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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. 報告流程. 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

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Mortgage-Backed Futures and Options

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  1. 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

  2. 報告流程 • 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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 (1/64th) 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) CBOT mortgage-backed futures options

  9. 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

  10. 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(1)

  11. 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 The effectiveness of various futures contracts in hedging GNMA mortgage-backed securities(2) *Is equal to the R-squared from the regression RS= ß0 +ß1Rf+ε, where RS is the five- or ten-day return on the GNMA MBSs (i.e. , Spot contract) and Rf is the corresponding return on the futures hedging instrument

  12. The Valuation of GNMA Futures and Futures Options • Underlying asset’s value • Interest rates

  13. 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

  14. 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

  15. 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

  16. 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

  17. 3. Scharfer and Schwarts(1987) Assumption: use the bond price as the single variable with a standard deviation of return proportional to duration Findings: (1) similar to those complicated calculation (2) lognormal process produces accurate option values

  18. 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.

  19. 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: jrise n-jfall ln(F*/F)= jlnU + (n-j)lnD = jln(U/D) + nlnD

  20. 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:

  21. 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)

  22. t<T • Intrinsic value (payoff from immediate exercise) • Value of the decision to postpone exercise Time value = W – I

  23. 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

  24. Model Versus Actual Futures-Options 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

  25. Model Versus Actual Futures-Options Prices ??? → 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

  26. Model Versus Actual Futures-Options Prices 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 GNMA MBSs

  27. 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 ^

  28. 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 + βp0 + ε } to {p0= α + βp0 + ε } ^ ^ ^

  29. 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 ^

  30. 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) ^

  31. 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 ^

  32. Regression of model put options actual prices • Regressing the model option price on the actual option price ^

  33. 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 ^

  34. 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) ^

  35. Summary and Conclusion • Empirically tests a simple valuation model • Compared to observed transaction prices • The ability of the CBOT MBF to hedge positions in current GNMA MBSs • Despite 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

  36. 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

  37. ~ The end ~ 休息一下 不要走開 馬上回來

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