ange Notes 2002-6-26 KAIST/
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
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$ > 08.50 or at 0% if the JPY/US$ 08.50. 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.
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)
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.)
Common Types of Structural Notes De-Leveraged and Leveraged Floaters De-leveraged floaters : 0.60 0-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 = 0.7 0-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% (6.7-0.5)*0.7.5 = 5.469% 5.255% 5.255% - 5.39%= -0.65% 0yr CMT % change 4.50% (6.7)*0.7.5 = 6.59% 6.066% 6.066% - 5.39%= 0.676%
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
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, 2002. 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.
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
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
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
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% %
ange Notes 6m LIBO range 3.50%.95% coupon 4.00%.00% time
ange Notes (eference ate) : CD : 3, 5, 5,, ange Notes., range,.
ange Notes criteria coupon BOND_ID BOND_NAME Maturity index low high low high Monitor K350907N5 89 3 2 3.75% 7.25% 4.00% 7.75% K3507027N53 3 2 0% 6.75% 2.50% 7.65% 2 K350807N53 42 3 2 0% 7.00% 2.50% 7.50% 2 K320037N5 774 3 2 0% 7.00% 2.50% 7.75% 2 K35807N52 92 3 2 0% 6.80% 2.00% 8.00% 2 K3204037N55 LG 94 3 2 0% 7.00% 2.50% 7.75% 2 K320047N59 776 3 2 0% 7.00% 2.50% 7.75% 2 total 7 * index_code: 2- CD * Monitor: : daily, 2: * criteria: low<= index_code range <= high
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
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 < 3.75 0 if CD 3.75 Digital Call (C) 3.75 if CD 7.25 0 if CD < 7.25 c c ange Note Payoff = Digital Call Digital Put
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
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 7.0 0 if CD < 7.0 ange Note Payoff = Digital Call ch c l
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
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
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%.5.5 2 Maturity (Years)
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%
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
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 (2002-5-23) Upper Bound
Properties of ange Notes H = 7.0%, L = 0%, c h = 7.75%, c h = 2.5% ITM : => OTM : =>
Properties of ange Notes ange digital call value => ange Notes
Properties of ange Notes Swap ate Curve
Properties of ange Notes : Swap ate => option value => ange Notes : Swap ate => option value => ange Notes
Properties of ange Notes Swap ate Curve ( parallel shift )
Properties of ange Notes
ange Note I H = 7.0%, L = 0%, c h = 7.75%, c h = 2.5% 0.9 : Cap/floor Volatility (2002/5/22)
ange Note I H = 7.0%, L = 0%, c h = 7.75%, c h = 2.5%
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
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
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
ange Note II European Digital Call Option ( T = 2, X = 0.02, = ) Index ate 2.0 % 2 2 Cash flow 2.24%.76% 2 2 2 2.45%.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) = = 0.02 0.7203
ange Note II European Digital Call Option ( T = 3, X = 0.02, = ) 2 2.75% ( 2.75% 2% ) D( T, uu) = > 2.0 % 2 2 2.24%.76% 2 2 2.45%.98%.60% 2 2 2.25%.84% ( 2.25% 2% ) D( T, uu) = > (.84% 2% ) D( T, du) = 0 < 2 2.5% (.5% 2% ) D( T, dd) = 0 < D( 0) = 0.4962
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
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.
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
isk Management Total Price = Digital Option * 0
isk Management Modified Duration = P / r P = SB / r DO P / r * 0
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
isk Management Total Price = Digital Option * 0
isk Management Modified Duration = P / r P = SB / r DO P / r * 0