Mortgage-Backed Futures and Options

Mortgage-Backed Futures and Options

組員 :

財研一 91357018 張容容財研一 91357023 王韻晴財研一 91357024 王敬智

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介紹 CDR ， Mortgage-backed futures ，futures-options

測試 MBF ,T-note ,T-bond hedge GNMA securities 之效果介紹 futures-options model

測試 simple contingent-claim model 所估計 option on MBF 之價格準確程度

GNMA Collateralized Depository Receipt(CDR)

在 1975,CBOT 引進 GNMA CDR futures 為首宗以 MBS 為合約標的的利率期貨合約，1970 年代， MBS 交易量大， CDR 提供其較佳的避險效果，故廣被 MBS dealers 及 mortgage originators 所使用，成交量達 2.3 million

但在 1980 年後， CDR 成交量遽減至 10,000 contracts

1987 年， CDR 下市

CDR 交易量遽減原因

次級市場與長期利率期貨市場的快速興起Treasury-bond futures 提供了更佳的避險效果期貨的賣方可依換算比例，調整償付債券的面額與利率，致使高利率的債券價值被高估低利率的債券較有避險的需求CDR futures contracts 之價格波動小

MBF & Options Contracts

在 1985 年 6 月引進 MBF 與 futures-options

Originator of fixed rate mortgages 可用 MBF 與 futures-options 來提前售出 GNMA securities 或 hedge 在外 mortgage 之價格波動在 1992 年 3 月，由於交易量不夠大，MBF 及 options 下市

MBF & Options Contracts 之特色

以一特定的 GNMA 債券為計價基礎以 15 家 dealer 之詢問價的中位數為結清價以現金結算，避免交割之債券有不同利率或到期期間等“質“的問題Options 與 futures contracts 之到期日相同

CBOT mortgage-backed futures

Trading Unit $100,000 par value

Coupons Traded Each month, the CBOT will list a new coupon four months in the future. The coupon for that month will be the newest GNMA coupon; trading nearest to par (100) but not greater than par.

Price Quotations In points and thirty-seconds of a point, e.g., 98-12 equals 98 and 12/32nds.

Trading Months Four consecutive months.

Daily Trading Limits 3 points (or $3,000 per contract) above or below the previous day’s settlement price (expandable to 4 1/2 points)

Last Trading Day At 1:00 p.m. on the Friday preceding the third Wednesday of the month

Settlement In cash on the last trading day based on the mortgage-backed Survey Price. The Survey Price shall be the median price obtained from a survey of dealers.

Trading Hours 7:20 a.m. to 2:00 p.m. (Chicago time).

CBOT mortgage-backed futures options

Underlying Instrument One CBOT MBF contract of a specified delivery month and coupon

Strike Prices Strike prices are set at multiples of one point ($1,000)

Price quotations Premiums are quoted in minimum increments of one sixty-fourth (1/64 th) of 1% of a $100,000 MBF contract, or $15.625 rounded up to the nearest penny.

Tick Size 1/64 of a point ($15.625 or $15.63 per contract)

Daily Price Limits Three points ($3,000)

Months Traded Four consecutive months

Last Trading Day Options cease trading at 1:00 p.m. Chicago time on the last day of trading in MBF in the corresponding delivery month

Expiration Unexercised options expire at 8:00 p.m. Chicago time on the last day of trading. In-the-money options are exercised automatically

Trading Hours 7:20 a.m. to 2;00 p.m. (Chicago time)

Hedging Effectiveness

避險效果可用被避險資產 (GNMA securities) 與期貨間相關係數的平方來衡量R s=0+1Rf+ 迴歸式中之斜率值為事後之最佳避險比例MBF 之避險效果優於 T-Note futures 及T-Bond futures

The effectiveness of various futures contracts in hedging GNMA mortgage-backed securities(1)

Five Day Hedges-Number of Observations=115

Hedging Instrument

Coefficient of Determination*

ß1

GNMA Futures (MBF)

0.952 0.990

T-Note Futures 0.830 1.108

T-Bond Futures 0.900 0.761

The effectiveness of various futures contracts in hedging GNMA mortgage-backed securities(2)

10 Day Hedges-Number of Observations=55

Hedging Instrument

Coefficient of Determination*

ß1

GNMA Futures (MBF)

0.965 0.972

T-Note Futures 0.828 1.028

T-Bond Futures 0.894 0.736

*Is equal to the R-squared from the regression RS= ß0 +ß1Rf+ε, where RS is the five- or te

n-day return on the GNMA MBSs (i.e. , Spot contract) and Rf is the corresponding return

on the futures hedging instrument

The Valuation of GNMA Futures and Futures Options

Underlying asset’s valueInterest rates

Bond Price Dynamics

1. Black and Scholes(1973) option pricing model

2. Equilibrium models of the term structure

3. Scharfer and Schwarts(1987) simple model

1. Black and Scholes(1973)

Assumption:

σ(variance of the rate of return on the underlying asset) is constant

r (short-term interest rate) is constant

Limitation:

σ is not constant for the bond or GNMA

interest rate uncertainty

2. Equilibrium Models of the Term Structure Cox, Ingersoll and Rose(1985)

Brennan and Schwarts(1982)

Courtadon(1982)

Vasicek(1977)

Assumption:

one or two interest rates follow exogenously determined stochastic processes

Limitation:

(1) require the estimation of the stochastic process for one or two interest rates

(2) require the estimation of utility-dependent parameters

(3) underlying bond or GNMA is not a state variable

(4) initial bond price = current market price

3. Scharfer and Schwarts(1987) Assumption: use the bond price as the single variable with a sta

ndard deviation of return proportional to duration Findings: (1) similar to those complicated calculation (2) lognormal process produces accurate option va

lues

Assumptions and Notation

(A1) Investors prefer more wealth and act as price takers. The MBF and futures-options markets are frictionless.

(A2) No taxes, all margin requirements can be met by posting interest-bearing securities.

(A3) r, the short-term interest rate on default-free securities, is a constant

(A4) Time is divided into a sequence of equal length discrete periods. All cash flows occur at these discrete points.

(A5) MBF price is a random variable following a lognormal process with a constant variance.

Generation of the MBF Price Lattice

Define MBF prices U: upward ratio probability: p D: downward ratio probability: 1-p N: no. of periods per year F: MBF price assume: jrise n-jfall ln(F*/F)= jlnU + (n-j)lnD = jln(U/D) + nlnD

E[ln(F*/F)] = E[j]ln(U/D) + nlnD

Var[ln(F*/F)] = Var[j][ln(U/D)]2

By binomial dist.: E[j] = np Var[j] = npq

We can get

μ=N[pln(U)+(1-p)ln(D)]

σ=N[(ln(U)-ln(D))2p(1-p)]

from which it follows that:]/)1()/(/exp[ ppNNNU

])1/()/(/exp[ pNpNND

Futures Option Valuation

Call options

expire at time T

j = node no. in the futures tree at T (from 1 to T+1)

FT,j = price of MBF

K = strike price

Θ = vector of parameters (μ,σand p)

]0,[),,;,( ,, KFMaxKrFC jTjTT

t<T

Intrinsic value (payoff from immediate exercise)

Value of the decision to postpone exercise

Time value = W – I

KFI jtjt ,,

],[),,;,( ,,, jtjtjtt WIMaxKrFC

)1/(])1([ 1,1,1. rCppCW jtjtjt

Model Versus Actual Futures-Options Prices

Underlying: GNMA 9.5% futures contract with expiration date prior to December of 1990

Inputs:– Closing price:9.5% GNMA MBF

– γ: short –term riskless rate of interest

– Υ: the period of time until expiration of

the future and option contracts

– μ: expected drift in MBF prices

– σ2 : variance of MBF prices

Assumptions:– γ: contemporaneous yield on 3-month

treasury bills– Υ: the number of days between observation

date and expiration date of the month– μ: equal to zero– σ2 : expected volatility of the underlying asset



???

→ Use a daily series of implied T-note volatilities → By the relationship between prices on T-note futures contracts and the prices of options written on these T-note futures

σ2 can’t be directly observed

Another ??? is

→ Use 70-day moving average of historical price volatilities on 9.5% GNMA MBSs and on T-note futures contracts→ Ratio × T-note volatility expected GNMA MBS and MBF price volatility Ratio is 0.684 ~ 0.912 , mean = 0.782

T-note future known to overstate the price of G

NMA MBSs


Tests of the Call Option Model

Summary statistics: call option contracts on MBF

Amounts are per $100 of current MBF price Time period : 1989.6 ~1990.11 Observations : 216

Mean Std. Dev. Min. Max.

Actual Option Price (Po) 1.03 0.80 0.02 5.03

Model Option Price ( Po) 1.01 0.79 0.00 4.97

Difference 0.014 0.087 -0.28 0.36

MBF Price-Strike Price 0.17 1.34 -4.59 5.03

Annual Volatility of MBF 0.053 0.01 0.039 0.075

^

Regression of model call options actual prices

Regressing the model option price on the actual option price

Reverse the dependent and independent variables

from { p0 = α + β p0 + ε } to { p0

= α + β p0 + ε }

Panel A : p0 = α + β p0 + ε

R2 Variable Coef. Std. Error T-Stat.

0.988 α 0.002 0.009 -0.2

β 0.987 0.007 133.2

^

^ ^

Regression ofmodel call options actual prices

Regressing the time value of the option, as

calculated by the model , on the actual time value of the option

Panel B : ( p0 – IV) = α + β( p0 – IV) + ε


0.91 α 0.004 0.010 0.4

β 0.956 0.020 47.4

^

Regression ofmodel call options actual prices

Regressing the difference between the actual and model option price

on IV ; the number of days remaining until expiration of option (γ) ;

and the annual standard deviation of MBF prices (σ)

Panel C : ( p0 – p0) = α + β1 IV +β2 γ + β3σ + ε


0.23 α 0.299 0.037 7.9

β1 0.004 0.003 1.1

β2 -0.000 0.000 -1.1

β3 -5.112 0.637 -8.0

^

Tests of the Put Option Model

Summary statistics: put option contracts on MBF

Amounts are per $100 of current MBF price Time period : 1989.6 ~1990.11 Observations : 332

Mean Std. Dev. Min. Max.

Actual Option Price (po) 0.68 0.58 0.02 4.06

Model Option Price ( po) 0.65 0.59 0.00 4.01

Difference 0.028 0.088 -0.31 0.27

MBF Price-Strike Price -0.54 1.22 -3.72 4.06

Annual Volatility of MBF 0.06 0.01 0.04 0.09

^

Regression of model put options actual prices

Regressing the model option price on

the actual option price

Panel A : p0 = α + β p0 + ε


0.98 α -0.035 0.007 -4.7

β 1.009 0.008 120.0

^

Regression ofmodel put options actual prices

Regressing the time value of the option, as

calculated by the model , on the actual time value of the option

Panel B : ( p0 – IV) = α + β( p0 – IV) + ε


0.91 α 0.014 0.008 -1.6

β 0.966 0.017 54.6

^

Regression of model put options actual prices

Regressing the difference between the actual and model option price

on IV ; the number of days remaining until expiration of option (γ) ;

and the annual standard deviation of MBF prices (σ)

Panel C : ( p0 – p0 ) = α + β1 IV +β2 γ + β3σ + ε


0.30 α 0.299 0.029 10.3

β1 -0.014 0.003 -4.2

β2 0.000 0.008 0.6

β3 -5.064 0.937 -10.7

^

Summary and Conclusion

Empirically tests a simple valuation modelCompared to observed transaction pricesThe ability of the CBOT MBF to hedge positions

in current GNMA MBSsDespite limitation/assumption, the model provide

an unbiased estimate of changes in the actual option price

The accuracy of the model may be further improved by in-sample calculations to infer expected GNMA MBS price volatility

Summary and Conclusion

The use of more complicated valuation models can only be rationalized if they provide more accurate estimates of actual prices than the relatively simple model tested in this paper

~ The end ~

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Mortgage-Backed Futures and Options