters independently, and then hope for this condition to be satisfied

• Historical covariance matrices

  • Straightforward to estimate:

    • Choose window size, and estimate parameters simultaneously

  • Drawbacks

    • Only accurate if true matrix constant
    • Can suffers from ghost effects

• Multivariate EWMA

  • This is more flexible and has smaller ghost effects
  • Must choose same decay factor for all terms
  • If we don’t, there is no guarantee that matrix will be PD or PSD

• Multivariate GARCH

  • This can be difficult

    • Can need a lot of parameters
    • High dimensions hard to handle
    • Problems of convergence of routines, etc.

  • Can use methods such as orthogonal GARCH to get around some of these problems

Generating PD or PSD covariance matrices

• One way to ensure PD or PSD matrices is to adjust eigenvalues, and then recover matrix from adjusted eigenvalues

  • If we want PD, eigenvalues must be positive
  • If we want PSD, eigenvalues must be non-negative

• Obtain eigenvalues, adjust any –ve (and maybe 0) ones using some rule

• Adjusted matrix satisfies our requirements

Computational problems

• Even if true matrix is PD (or PSD), estimated matrix might not be

• Risk factors might be highly correlated

• This can produce 0 or –ve estimated eigenvalues

• These problems can be aggravated if covariance matrix is used for trading or risk management

• Possible answers:

  • Choose risk factors that are not too highly correlated
  • Don't choose too many risk factors
  • Alternatively, can adjust eigenvalues

Variance Swaps

Variance Swaps

• A variance swap is a forward contract on the realized variance, the payoff is

• it can be written on any asset (usually equities or equity indices)
• A correlation swap is similar to a variance swap, however its payoff is tied to the realized average correlation in a portfolio over the selected period

• e.g. A one yield contract on S&P 500 index: , . If , the payoff to the long position is

• Market value of an variance swap

• Correlation trading

Dynamic Trading

Dynamic Optoin Replication

Holding a call option is equivalent to holding a fraction of underlying asset

Dynamic replication of a put

Static Optoin Replication

Implications for Trading

• Dynamic replication of a long option is bound to loss money

  • it buys the asset after the price has gong up (too late)
  • the loss of each transaction will culmulate to an option premium, which is driven by the realized valotility

• Selling an option and dynamically hedging it using the underlying instrument

  • the strategy is delta-neutral
  • revenue from selling an option: a function of implied volatility
  • cost from dynamic hedging: a function of realized volatility
  • in equity markets, implied volatility tends to be greater than the realized volatility

• More general implications

  • large scale automatic trading system have the potential to be destabilizing
  • selling an asset after its price has gong down is similar to prudent risk-management practices

Mortgage-Backed Securities Risk

Prepayment Risk

Mortgage as Annuities

• Mortgages can be structured to have fixed or floating payments. Assuming a fixed monthly payment ,the price-yield relationship is

• Life of bond

  • maturity
  • average life (AL):
  • duration:

• When dealing with pool of mortgages, we use

  • weighted average maturity (WAM)
  • weighted average maturity coupon (WAC)
  • weighted average maturity life (WAL)

Prepayment Speed

• Factors which affect mortgage refinancing patterns / prepayment speed

  • spread between the mortgage rate and current rates
  • age of the loan
  • refinancing incentives
  • previous path of interest rates
  • level of mortgage rate
  • economic activity
  • seasonal effects

• Conditional prepayment rate (CPR) & single month mortality (SMM)

• Public Securities Association (PSA) Prepayment Model:

By converntion, prepayment patterns are expressed as a percentage of the PSA speed.

Project cash flows based on the prepayment speed pattern.

Measuring Prepayment Risk

• Prepayment risk

  • contraction risk: low rate more prepayment shorter average life
  • extension risk: high rate less prepayment longer average life

• Dealing with the changing cash-flow pattern

  • effective duration
  • effective convexity

• The option component

  • prepayment represents a long position in option for the borrower
  • prepayment represents a short position in option for the lender

• Option-Adjusted Spread (OAS)

  • static spread (SS): the difference between the yield of the MBS and that of a Treasury note with the same weighted average life
  • zero spread (ZS): a fixed spread added to zero-coupon rate so that the discounted value of the projected cash flows equals the current price
  • The OAS method involves running simulations of various scenarios and prepayments to establish the option cost

Securitization

Priciples of Securitization

• Procedures of securitization (off-balance-sheet)

  • create a special purpose vehicle (SPV), or special-purpose entity (SPE)
  • the originator pools a group of assets and sells them to the SPV
  • SPV issues tradable claims, or securities, that are backed by the financial assets

• Advantage of this structure

  • it shields the asset-backed security (ABS) investor from the credit risk of the originator
  • pooling offers ready-made diversification across many assets

• All sorts of assets (collateral) can be included in ABSs

  • mortgage loans, student loans, credit card receivables, account receivables, debt obligations, and etc.
  • RMBSs: backed by residential mortgage loan
  • CMBSs: backed by commercial mortgage loan
  • the cash flow of RMBSs and CMBSs is subject to interest rate risk, prepayment risk, and default risk

• Structure of securitization

  • pass-through: the SPV issues single class of security
  • tranching: the SPV issues different classes of securities

• On-balance-sheet securitizations (covered bonds or Pfandbriede in Germany)

  • banks originates the loans and issues securities secured by these loans, which are kept on its books
  • investors have recourse against the bank in the case of defaults on the motgage
  • banks provides a guarantee against credit risk

Issues with Securitization

• The moral hazard problem

• The adverse selection problem

• cerdit rating fails for securitization that with complex structures

• Securitization pushes housing prices away from their fundamental values

• When the securitization markets froze, many banks and loan originators were stuck with loans that were warehoused, or held in a pipeline that was supposed to be temporary

Rethinking Securitization

Tranching

Concept

• Cash flows are redistributed to fit investors' need

  • however, cash flows and risks are fully preserved
  • they are only redistributed across tranches
  • weighted duration & convexity of the portfolio of tranches must add up to the original duration and convexity

• This structure applies to collateralized mortgage obligations (CMOs), collateralized bond obligations (CBOs) collateralized loan obligations (CLOs), collateralized debt obligations (CDOs)

Inverse Floaters

• Try to verify that the outgoing cash flows exactly add up to the incoming cash flows
• Suppose the dollar duration of the original five-year note is 4.5 years, try to analyze the duration of the floater and the inverse floater just before a reset
• What if the coupon is tied to twice LIBOR, i.e. ?

CMOs

• Sequential-pay tranches: defined by prioritizing the payment of principal into different tranches

• Planned amortization class (PAC)

  • PAC bonds offer a fixed redemption schedule as long as prepayments on the collateral stays within a specified PSA range (say 100 to 250 PSA), called the PSA collar
  • the principal payment is set at the minimum payment of these two extreme values for every month of its life
  • all prepayment risk is transfered to other bonds in the CMO structure, called support bonds.

• IO/PO structure: strips the MBS into two components

  • the interest-only (IO) tranche receives only the interest payments on the underlying MBS
  • the principal-only (PO) tranche receives only the principal payments on the underlying MBS
  • interest rate fall principal payments come early PO appreciate in value while IO depreciate in value