Bootstrapping the interest-rate term structure Marco Marchioro www.marchioro.org October 20th, 2012 Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 1 Summary (1/2) • Market quotes of deposit rates, IR futures, and swaps • Need for a consistent interest-rate curve • Instantaneous forward rate • Parametric form of discount curves • Choice of curve nodes Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 2 Summary (2/2) • Bootstrapping quoted deposit rates • Bootstrapping using quoted interest-rate futures • Bootstrapping using quoted swap rates • QuantLib, bootstrapping, and rate helpers • Derivatives on foreign-exchange rates • Sensitivities of interest-rate portfolios (DV01) • Hedging portfolio with interest-rate risk Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 3 Major liquid quoted interest-rate derivatives For any given major currency (EUR, USD, GBP, JPY, ...) • Deposit rates • Interest-rate futures (FRA not reliable!) • Interest-rate swaps Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 4 Quotes from Financial Times Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 5 Consistent interest-rate curve We need a consistent interest-rate curve in order to • Understand the current market conditions (e.g. forward rates) • Compute the at-the-money strikes for Caps, Floor, and Swaptions • Compute the NPV of exotic derivatives • Determine the \fair" forward currency-exchange rate • Hedge portfolio exposure to interest rates • ... (many more reasons) ... Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 6 One forward rate does not fit all (1/2) Assume a continuously compounded discount rate from a flat rate r D(t) = e−r t (1) Matching exactly the implied discount for the first deposit rate 1 −r T = D(T1) = e 1 (2) 1 + T1 rfix(1) and for the second deposit rate 1 −r T = D(T2) = e 2 (3) 1 + T1 rfix(2) Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 7 One forward rate does not fit all (2/2) Yielding 1 r = log 1 + T1 rfix(1) (4) T1 and 1 r = log 1 + T2 rfix(2) (5) T2 which would imply two values for the same r. Hence, a single constant rate is not consistent with all market quotes! Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 8 Instantaneous forward rate (1/2) Given two future dates d1 and d2, the forward rate was defined as, " # 1 D (d1) − D (d2) rfwd(d1; d2) = (6) T (d1; d2) D (d2) We define the instantaneous forward rate f(d1) as the limit, f(d1) = lim rfwd(d1; d2) (7) d2!d1 Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 9 Instantaneous forward rate (2/2) Given certain day-conventions, set T = T (d0; d) then after preforming a change of variable from d to T we have, 1 "D(T ) − D(T + ∆t)# f(T ) = lim (8) ∆t!0 ∆t D(T + ∆t) It can be shown that 1 @D(T ) @ log [D(T )] f(T ) = − = − (9) D(T ) @T @T Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 10 Instantaneous forward rate for flat curve Consider a continuously-compounded flat-forward curve D(d) = e−z T (d0;d) () D(T ) = e−z T (10) with a given zero rate z, then h −z T i @ log [D(T )] @ log e f(T ) = − = − @T @T @ [−z T ] = − = z @T is the instantaneous forward rate Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 11 Discount from instantaneous forward rate Integrating the expression for the instantaneous forward rate Z @ log [D(t)] Z Z T dt = − f(t)dt () log [D(T )] = − f(t)dt @T 0 and taking the exponential we obtain ( Z T ) D(T ) = exp − f(t)dt 0 so that choosing f(t) results in a discount factor Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 12 Forward expectations Recall R T ( )d R T ( ) D(T ) = E e− 0 r t t = e− 0 f t dt (11) Similarly in the forward measure (see Brigo Mercurio) " Z T # T 1 0 0 rfwd(t; T ) = E r(t )dt (12) T − t t and f(T ) = ET [r(t)dt] (13) Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 13 Piecewise-flat forward curve (1/2) Given a number of nodes, T1 < T2 < T3, define the instantaneous forward rate as f(t) = f1 for t ≤ T1 (14) f(t) = f2 for T1 < t ≤ T2 (15) f(t) = f3 for T2 < t ≤ T3 (16) f(t) = ::: until the last node Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 14 Piecewise-flat forward curve (2/2) We determine the discount factor D(T ) using equation ( Z T ) D(T ) = exp − f(t)dt 0 It can be shown that −f (T −T ) D(T ) = 1 · e 1 0 for T ≤ T1 (17) −f (T −T ) D(T ) = D(T1) e 2 1 for T1 < T ≤ T2 (18) ::: = ::: (19) −f +1(T −Ti) D(T ) = D(Ti) e i for Ti < T ≤ Ti+1 (20) Recall that T0 = 0 Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 15 Questions? Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 16 (The art of) choosing the curve nodes • Choose d0 the earliest settlement date • First few nodes to fit deposit rates (until 1st futures?) • Some nodes to fit futures until about 2 years • Final nodes to fit swap rates Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 17 Why discard long-maturity deposit rates? Compare cash flows of a deposit and a one-year payer swap for a notional of 100,000$ Date Deposit IRS Fixed Leg IRS Ibor Leg Today - 100,000$ 0$ 0$ Today + 6m 0$ 0$ 1,200$ Today + 12m 102,400$ -2,500$ 1,280∗$ For maturities longer than 6 months credit risk is not negligible *Estimated by the forward rate Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 18 Talking to the trader: bootstrap • Deposit rates are unreliable: quoted rates may not be tradable • Libor fixings are better but fixed once a day (great for risk- management purposes!) • FRA quotes are even more unreliable than deposit rates Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 19 Boostrap of the USD curve using different helper lists 0.9 0.8 0.7 0.6 0.5 Zero rates (%) 0.4 Depo1Y + Swaps 0.3 Depo6m + Swaps Depo3m + Swaps 0.2 Depo3m + Futs + Swaps Depo2m + Futs + Swaps 0.1 0 0.5 1 1.5 2 2.5 3 3.5 time to maturity Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 20 Boostrap of the USD curve using different helper lists 0.4 Depo1Y + Swaps 0.35 Depo6m + Swaps Depo3m + Swaps 0.3 Depo3m + Futs + Swaps 0.25 0.2 0.15 0.1 Spread over risk free (%) 0.05 0 -0.05 0 0.5 1 1.5 2 2.5 3 3.5 time to maturity Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 21 Discount interpolation Taking the logarithm in the piecewise-flat forward curve log [D(T )] = log D(Ti−1) − (T − Ti)fi+1 (21) discount factors are interpolated log linearly • Other interpolations are possible and give slightly different results between nodes (see QuantLib for a list) • Important: use the same type of interpolation for all curves! Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 22 Bootstrapping the first node (1/2) Set the first node to the maturity of the first depo rate. Recalling equation (2) for f1 = r, −f T 1 D(T1) = e 1 1 = (22) 1 + T1 rfix(1) This equation can be solved for f1 to give, 1 f1 = log 1 + T1 rfix(1) (23) T1 we obtain the value of f1. Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 23 Bootstrapping the first node (2/2) 6.0% 6 5.0% 4.0% 3.0% 2.0% 1.0% f 1 • • - 3m 6m 1y 2y 3y 4y 5y 7y 10y Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 24 Bootstrapping the second node (1/2) Set the second node to the maturity of the second depo rate. The equivalent equation for the second node gives, −f T −f (T −T ) 1 D(T2) = e 1 1 e 2 2 1 = (24) 1 + T2 rfix(2) from which we obtain log 1 + T2 rfix(2) − f1 T1 f2 = (25) T2 − T1 Continue for all deposit rates to be included in the term structure Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 25 Bootstrapping the second node (2/2) 6.0% 6 5.0% 4.0% 3.0% f 2.0% 2 • 1.0% f 1 • • - 3m 6m 1y 2y 3y 4y 5y 7y 10y Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 26 Bootstrapping from quoted futures (1/2) For each futures included in the term structure • Add the futures maturity + tenor date to the node list • Solve for the appropriate forward rates that reprice the futures Note: futures are great hedging tools Advanced Derivatives, Interest Rate Models 2010-2012 c Marco Marchioro Bootstrapping the interest-rate term structure 27 Bootstrapping