Derivatives Swaps. Professor André Farber Solvay Business School Université Libre de Bruxelles

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1 Derivatives Swaps Professor André Farber Solvay Business School Université Libre de Bruxelles

2 Interest Rate Derivatives Forward rate agreement (FRA): OTC contract that allows the user to "lock in" the current forward rate. Treasury Bill futures: a futures contract on 90 days Treasury Bills Interest Rate Futures (IRF): exchange traded futures contract for which the underlying interest rate (Dollar LIBOR, Euribor,..) has a maturity of 3 months Government bonds futures: exchange traded futures contracts for which the underlying instrument is a government bond. Interest Rate swaps: OTC contract used to convert exposure from fixed to floating or vice versa. Derivatives 05 Swaps 2

3 Swaps: Introduction Contract whereby parties are committed: To exchange cash flows At future dates Two most common contracts: Interest rate swaps Currency swaps Derivatives 05 Swaps 3

4 Plain vanilla interest rate swap Contract by which Buyer (long) committed to pay fixed rate R Seller (short) committed to pay variable r (Ex:LIBOR) on notional amount M No exchange of principal at future dates set in advance t + Δt, t + 2 Δt, t + 3Δt, t+ 4 Δt,... Most common swap : 6-month LIBOR Derivatives 05 Swaps 4

5 Interest Rate Swap: Example Objective Borrowing conditions Fix Var A Fix 5% Libor + 1% B Var 4% Libor+ 0.5% Swap: Libor+1% 3.80% 3.70% A Bank B 4% Gains for each company A B Outflow Libor+1% 4% 3.80% Libor Inflow Libor 3.70% Total 4.80% Libor+0.3% Saving 0.20% 0.20% A free lunch? Libor Libor Derivatives 05 Swaps 5

6 Payoffs Periodic payments (i=1, 2,..,n) at time t+δt, t+2δt,..t+iδt,..,t= t+nδt Time of payment i: t i = t + i Δt Long position: Pays fix, receives floating Cash flow i (at time ti): Difference between a floating rate (set at time t i-1 =t+ (i-1) Δt) and a fixed rate R adjusted for the length of the period (Δt) and multiplied by notional amount M CF i = M (r i-1 - R) Δt where r i-1 is the floating rate at time t i-1 Derivatives 05 Swaps 6

7 IRS Decompositions IRS:Cash Flows (Notional amount = 1, τ= Δt ) TIME 0 τ 2τ... (n-1)τ n τ Inflow r 0 τ r 1 τ... r n-2 τ r n-1 τ Outflow R τ R τ... R τ R τ Decomposition 1: 2 bonds, Long Floating Rate, Short Fixed Rate TIME 0 τ 2τ (n-1)τ n τ Inflow r 0 τ r 1 τ... r n-2 τ 1+r n-1 τ Outflow R τ R τ... R τ 1+R τ Decomposition 2: n FRAs TIME 0 τ 2τ (n-1)τ n τ Cash flow (r 0 - R)τ (r 1 -R)τ (r n-2 -R)τ (r n-1 - R) Derivatives 05 Swaps 7

8 Valuation of an IR swap Since a long position position of a swap is equivalent to: a long position on a floating rate note a short position on a fix rate note Value of swap ( V swap ) equals: Value of FR note V float - Value of fixed rate bond V fix V swap = V float - V fix Fix rate R set so that Vswap = 0 Derivatives 05 Swaps 8

9 Valuation (i) IR Swap = Long floating rate note + Short fixed rate note (ii) IR Swap = Portfolio of n FRAs (iii) Swap valuation based on forward rates (for given swap rate R): (iv) Swap valuation based on current swap rate for same maturity Derivatives 05 Swaps 9

10 Valuation of a floating rate note The value of a floating rate note is equal to its face value at each payment date (ex interest). Assume face value = 100 At time n: V float, n = 100 At time n-1: V float,n-1 = 100 (1+r n-1 τ)/ (1+r n-1 τ) = 100 At time n-2: V float,n-2 = (V float,n r n-2 τ)/ (1+r n-2 τ) = 100 and so on and on. V float 100 Time Derivatives 05 Swaps 10

11 IR Swap = Long floating rate note + Short fixed rate note n f = M M ( R Δ t) DF + DF Swap i n t= 1 Value of swap = f swap = V float - V fix Fixed rate R set initially to achieve f swap = 0 Derivatives 05 Swaps 11

12 (ii) IR Swap = Portfolio of n FRAs n n [ (1 ) ] = = + Δ f f M DF M R t DF swap FRA, i i 1 i i= 1 i= 1 Value of FRA f FRA,i = M DF i-1 - M (1+ R Δt) DF i n n n f = f, = [ M DF 1 M(1 + RΔ t) DF] = M M RΔ t DF + DF swap FRA i i i i n i= 1 i= 1 i= 1 Derivatives 05 Swaps 12

13 FRA Review i -1 Δt i M ( r R) Δt i 1 (1 + r Δt) i 1 M (1 + ri 1Δt) (1 + RΔt) (1 + r Δt) i 1 M M(1 + RΔt) Value of FRA f FRA,i = M DF i-1 - M (1+ R Δt) DF i Derivatives 05 Swaps 13

14 (iii) Swap valuation based on forward rates n f = M ( Rˆ R) Δt DF swap i i t= 1 Rewrite the value of a FRA as: f DF = M (1 + RΔ t) DF = M ( Rˆ R) Δt DF i 1 FRA, i i i i DF i Derivatives 05 Swaps 14

15 (iv) Swap valuation based on current swap rate n f = M ( R R) Δt DF swap swap i i= 1 As: n M R Δ t DF = M M DF i= 1 swap i n n f = M M ( R Δ t) DF + DF = V V Swap i n float fix t= 1 Derivatives 05 Swaps 15

16 Swap Rate Calculation Value of swap: f swap =V float - V fix = M - M [R Σ d i + d n ] where d t = discount factor Set R so that f swap = 0 R = (1-d n )/(Σ d i ) Example 3-year swap - Notional principal = 100 Spot rates (continuous) Maturity Spot rate 4.00% 4.50% 5.00% Discount factor R = ( )/( ) = 5.09% Derivatives 05 Swaps 16

17 Swap: portfolio of FRAs Consider cash flow i : M (r i-1 - R) Δt Same as for FRA with settlement date at i-1 Value of cash flow i = M d i-1 - M(1+ RΔt) d i Reminder: V fra = 0 if R fra = forward rate F i-1,i V fra t-1 > 0 If swap rate R > fwd rate F t-1,t = 0 If swap rate R = fwd rate F t-1,t <0 If swap rate R < fwd rate F t-1,t => SWAP VALUE = Σ V fra t Derivatives 05 Swaps 17

Derivatives Options on Bonds and Interest Rates. Professor André Farber Solvay Business School Université Libre de Bruxelles

Derivatives Options on Bonds and Interest Rates Professor André Farber Solvay Business School Université Libre de Bruxelles Caps Floors Swaption Options on IR futures Options on Government bond futures