Derivatives Swaps
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DerivativesSwaps
Professor André FarberSolvay Business SchoolUniversité Libre de Bruxelles
Derivatives 05 Swaps |2April 22, 2023
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 |3April 22, 2023
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 |4April 22, 2023
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 + t, t + 2 t, t + 3t , t+ 4 t, ...
• Most common swap : 6-month LIBOR
Derivatives 05 Swaps |5April 22, 2023
Interest Rate Swap: Example
Objective Borrowing conditions
Fix VarA Fix 5% Libor + 1%B Var 4% Libor+ 0.5%
Swap:
• Gains for each company• A BOutflow Libor+1% 4% 3.80% LiborInflow Libor 3.70%Total 4.80% Libor+0.3% Saving 0.20% 0.20%
A free lunch ?
A Bank BLibor Libor
4%Libor+1%3.80% 3.70%
Derivatives 05 Swaps |6April 22, 2023
Payoffs
• Periodic payments (i=1, 2, ..,n) at time t+t, t+2t, ..t+it, ..,T= t+nt • Time of payment i: ti = t + i t
• Long position: Pays fix, receives floating
• Cash flow i (at time ti): Difference between • a floating rate (set at time ti-1=t+ (i-1) t) and • a fixed rate R • adjusted for the length of the period (t) and • multiplied by notional amount M• CFi = M (ri-1 - R) t
• where ri-1 is the floating rate at time ti-1
Derivatives 05 Swaps |7April 22, 2023
IRS Decompositions
• IRS:Cash Flows (Notional amount = 1, = t )TIME 0 2 ... (n-1) n Inflow r0 r1 ... rn-2 rn-1
Outflow R R ... R R
• Decomposition 1: 2 bonds, Long Floating Rate, Short Fixed RateTIME 0 2 … (n-1) n Inflow r0 r1 ... rn-2 1+rn-1
Outflow R R ... R 1+R
• Decomposition 2: n FRAs• TIME 0 2 … (n-1) n • Cash flow (r0 - R) (r1 -R) … (rn-2 -R) (rn-1- R)
Derivatives 05 Swaps |8April 22, 2023
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 ( Vswap ) equals:
– Value of FR note Vfloat - Value of fixed rate bond Vfix
Vswap = Vfloat - Vfix
• Fix rate R set so that Vswap = 0
Derivatives 05 Swaps |9April 22, 2023
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 |10April 22, 2023
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: Vfloat, n = 100
• At time n-1: Vfloat,n-1 = 100 (1+rn-1)/ (1+rn-1) = 100
• At time n-2: Vfloat,n-2 = (Vfloat,n-1+ 100rn-2)/ (1+rn-2) = 100
• and so on and on….
Vfloat
Time
100
Derivatives 05 Swaps |11April 22, 2023
IR Swap = Long floating rate note + Short fixed rate note
Value of swap = fswap = Vfloat - Vfix
1
( )n
Swap i nt
f M M R t DF DF
Fixed rate R set initially to achieve fswap = 0
Derivatives 05 Swaps |12April 22, 2023
(ii) IR Swap = Portfolio of n FRAs
Value of FRA fFRA,i = M DFi-1 - M (1+ R t) DFi
, 11 1 1
(1 )n n n
swap FRA i i i i ni i i
f f M DF M R t DF M M R t DF DF
, 11 1
(1 )n n
swap FRA i i ii i
f f M DF M R t DF
Derivatives 05 Swaps |13April 22, 2023
FRA Review
i -1 iΔt
1
1
( )(1 )
i
i
r R tMr t
1
1
(1 ) (1 )(1 )
i
i
r t R tMr t
M (1 )M R t
Value of FRA fFRA,i = M DFi-1 - M (1+ R t) DFi
Derivatives 05 Swaps |14April 22, 2023
(iii) Swap valuation based on forward rates
1,
ˆ(1 ) ( )iFRA i i i i
i
DFf M R t DF M R R t DFDF
Rewrite the value of a FRA as:
1
ˆ( )n
swap i it
f M R R t DF
Derivatives 05 Swaps |15April 22, 2023
(iv) Swap valuation based on current swap rate
1
( )n
swap swap ii
f M R R t DF
1
n
swap i ni
M R t DF M M DF
As:
1
( )n
Swap i n float fixt
f M M R t DF DF V V
Derivatives 05 Swaps |16April 22, 2023
Swap Rate Calculation
• Value of swap: fswap =Vfloat - Vfix = M - M [R di + dn]
where dt = discount factor
• Set R so that fswap = 0 R = (1-dn)/(di)• Example 3-year swap - Notional principal = 100
Spot rates (continuous)Maturity 1 2 3Spot rate 4.00% 4.50% 5.00%Discount factor 0.961 0.914 0.861
R = (1- 0.861)/(0.961 + 0.914 + 0.861) = 5.09%
Derivatives 05 Swaps |17April 22, 2023
Swap: portfolio of FRAs
• Consider cash flow i : M (ri-1 - R) t– Same as for FRA with settlement date at i-1
• Value of cash flow i = M di-1- M(1+ Rt) di
• Reminder: Vfra = 0 if Rfra = forward rate Fi-1,I
• Vfra t-1
• > 0 If swap rate R > fwd rate Ft-1,t
• = 0 If swap rate R = fwd rate Ft-1,t
• <0 If swap rate R < fwd rate Ft-1,t
• => SWAP VALUE = Vfra t
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