Range Notes KAIST/

Transcription

1 ange Notes KAIST/

2 What are structural notes? fixed coupon floating coupon. Straight Debt Interest ate Derivatives (Embeddos) ( ) The Federal Home Loan Bank (FHLB) Federal National Mortgage Association (FNMA) Federal Home Loan Mortgage Corporation (FHLMC) => Although the credit risk of these securities is minimal, other risks such as interest rate risk, market (price) risk, and liquidity risk can be material

3 Why have structural notes become popular?. They offer the potential for greater returns than prevailing market rates.. They can serve to hedge risks faced by the investor. A company which is long (owns) Japanese yen is exposed to the risk of yen depreciation. The FHLB issued a one-year structured range note which accrued interest daily at 7% if the JPY/US$ > or at 0% if the JPY/US$ If the yen depreciates, the note accrues interest at an above-market rate. Meanwhile, the company s yen holdings will decline in value.. To take a market view. In the above example an investor who believed that the dollar would appreciate against the yen would be compensated handsomely for being correct. However, if the investor s view proved incorrect, he would be locked into an investment which paid no interest for a period of up to one year.

4 Common Types of Structural Notes Inverse Floating-ate Notes Dual-Index Notes Index-Amortizing Notes (IAN) De-Leveraged and Leveraged Floaters ange Notes (ange Floaters)

5 Common Types of Structural Notes Inverse Floating-ate Notes coupon (reference rate) Ex : 3yr, semi-annual coupon = 3% - LIBO, floor 0% Dual-Index Notes Coupon2 market index Market Index : the prime rate, LIBO, CMT yields Ex : prime rate 250 basis points 6m LIBO Index-Amortizing Notes (IAN) An IAN is a form of structured note for which the outstanding principal or note amortizes according to a predetermined schedule. The predetermined amortization schedule is linked to the level of a designated index (such as LIBO, CMT, etc.)

6 Common Types of Structural Notes De-Leveraged and Leveraged Floaters De-leveraged floaters : year CMT 00 basis points Leveraged floaters :.25 LIBO 00 basis points Leverage interest rate risk. Ex : yr cpn 4.5%, 5yr CMT 5.39%, 0yr CMT 6.7% 2-5yr cpn = year CMT 50 basis points Performance yr cpn 2-5yr cpn YTM (Par Yield) Performance ( YTM 5yr CMT) 0yr CMT 0.5% change 4.50% ( )*0.7.5 = 5.469% 5.255% 5.255% %= -0.65% 0yr CMT % change 4.50% (6.7)*0.7.5 = 6.59% 6.066% 6.066% %= 0.676%

7 ange Notes CouponMarket Index ange Notes2 coupon level. Higher coupon rate : the index remains within a designated range Lower coupon rate : the index falls outside the range => Digital Options (Binary Options) Coupon = c c h l : if : if H L L < or < H Market Index : interest rates, currencies, commodities, and equities

8 Digital Options (Binary Options) Option that pays off a fixed amount if the value of the underlying is beyond a specified level (strike) at a point in time (the expiry date). Cash or nothing digital option Asset or nothing digital option. Digital options are sometimes called binary options. Cash-or-nothing call option A digital that pays $00,000 if the S&P 500 Index exceeds,300 on December 3, It pays nothing otherwise. Asset-or-nothing call option A digital that pays a share of IBM if the price of the IBM share exceeds $00.

9 Digital Options (Binary Options) Payoff diagram Forward Standard Call Digital Put F F if S T < K2 0 if S T K2 K 2 f K Digital Call F if S T K 0 if S T < K Underlying price

10 Digital Options (Binary Options) Cash-or-nothing call (put) option F if S T K ( if ST < K) 0 if S < T K ( if ST K) rt call premium = F e N ( d2 ) where ln( S0 / K ) ( r d2 = σ T N ( ) : cumulative normal 2 σ 2 ) T, F distribution Asset-or-nothing call (put) option S if S K ( S K) T 0 T if S T < K if T < ( if ST K) : fixed function payoff

11 ange Notes Ex : 2yr maturity, semi-annual coupon If (.95% 6m LIBO 3.5%) coupon = 4% If (.95% < 6m LIBO or > 3.5%) coupon = % coupon 4% %.95% 3.5% 6m LIBO

12 ange Notes Ex : 2yr maturity, semi-annual coupon If (.95% 6m LIBO 3.5%) coupon = 4% If (.95% < 6m LIBO or > 3.5%) coupon = % Cash Flow 6m LIBO Actual Coupon Best Case Worst Case st 2.80% 4.0% 4.0%.0% 2st 3.00% 4.0% 4.0%.0% 3st 3.60%.0% 4.0%.0% 4st 3.70%.0% 4.0%.0% Par yield ange Notes 2.52% 4% %

13 ange Notes 6m LIBO range 3.50%.95% coupon 4.00%.00% time

14 ange Notes (eference ate) : CD : 3, 5, 5,, ange Notes., range,.

15 ange Notes criteria coupon BOND_ID BOND_NAME Maturity index low high low high Monitor K350907N % 7.25% 4.00% 7.75% K N % 6.75% 2.50% 7.65% 2 K350807N % 7.00% 2.50% 7.50% 2 K320037N % 7.00% 2.50% 7.75% 2 K35807N % 6.80% 2.00% 8.00% 2 K N55 LG % 7.00% 2.50% 7.75% 2 K320047N % 7.00% 2.50% 7.75% 2 total 7 * index_code: 2- CD * Monitor: : daily, 2: * criteria: low<= index_code range <= high

16 ange Notes Ex : K350907N5 89, 2002/5/3, 3 If (3.75% CD 7.25%) coupon = 7.75% If (CD < 3.75% or CD > 7%) coupon = 4.0% cpn 7.75% 4.0% 3.75% 7.25% CD

17 ange Notes H = 7.25%, L = 3.75%, c h = 7.75%, c h = 4.0% cpn 7.75% cpn 7.75% ABC 4.0% 4.0% 3.75% 7.25% CD 3.75% 7.25% CD -3.75% Digital Put (B) 3.75 if CD < if CD 3.75 Digital Call (C) 3.75 if CD if CD < 7.25 c c ange Note Payoff = Digital Call Digital Put

18 ange Notes Ex : K320037N5 774, 2002/5/23, 3 If (0% CD 7%) coupon = 7.75% If (CD > 7%) coupon = 2.5% cpn 7.75% 2.5% 7.0% CD

19 ange Notes H = 7.0%, L = 0%, c h = 7.75%, c h = 2.5% cpn 7.75% cpn 7.75% AB 2.5% 7.0% 2.5% CD 7.0% CD -5.25% Digital Call (B) 5.25 if CD if CD < 7.0 ange Note Payoff = Digital Call ch c l

20 ange Notes ange Notes Payoff ange Notes (A) digital option(b) ange Notes = Straight Bond Value n PV { Digital _ Call _ Option i } i = n PV { Digital _ Put _ Option i } i = n= total number of coupon payment

21 ange Notes I Digital Call Option = L ( ch cl ) P(0, τ ) N( d 2, ) Digital Put Option = L ( ch cl ) P(0, τ ) N ( d 2, 2) ln( f / H ) σ 2 fτ ln( f / L ) σ d 2 2, = d 2 2,2 = σ τ σ τ L : face f (0, τ ) : value P(0, τ ) : zero, f forward coupon : coupon rate at time bond period 0 for price at the time future 0 f 2 time [ τ, τ 0 L τ τ L f τ ] f ( 0, τ) cash flow = L ( c 0 h c l ) if if f (0, τ ) > f (0, τ ) H H

22 ange Notes I Forward ate from Swap Curve 5.60% 5.47% Swap ate (%) 5.40% 5.20% 5.30% 5.00% 5.00% 5.% 4.80% Maturity (Years)

23 ange Notes I Forward ate from Swap Curve Swap Curve (Par bond yield curve) r Y = 5.0% r Y2 = 5.% Zero ate (discount function) r ZC2 = 5.3% r ZC3 = 5.32% Implied Forward ate f,2 = 5.226% f 2,3 = 5.7%

24 Properties of ange Notes ange Notes ange Notes =. ange Criteria 2. Volatility of the reference rate 3. Shape of the forward yield curve as implied by the current interest rate term structure

25 Properties of ange Notes Ex: H = 7.0%, L = 0%, c h = 7.5%, c h = 2.5% (K350807N53) Swap ate, Zero ate, Forward ate ( ) Upper Bound

26 Properties of ange Notes H = 7.0%, L = 0%, c h = 7.75%, c h = 2.5% ITM : => OTM : =>

27 Properties of ange Notes ange digital call value => ange Notes

28 Properties of ange Notes Swap ate Curve

29 Properties of ange Notes : Swap ate => option value => ange Notes : Swap ate => option value => ange Notes

30 Properties of ange Notes Swap ate Curve ( parallel shift )

31 Properties of ange Notes

32 ange Note I H = 7.0%, L = 0%, c h = 7.75%, c h = 2.5% 0.9 : Cap/floor Volatility (2002/5/22)

33 ange Note I H = 7.0%, L = 0%, c h = 7.75%, c h = 2.5%

34 ange Note I Implied Volatility volatility digital call value Implied Volatility > cap/floor volatility => digital option => ange Notes Implied Upper Bound ange digital call value Implied upper bound > actual bound => digital option => ange Notes

35 ange Note II term structure tree node index rate digital option value Interest ate Model Equilibrium model Vasicek : CI : dr dr = ( b ar ) dt = ( b ar ) dt σ σ dz r dz No-arbitrage model BDT : Extended Vasicek : Extended CI : dr = a ( t ) rdt b ( t ) rdz dr = ( Φ ( t ) a ( t ) r ) dt dr = ( Φ ( t ) a ( t ) r ) dt σ σ dz r dz

36 ange Note II Digital Call Option on the Interest ates Payoff at time T isk-neutral Valuation < = X T T if X T T if T D ), ( 0 ), ( ) ( ) )(, ( ), ( t T T t T t P where = [ ] ) ( ) (0, (0 ) ^ T D E T P D = 0 ):, ( time at couponprice zero T t P where < = X T T P if X T T P if D T ), ( 0 ), ( ) ( Digital Put on the Zero Bond

37 ange Note II European Digital Call Option ( T = 2, X = 0.02, = ) Index ate 2.0 % 2 2 Cash flow 2.24%.76% %.98%.60% ( 2.45% 2% ) D( T, uu) = > (.98% 2% ) D( T, ud ) = < (.60% 2% ) D( T, dd) = 0 < 0.5D(, u) 0.5D(, d ) D( 0) = =

38 ange Note II European Digital Call Option ( T = 3, X = 0.02, = ) % ( 2.75% 2% ) D( T, uu) = > 2.0 % %.76% %.98%.60% %.84% ( 2.25% 2% ) D( T, uu) = > (.84% 2% ) D( T, du) = 0 < 2 2.5% (.5% 2% ) D( T, dd) = 0 < D( 0) =

39 ange Note II ange Notes Payoff at time t Cash Flows from time 0 until maturity isk-neutral Valuation > < = u L L u l H t t t t if c L t t if c L t D ), (, ), ( ), ( ) ( { } { } [ ] = > < 0 ), (, ), ( ), ( T j j j j j L j j H u L u l c L c L { } { } ( = = > < 0 ), (, ), ( ), ( ^ ) (0, (0) T j j j j j L j j H u L u l c L c E L j P N = o if A istru if A / 0

40 isk Management Interest rate risk of range notes digital option. ange Notes P = SB DO = Straight Bond Value Digital Option Value Duration : % Modified Duration = P / r P = SB / r DO / r P Digital Option payoff option.

41 isk Management Ex : K320037N5 774, 2002/5/23, 3 If (0% CD 7%) coupon = 7.75% If (CD > 7%) coupon = 2.5% cpn 7.75% 2.5% 7.0% CD

42 isk Management Total Price = Digital Option * 0

43 isk Management Modified Duration = P / r P = SB / r DO P / r * 0

44 isk Management Ex : K350907N5 89, 2002/5/3, 3 If (3.75% CD 7.25%) coupon = 7.75% If (CD < 3.75% or CD > 7%) coupon = 4.0% cpn 7.75% 4.0% 3.75% 7.25% CD

45 isk Management Total Price = Digital Option * 0

46 isk Management Modified Duration = P / r P = SB / r DO P / r * 0