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Chapter 14 Pricing of Interest Rate Derivatives In this chapter we consider the pricing of caplets, caps, and swaptions, using change of numéraire and forward swap measures. 14.1 Forward Measures and Tenor Structure The maturity dates are arranged according to a discrete tenor structure {0 = T0 < T1 < T2 < · · · < Tn }. An example of forward interest rate curve data is given in the table of Figure 14.1, which contains the values of (T1 , T2 , . . . , T23 ) and of {f (t, t + Ti , t + Ti + δ)}i=1,2,...,23 , with t = 07/05/2003 and δ = six months. 2D 2.55 8Y 3.88 1W 2.53 9Y 4.02 1M 2.56 10Y 4.14 2M 2.52 11Y 4.23 3M 2.48 12Y 4.33 1Y 2.34 13Y 4.40 2Y 2.49 14Y 4.47 3Y 2.79 15Y 4.54 4Y 3.07 20Y 4.74 5Y 3.31 25Y 4.83 6Y 7Y 3.52 3.71 30Y 4.86 Fig. 14.1: Forward rates arranged according to a tenor structure. Recall that by definition of P (t, Ti ) and absence of arbitrage the process t 7−→ e− rt 0 rs ds P (t, Ti ), 0 ≤ t ≤ Ti , i = 1, 2, . . . , n, is an Ft -martingale under P = P, and as a consequence (P (t, Ti ))t∈[0,Ti ] can be taken as numéraire in the definition ∗ r Ti dP̂i 1 = e− 0 rs ds dP P (0, Ti ) " (14.1) N. Privault of the forward measure P̂i . The following proposition will allow us to price contingent claims using the forward measure P̂i , it is a direct consequence of Proposition 12.2, noting that here we have P (Ti , Ti ) = 1. Proposition 14.1. For all sufficiently integrable random variables F we have i h r Ti ˆ i [F | Ft ], 0 ≤ t ≤ T, i = 1, 2, . . . , n. IE∗ F e− t rs ds Ft = P (t, Ti )IE (14.2) Recall that for all Ti , Tj ≥ 0, the deflated process t 7−→ P (t, Tj ) , P (t, Ti ) 0 ≤ t ≤ min(Ti , Tj ), is an Ft -martingale under P̂i , cf. Proposition 12.3. Dynamics under the forward measure In order to apply Proposition 14.1 and to compute the price i h rT ˆ i [F | Ft ], IE∗ e− t rs ds F Ft = P (t, Ti )IE it can be useful to determine the dynamics of the underlying processes rt , f (t, T, S), and P (t, T ) under the forward measure P̂i . Let us assume that the dynamics of the bond price P (t, Ti ) is given by dP (t, Ti ) = rt dt + ζi (t)dWt , P (t, Ti ) (14.3) for i = 1, 2, . . . , n, where (Wt )t∈R+ is a standard Brownian motion under P and (rt )t∈R+ and (ζi (t))t∈R+ are adapted processes with respect to the filtration (Ft )t∈R+ generated by (Wt )t∈R+ , i.e. w wt t 1wt |ζi (s)|2 ds , P (t, Ti ) = P (0, Ti ) exp rs ds + ζi (s)dWs − 0 0 2 0 t ∈ [0, Ti ], i = 1, 2, . . . , n. Forward Brownian motions By the Girsanov Theorem 12.4, wt Ŵti := Wt − ζi (s)ds, 0 0 ≤ t ≤ Ti , 410 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html (14.4) " Pricing of Interest Rate Derivatives is a standard Brownian motion under P̂i for all i = 1, 2, . . . , n, cf. e.g. (12.10), hence we have dŴti = dWt − ζi (t)dt, i = 1, 2, . . . , n, and dŴtj = dWt − ζj (t)dt = dŴti + (ζi (t) − ζj (t))dt, i, j = 1, 2, . . . , n, (Ŵtj )t∈R+ which shows that has drift (ζi (t) − ζj (t))t∈R+ under P̂i . Hence the dynamics of t 7−→ P (t, Tj ) under P̂j is given by dP (t, Tj ) = rt dt + |ζj (t)|2 dt + ζj (t)dŴtj , P (t, Tj ) i, j = 1, 2, . . . , n, (14.5) where (Ŵtj )t∈R+ is a standard Brownian motion under P̂j , and we have w wt t 1wt P (t, Tj ) = P (0, Tj ) exp rs ds + ζj (s)dŴsj + |ζj (s)|2 ds 0 0 2 0 w wt wt t 1wt = P (0, Tj ) exp rs ds + ζj (s)dŴsi + ζj (s)ζi (s)ds − |ζj (s)|2 ds 0 0 0 2 0 w wt t 1wt 1wt i = P (0, Tj ) exp rs ds + ζj (s)dŴs − |ζj (s) − ζi (s)|2 ds + |ζi (s)|2 ds , 0 0 2 0 2 0 t ∈ [0, Tj ], i, j = 1, 2, . . . , n. Consequently, the forward price P (t, Tj )/P (t, Ti ) can be written as w t P (t, Tj ) P (0, Tj ) 1wt = exp (ζj (s) − ζi (s))dŴsj + |ζj (s) − ζi (s)|2 ds 0 P (t, Ti ) P (0, Ti ) 2 0 w t P (0, Tj ) 1wt i = |ζi (s) − ζj (s)|2 ds , exp (ζj (s) − ζi (s))dŴs − 0 P (0, Ti ) 2 0 t ∈ [0, Tj ], i, j = 1, 2, . . . , n, (14.6) which also follows from Proposition 12.5. In case the short rate process (rt )t∈R+ is given as the (Markovian) solution to the stochastic differential equation drt = µ(t, rt )dt + σ(t, rt )dWt , its dynamics will be given under P̂i by drt = µ(t, rt )dt + σ(t, rt )ζi (t)dt + σ(t, rt )dŴti . (14.7) In the Vasicek case, by (13.15) we have " 411 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault drt = (a − brt )dt + σdWt , and σ ζi (t) = − (1 − e−b(Ti −t) ), b hence from (14.7) we have drt = (a − brt )dt − 0 ≤ t ≤ Ti , σ2 (1 − e−b(Ti −t) )dt + σdŴti b (14.8) and we obtain dP (t, Ti ) σ2 σ = rt dt + 2 (1 − e−b(Ti −t) )2 dt − (1 − e−b(Ti −t) )dŴti , P (t, Ti ) b b from (13.15). 14.2 Bond Options The next proposition can be obtained as an application of the Margrabe formula (12.26) of Proposition 12.10 by taking Xt = P (t, Tj ), Nt = P (t, Ti ), and X̂t = Xt /Nt = P (t, Tj )/P (t, Ti ). In the Vasicek model, this formula has been first obtained in [58]. We work with a standard Brownian motion (Wt )t∈R+ under P, generating the filtration (Ft )t∈R+ , and an Ft -adapted short rate process (rt )t∈R+ . Proposition 14.2. Assume that the dynamics of the bond prices P (t, Ti ), P (t, Tj ) under P∗ are given by dP (t, Ti ) = rt dt + ζi (t)dWt , P (t, Ti ) dP (t, Tj ) = rt dt + ζj (t)dWt , P (t, Tj ) where (ζi (t))t∈R+ and (ζj (t))t∈R+ are deterministic functions. Then the price of a bond call option on P (Ti , Tj ) with payoff F = (P (Ti , Tj ) − κ)+ can be written as i h r Ti IE∗ e− t rs ds (P (Ti , Tj ) − κ)+ Ft (14.9) v 1 P (t, Tj ) v 1 P (t, Tj ) = P (t, Tj )Φ + log − κP (t, Ti )Φ − + log , 2 v κP (t, Ti ) 2 v κP (t, Ti ) with v 2 = w Ti t |ζj (s) − ζi (s)|2 ds. Proof. First we note that using Nt = P (t, Ti ) as a numéraire the price of a bond call option on P (Ti , Tj ) with payoff F = (P (Ti , Tj )−κ)+ can be written 412 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives from Proposition 12.2 using the forward measure P̂, or directly by (12.5), as i i h h r Ti ˆ i (P (Ti , Tj ) − κ)+ Ft . IE∗ e− t rs ds (P (Ti , Tj ) − κ)+ Ft = P (t, Ti )IE Next, by (14.6) or by solving (12.12) in Proposition 12.5 we can write P (Ti , Tj ) as the geometric Brownian motion w Ti P (t, Tj ) 1 w Ti P (Ti , Tj ) = (ζj (s) − ζi (s))dŴsi − |ζi (s) − ζj (s)|2 ds , exp t P (t, Ti ) 2 t under the forward measure P̂. Since (ζi (s))s∈[0,Ti ] and (ζj (s))s∈[0,Tj ] in (14.3) are deterministic functions, P (Ti , Tj ) is a lognormal random variable given Ft under P̂ and we can use Lemma 6.6 to price the bond option by the Black-Scholes formula p Bl κP (0, Ti ), P (t, Tj ), v/ Ti − t, 0, Ti − t √ with strike price κP (0, Ti ), underlying P (t, Tj ), volatility parameter v/ Ti − t, time to maturity Ti − t and zero interest rate, which yields (14.9). Note that from Corollary 12.12 the decomposition (14.9) gives the selffinancing portfolio in the assets P (t, Ti ) and P (t, Tj ) for the claim with payoff (P (Ti , Tj ) − κ)+ . In the Vasicek case the above bond option price could also be computed rT from the joint law of rT , t rs ds , which is Gaussian, or from the dynamics (14.5)-(14.8) of P (t, T ) and rt under P̂i , cf. § 7.3 of [84], and [66] for the CIR and other short rate models with correlated Brownian motions. 14.3 Caplet Pricing The caplet on the spot forward rate f (Ti , Ti , Ti+1 ) with strike price κ is a contract with payoff (f (Ti , Ti , Ti+1 ) − κ)+ , priced at time t ∈ [0, Ti ] from Proposition 12.2 using the forward measure P̂i as i h r Ti+1 rs ds IE∗ e− t (f (Ti , Ti , Ti+1 ) − κ)+ Ft (14.10) + ˆ = P (t, Ti+1 )IEi+1 (f (Ti , Ti , Ti+1 ) − κ) | Ft , by taking Nt = P (t, Ti+1 ) as a numéraire. Next, we consider the caplet with payoff (L(Ti , Ti , Ti+1 ) − κ)+ " 413 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault on the LIBOR rate L(t, Ti , Ti+1 ) = 1 Ti+1 − Ti P (t, Ti ) −1 , P (t, Ti+1 ) 0 ≤ t ≤ Ti < Ti+1 , which is a martingale under P̂i+1 defined in (14.1), from Proposition 12.3. The next formula (14.12) is known as the Black caplet formula. We assume that L(t, Ti , Ti+1 ) is modeled in the BGM model of Section 13.7. Proposition 14.3. Assume that L(t, Ti , Ti+1 ) is modeled as dL(t, Ti , Ti+1 ) = γi (t)dB̂ti+1 , L(t, Ti , Ti+1 ) (14.11) 0 ≤ t ≤ Ti , i = 1, 2, . . . , n − 1, where t 7−→ γi (t) is a deterministic function, i = 1, 2, . . . , n − 1. The caplet on L(Ti , Ti , Ti+1 ) is priced as time t ∈ [0, Ti ] as i h r Ti+1 rs ds IE∗ e− t (L(Ti , Ti , Ti+1 ) − κ)+ Ft = P (t, Ti+1 )L(t, Ti , Ti+1 )Φ(d+ ) − κP (t, Ti+1 )Φ(d− ), where d+ = log(L(t, Ti , Ti+1 )/κ) + σi2 (t)(Ti − t)/2 √ , σi (t) Ti − t d− = log(L(t, Ti , Ti+1 )/κ) − σi2 (t)(Ti − t)/2 √ , σi (t) Ti − t and and |σi (t)|2 = (14.12) 0 ≤ t ≤ Ti+1 , 1 w Ti |γi |2 (s)ds. Ti − t t Proof. Taking P (t, Ti+1 ) as a numéraire, the forward price X̂t := P (t, Ti ) = 1 + (Ti+1 − Ti )L(Ti , Ti , Ti+1 ) P (t, Ti+1 ) is a martingale under P̂i+1 by Proposition 12.3. More precisely, by (14.11) we have w Ti 1 w Ti L(Ti , Ti , Ti+1 ) = L(t, Ti , Ti+1 ) exp γi (s)dB̂si+1 − |γi (s)|2 ds , t 2 t 414 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives 0 ≤ t ≤ Ti , i.e. t 7−→ L(t, Ti , Ti+1 ) is a geometric Brownian motion with volatility γi (t) under P̂i+1 . Hence by (14.10) we have i h r Ti+1 rs ds IE∗ e− t (L(Ti , Ti , Ti+1 ) − κ)+ Ft = P (t, Ti+1 ) IEi+1 (L(Ti , Ti , Ti+1 ) − κ)+ | Ft = P (t, Ti+1 )Bl(κ, L(t, Ti , Ti+1 ), σi (t), 0, Ti − t), t ∈ [0, Ti ], where Bl(κ, x, σ, r, τ ) = xΦ(d+ ) − κe−rτ Φ(d− ) is the Black-Scholes function with |σi (t)|2 = 1 w Ti |γi |2 (s)ds. Ti − t t The formula (14.12) is also known as the Black (1976) formula when applied to options on underlying futures or forward contracts on commodities, which are modeled according to (14.11). In this case, the bond price P (t, Ti+1 ) can be simply modeled as P (t, Ti+1 ) = e−r(Ti+1 −t) and (14.12) becomes e−r(Ti+1 −t) L(t, Ti , Ti+1 )Φ(d+ ) − κe−r(Ti+1 −t) Φ(d− ), where L(t, Ti , Ti+1 ) is the underlying future price. In general we may also write (14.12) as i h r Ti+1 rs ds (Ti+1 − Ti ) IE∗ e− t (L(Ti , Ti , Ti+1 ) − κ)+ Ft P (t, Ti ) = P (t, Ti+1 ) − 1 Φ(d+ ) − κ(Ti+1 − Ti )P (t, Ti+1 )Φ(d− ) P (t, Ti+1 ) = (P (t, Ti ) − P (t, Ti+1 ))Φ(d+ ) − κ(Ti+1 − Ti )P (t, Ti+1 )Φ(d− ), and by Corollary 12.12 this gives the self-financing portfolio strategy (Φ(d+ ), −Φ(d+ ) − κ(Ti+1 − Ti )Φ(d− )) in the bonds (P (t, Ti ), P (t, Ti+1 )) with maturities Ti and Ti+1 , cf. Corollary 12.13 and [88]. Floorlets Similarly, a floorlet on f (Ti , Ti , Ti+1 ) with strike price κ is a contract with payoff (κ − f (Ti , Ti , Ti+1 ))+ , priced at time t ∈ [0, Ti ] as " 415 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault i h r Ti+1 rs ds IE∗ e− t (κ − f (Ti , Ti , Ti+1 ))+ Ft ˆ i+1 (κ − f (Ti , Ti , Ti+1 ))+ | Ft , = P (t, Ti+1 )IE 0 ≤ t ≤ Ti+1 . Floorlets are analog to put options and can be similarly priced by the call/put parity in the Black-Scholes formula. Cap Pricing More generally one can consider caps that are relative to a given tenor structure {T1 , . . . , Tn }, with discounted payoff n−1 X (Tk+1 − Tk )e− r Tk+1 t rs ds (f (Tk , Tk , Tk+1 ) − κ)+ . k=1 Pricing formulas for caps are easily deduced from analog formulas for caplets, since the payoff of a cap can be decomposed into a sum of caplet payoffs. Thus the price of a cap at time t ∈ [0, T1 ] is given by "n−1 # rT X ∗ − t k+1 rs ds + IE (Tk+1 − Tk )e (f (Tk , Tk , Tk+1 ) − κ) Ft k=1 = n−1 X rT k+1 rs ds (Tk+1 − Tk ) IE∗ e− t (f (Tk , Tk , Tk+1 ) − κ)+ Ft k=1 = n−1 X i h ˆ k+1 (f (Tk , Tk , Tk+1 ) − κ)+ Ft . (Tk+1 − Tk )P (t, Tk+1 )IE k=1 (14.13) In the BGM model (14.11) the cap with payoff n−1 X (Tk+1 − Tk )(L(Tk , Tk , Tk+1 ) − κ)+ k=1 can be priced at time t ∈ [0, T1 ] by the Black formula n−1 X (Tk+1 − Tk )P (t, Tk+1 )Bl(κ, L(t, Tk , Tk+1 ), σk (t), 0, Tk − t). k=1 14.4 Forward Swap Measures In this section we introduce the forward measures to be used for the pricing of swaptions, and we study their properties. We start with the definition of 416 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives the annuity numéraire P (t, Ti , Tj ) = j−1 X (Tk+1 − Tk )P (t, Tk+1 ), 0 ≤ t ≤ Ti , (14.14) k=i with in particular P (t, Ti , Ti+1 ) = (Ti+1 − Ti )P (t, Ti+1 ), 0 ≤ t ≤ Ti . 1 ≤ i < n. The annuity numéraire satisfies the following martingale property, rt which can be proved by linearity and the fact that t 7−→ e− 0 rs ds P (t, Tk ) is a martingale for all k = 1, 2, . . . , n, under Assumption (A). Remark 14.4. The discounted annuity numéraire t 7−→ e− rt 0 rs ds P (t, Ti , Tj ) = e− rt 0 rs ds j−1 X (Tk+1 −Tk )P (t, Tk+1 ), 0 ≤ t ≤ Ti , k=i is a martingale under P = P∗ . The forward swap measure P̂i,j is defined by r Ti P (Ti , Ti , Tj ) dP̂i,j = e− 0 rs ds , dP P (0, Ti , Tj ) (14.15) 1 ≤ i < j ≤ n. We have " # i h r Ti dP̂i,j 1 IE∗ IE∗ e− 0 rs ds P (Ti , Ti , Tj ) Ft Ft = dP P (0, Ti , Tj ) = P (t, Ti , Tj ) − r t rs ds e 0 , P (0, Ti , Tj ) 0 ≤ t ≤ Ti , by Remark 14.4, and r Ti dP̂i,j|Ft P (Ti , Ti , Tj ) = e− t rs ds , dP|Ft P (t, Ti , Tj ) 0 ≤ t ≤ Ti+1 , (14.16) by Relation (12.3) in Lemma 12.1. We also know that the process t 7−→ vki,j (t) := P (t, Tk ) , P (t, Ti , Tj ) i, j, k = 1, 2, . . . , n, is an Ft -martingale under P̂i,j by Proposition 12.3. It follows that the LIBOR swap rate " 417 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault S(t, Ti , Tj ) := P (t, Ti ) − P (t, Tj ) = vii,j (t) − vji,j (t), P (t, Ti , Tj ) 0 ≤ t ≤ Ti , defined in Proposition 13.7 is also a martingale under P̂i,j . Using the forward swap measure we obtain the following pricing formula for a given integrable claim with payoff of the form P (Ti , Ti , Tj )F : # " i h r Ti dP̂i,j|Ft IE∗ e− t rs ds P (Ti , Ti , Tj )F Ft = P (t, Ti , Tj ) IE∗ F Ft dP|Ft h i ˆ = P (t, Ti , Tj )IEi,j F Ft , (14.17) after applying (14.15) and (14.16) on the last line, or Proposition 12.2. 14.5 Swaption Pricing on the LIBOR A swaption on the forward rate f (T1 , Tk , Tk+1 ) is a contract meant for protection against a risk based on an interest rate swap, and has payoff j−1 rT X − k+1 rs ds (Tk+1 − Tk )e Ti (f (Ti , Tk , Tk+1 ) − κ) !+ , k=i at time Ti . This swaption can be priced at time t ∈ [0, Ti ] under the riskneutral measure P∗ as !+ j−1 rT r Ti X − k+1 rs ds IE∗ e− t rs ds (Tk+1 − Tk )e Ti (f (Ti , Tk , Tk+1 ) − κ) F t . k=i (14.18) In the sequel and in practice the price (14.18) of the swaption will be evaluated as !+ j−1 rT X ∗ − t i rs ds IE e (Tk+1 − Tk )P (Ti , Tk+1 )(f (Ti , Tk , Tk+1 ) − κ) Ft , k=i t ∈ [0, Ti ], where we approximate the discount factor e ditional expectation P (Ti , Tk+1 ) given FTi . − r Tk+1 Ti (14.19) rs ds by its con- Using the inequality + + (x1 + · · · + xm ) ≤ x+ 1 + · · · xm , 418 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html x1 , . . . , xm ∈ R, " Pricing of Interest Rate Derivatives the above term (14.19) can be upper bounded by the cap price (14.13) written as !+ j−1 rT X ∗ − t i rs ds IE e (Tk+1 − Tk )P (Ti , Tk+1 )(f (Ti , Tk , Tk+1 ) − κ) Ft k=i " ≤ IE∗ e− r Ti t rs ds j−1 X + (Tk+1 − Tk )P (Ti , Tk+1 ) (f (Ti , Tk , Tk+1 ) − κ) Ft # k=i = j−1 X i h r Ti + (Tk+1 − Tk ) IE∗ e− t rs ds P (Ti , Tk+1 ) (f (Ti , Tk , Tk+1 ) − κ) Ft k=i = j−1 X r rT Ti − k+1 rs ds + (Tk+1 − Tk ) IE∗ e− t rs ds IE∗ e Ti FTi (f (Ti , Tk , Tk+1 ) − κ) Ft k=i = j−1 X rT k+1 + rs ds (Tk+1 − Tk ) IE∗ IE∗ e− t (f (Ti , Tk , Tk+1 ) − κ) FTi Ft k=i = j−1 X rT k+1 + rs ds (Tk+1 − Tk ) IE∗ e− t (f (Ti , Tk , Tk+1 ) − κ) Ft k=i = IE∗ "j−1 X (Tk+1 − Tk )e− r Tk+1 t rs ds (f (Ti , Tk , Tk+1 ) − κ) + # Ft . k=i In addition, when j = i + 1, the swaption price (14.19) coincides with the price at time t of a caplet on [Ti , Ti+1 ] since i h r Ti + IE∗ e− t rs ds ((Ti+1 − Ti )P (Ti , Ti+1 )(f (Ti , Ti , Ti+1 ) − κ)) Ft i h r Ti + = (Ti+1 − Ti ) IE∗ e− t rs ds P (Ti , Ti+1 ) ((f (Ti , Ti , Ti+1 ) − κ)) Ft r rT Ti − i+1 rs ds + = (Ti+1 − Ti ) IE∗ e− t rs ds IE∗ e Ti FTi ((f (Ti , Ti , Ti+1 ) − κ)) r rT Ti − i+1 rs ds + = (Ti+1 − Ti ) IE∗ IE∗ e− t rs ds e Ti ((f (Ti , Ti , Ti+1 ) − κ)) FTi i h r Ti+1 + rs ds = (Ti+1 − Ti ) IE∗ e− t (f (Ti , Ti , Ti+1 ) − κ) Ft , 0 ≤ t ≤ Ti , which coincides with the caplet price (14.10) up to the factor Ti+1 − Ti . In case we replace the forward rate f (t, T, S) with the LIBOR rate L(t, T, S) defined in Proposition 13.7, the payoff of the swaption can be rewritten as in the following lemma which is a direct consequence of the definition of the swap rate S(Ti , Ti , Tj ). " 419 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html Ft Ft N. Privault Lemma 14.5. The payoff of the swaption in (14.19) can be rewritten as !+ j−1 X (Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ) k=i = (P (Ti , Ti ) − P (Ti , Tj ) − κP (Ti , Ti , Tj )) + + = P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) . (14.20) (14.21) Proof. The relation j−1 X (Tk+1 − Tk )P (t, Tk+1 )(L(t, Tk , Tk+1 ) − S(t, Ti , Tj )) = 0 k=i that defines the forward swap rate S(t, Ti , Tj ) shows that j−1 X (Tk+1 − Tk )P (t, Tk+1 )L(t, Tk , Tk+1 ) k=i = S(t, Ti , Tj ) j−1 X (Tk+1 − Tk )P (t, Tk+1 ) k=i = P (t, Ti , Tj )S(t, Ti , Tj ) = P (t, Ti ) − P (t, Tj ), cf. Proposition 13.7, hence by the definition (14.14) of P (t, Ti , Tj ) we have j−1 X (Tk+1 − Tk )P (t, Tk+1 )(L(t, Tk , Tk+1 ) − κ) k=i = P (t, Ti ) − P (t, Tj ) − κP (t, Ti , Tj ) = P (t, Ti , Tj ) (S(t, Ti , Tj ) − κ) , and for t = Ti we get !+ j−1 X (Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ) k=i + = P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) . The next proposition simply states that a swaption on the LIBOR rate can be priced as a European call option on the swap rate S(Ti , Ti , Tj ) under the forward swap measure P̂i,j . Proposition 14.6. The price (14.19) of the European swaption with payoff 420 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives !+ j−1 X (Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ) (14.22) k=i on the LIBOR market can be written under the forward swap measure P̂i,j as i h ˆ i,j (S(Ti , Ti , Tj ) − κ)+ Ft , 0 ≤ t ≤ Ti . P (t, Ti , Tj )IE Proof. As a consequence of (14.17) and Lemma 14.5 we find !+ j−1 rT X ∗ − t i rs ds IE e (Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ) Ft k=i = = = = i h r Ti + IE e− t rs ds (P (Ti , Ti ) − P (Ti , Tj ) − κP (Ti , Ti , Tj )) Ft (14.23) i h r Ti + IE∗ e− t rs ds P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) Ft # " 1 + ∗ dP̂i,j|Ft IE (S(Ti , Ti , Tj ) − κ) Ft P (t, Ti , Tj ) dP|Ft i h ˆ i,j (S(Ti , Ti , Tj ) − κ)+ Ft . P (t, Ti , Tj )IE (14.24) ∗ In the next proposition we price a swaption with payoff (14.22) or equivalently (14.21). Proposition 14.7. Assume that the LIBOR swap rate (13.34) is modeled as a geometric Brownian motion under Pi,j , i.e. dS(t, Ti , Tj ) = S(t, Ti , Tj )σ̂(t)dŴti,j , where (σ̂(t))t∈R+ is a deterministic function. Then the swaption with payoff + (P (T, Ti ) − P (T, Tj ) − κP (Ti , Ti , Tj ))+ = P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) can be priced using the Black-Scholes formula as i h r Ti + IE∗ e− t rs ds P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) Ft = (P (t, Ti ) − P (t, Tj ))Φ+ (t, S(t, Ti , Tj )) −κΦ− (t, S(t, Ti , Tj )) j−1 X (Tk+1 − Tk )P (t, Tk+1 ), k=i where d+ = " 2 log(S(t, Ti , Tj )/κ) + σi,j (t)(Ti − t)/2 √ , σi,j (t) Ti − t 421 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault and d− = 2 log(S(t, Ti , Tj )/κ) − σi,j (t)(Ti − t)/2 √ , σi,j (t) Ti − t and |σi,j (t)|2 = 1 w Ti (σ̂(s))2 ds. Ti − t t Proof. Since S(t, Ti , Tj ) is a geometric Brownian motion with volatility function (σ̂(t))t∈R+ under P̂i,j , by (14.20)-(14.21) and (14.23)-(14.24) we have i h r Ti + IE∗ e− t rs ds P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) Ft i h rT = IE∗ e− t rs ds (P (T, Ti ) − P (T, Tj ) − κP (Ti , Ti , Tj ))+ Ft i h ˆ i,j (S(Ti , Ti , Tj ) − κ)+ Ft = P (t, Ti , Tj )IE = P (t, Ti , Tj )Bl(κ, S(Ti , Ti , Tj ), σi,j (t), 0, Ti − t) = P (t, Ti , Tj ) (S(t, Ti , Tj )Φ+ (t, S(t, Ti , Tj )) − κΦ− (t, S(t, Ti , Tj ))) = (P (t, Ti ) − P (t, Tj ))Φ+ (t, S(t, Ti , Tj )) − κP (t, Ti , Tj )Φ− (t, S(t, Ti , Tj )) = (P (t, Ti ) − P (t, Tj ))Φ+ (t, S(t, Ti , Tj )) −κΦ− (t, S(t, Ti , Tj )) j−1 X (Tk+1 − Tk )P (t, Tk+1 ). k=i In addition the hedging strategy (Φ+ (t, S(t, Ti , Tj )), −κΦ− (t, S(t, Ti , Tj ))(Ti+1 − Ti ), . . . . . . , −κΦ− (t, S(t, Ti , Tj ))(Tj−1 − Tj−2 ), −Φ+ (t, S(t, Ti , Tj ))) based on the assets (P (t, Ti ), . . . , P (t, Tj )) is self-financing by Corollary 12.13, cf. also [88]. Swaption prices can also be computed by an approximation formula, from the exact dynamics of the swap rate S(t, Ti , Tj ) under P̂i,j , based on the bond price dynamics of the form (14.3), cf. [102], page 17. Swaption volatilities can be estimated from swaption prices as implied volatilities from the Black pricing formula: 422 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives 0.2 0.18 0.16 0.14 0.12 9 8 7 0.1 6 5 0.08 4 0 i 3 1 2 3 2 4 5 6 7 j 1 8 9 0 Fig. 14.2: Implied swaption volatilities. Bermudan swaption pricing in Quantlib The Bermudan swaption on the tenor structure {Ti , . . . , Tj } is priced as the supremum !+ j−1 r X ∗ − tτ rs ds sup IE e (Tk+1 − Tk )P (τ, Tk+1 )(L(τ, Tk , Tk+1 ) − κ) Ft τ ∈{Ti ,...,Tj−1 } = sup ∗ IE h e − k=τ rτ rs ds t (P (τ, τ ) − P (τ, Tj ) − κP (τ, τ, Tj )) + τ ∈{Ti ,...,Tj−1 } = sup τ ∈{Ti ,...,Tj−1 } i Ft i h rτ + IE∗ e− t rs ds P (τ, τ, Tj ) (S(τ, τ, Tj ) − κ) Ft , where the supremum is over all stopping times τ taking values in {Ti , . . . , Tj }. Bermudan swaptions can be priced using this code in (R)quantlib, with the following output: Summary of pricing results for Bermudan Swaption Price (in bp) of Bermudan swaption is 24.92137 Stike is NULL (ATM strike is 0.05 ) Model used is: Hull-White using analytic formulas Calibrated model parameters are: a = 0.04641 sigma = 0.005869 This modified code can be used in particular the pricing of ordinary swaptions with the output: Summary of pricing results for Bermudan Swaption " 423 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault Price (in bp) of Bermudan swaption is 22.45436 Stike is NULL (ATM strike is 0.05 ) Model used is: Hull-White using analytic formulas Calibrated model parameters are: a = 0.07107 sigma = 0.006018 Exercises Exercise 14.1 Consider two bonds with maturities T1 and T2 , T1 < T2 , which follow the stochastic differential equations dP (t, T1 ) = rt P (t, T1 )dt + ζ1 (t)P (t, T1 )dWt and dP (t, T2 ) = rt P (t, T2 )dt + ζ2 (t)P (t, T2 )dWt . a) Using Itô calculus, show that the forward process P (t, T2 )/P (t, T1 ) is a driftless geometric Brownian motion driven by dŴt := dWt −ζ1 (t)dt under the T1 -forward measure h P̂.r T1 i b) Compute the price IE∗ e− 0 rs ds (K − P (T1 , T2 ))+ of a bond put option using change of numeraire and the Black-Scholes formula. Hint: Given X a centered Gaussian random variable with variance v 2 , we have: IE∗ [(κ−xeX−v 2 ) ] = κΦ(v/2+(log(κ/x))/v)−xΦ(−v/2+(log(κ/x))/v). /2 + Exercise 14.2 Given two bonds with maturities T , S and prices P (t, T ), P (t, S), consider the LIBOR rate L(t, T, S) = P (t, T ) − P (t, S) (S − T )P (t, S) at time t, modeled as dL(t, T, S) = µt L(t, T, S)dt + σL(t, T, S)dWt , 0 ≤ t ≤ T, (14.25) where (Wt )t∈[0,T ] is a standard Brownian motion under the risk-neutral measure P∗ , σ > 0 is a constant, and (µt )t∈[0,T ] is an adapted process. Let i h rS Ft = IE∗ e− t rs ds (κ − L(T, T, S))+ Ft 424 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives denote the price at time t of a floorlet option with strike price κ, maturity T , and payment date S. a) Rewrite the value of Ft using the forward measure P̂S with maturity S. b) What is the dynamics of L(t, T, S) under the forward measure P̂S ? c) Write down the value of Ft using the Black-Scholes formula. Hint. Given X a centered Gaussian random variable with variance v 2 we have IE∗ [(κ − em+X )+ ] = κΦ(−(m − log κ)/v) − em+ v2 2 Φ(−v − (m − log κ)/v), where Φ denotes the Gaussian cumulative distribution function. Exercise 14.3 Consider a European swaption giving its holder the right to enter into a 3-year annual pay swap in four years, where a fixed rate of 5% is paid and the LIBOR rate is received. Assume that the yield curve is flat at 5% per annum with annual compounding and the volatility of the swap rate is 20%. The notional principal is $10 millions and is quoted in basis points (one basis point = .01%). a) What are the key assumptions in order to apply Black’s formula to value this swaption? b) Compute the price of this swaption using Black’s formula as an application of Proposition 14.7, using the functions Φ(d+ ) and Φ(di ). Exercise 14.4 We work in the short rate model drt = σdBt , where (Bt )t∈R+ is a standard Brownian motion under P, and P2 is the forward measure defined by r T2 dP2 1 = e− 0 rs ds . dP P (0, T2 ) a) State the expressions of ζ1 (t) and ζ2 (t) in dP (t, Ti ) = rt dt + ζi (t)dBt , P (t, Ti ) i = 1, 2, and the dynamics of the P (t, T1 )/P (t, T2 ) under P2 , where P (t, T1 ) and P (t, T2 ) are bond prices with maturities T1 and T2 . b) State the expression of the forward rate f (t, T1 , T2 ). c) Compute the dynamics of f (t, T1 , T2 ) under the forward measure P2 with r T2 1 dP2 = e− 0 rs ds . dP P (0, T2 ) " 425 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault d) Compute the price i h r T2 (T2 − T1 ) IE∗ e− t rs ds (f (T1 , T1 , T2 ) − κ)+ Ft of a cap at time t ∈ [0, T1 ], using the expectation under the forward measure P2 . e) Compute the dynamics of the swap rate process S(t, T1 , T2 ) = P (t, T1 ) − P (t, T2 ) , (T2 − T1 )P (t, T2 ) t ∈ [0, T1 ], under P2 . f) Compute the swaption price i h r T1 (T2 − T1 ) IE∗ e− t rs ds P (T1 , T2 )(S(T1 , T1 , T2 ) − κ)+ Ft on the swap rate S(T1 , T1 , T2 ) using the expectation under the forward swap measure P1,2 . Exercise 14.5 Consider three zero-coupon bonds P (t, T1 ), P (t, T2 ) and P (t, T3 ) with maturities T1 = δ, T2 = 2δ and T3 = 3δ respectively, and the forward LIBOR L(t, T1 , T2 ) and L(t, T2 , T3 ) defined by 1 P (t, Ti ) L(t, Ti , Ti+1 ) = −1 , i = 1, 2. δ P (t, Ti+1 ) Assume that L(t, T1 , T2 ) and L(t, T2 , T3 ) are modeled in the BGM model by dL(t, T1 , T2 ) = e−at dŴt2 , L(t, T1 , T2 ) 0 ≤ t ≤ T1 , (14.26) and L(t, T2 , T3 ) = b, 0 ≤ t ≤ T2 , for some constants a, b > 0, where Ŵt2 is a standard Brownian motion under the forward rate measure P2 defined by r T2 e− 0 rs ds dP2 = . dP P (0, T2 ) a) Compute L(t, T1 , T2 ), 0 ≤ t ≤ T2 by solving Equation (14.26). b) Show that the price at time t of the caplet with strike price κ can be written as i h r T2 ∗ ˆ 2 (L(T1 , T1 , T2 ) − κ)+ | Ft , IE e− t rs ds (L(T1 , T1 , T2 ) − κ)+ Ft = P (t, T2 )IE ˆ 2 denotes the expectation under the forward measure P2 . where IE 426 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives c) Using the hint below, compute the price at time t of the caplet with strike price κ on L(T1 , T1 , T2 ). d) Compute P (t, T1 ) , P (t, T1 , T3 ) 0 ≤ t ≤ T1 , and P (t, T3 ) , P (t, T1 , T3 ) 0 ≤ t ≤ T2 , in terms of b and L(t, T1 , T2 ), where P (t, T1 , T3 ) is the annuity numéraire P (t, T1 , T3 ) = δP (t, T2 ) + δP (t, T3 ), 0 ≤ t ≤ T2 . e) Compute the dynamics of the swap rate t 7−→ S(t, T1 , T3 ) = P (t, T1 ) − P (t, T3 ) , P (t, T1 , T3 ) 0 ≤ t ≤ T1 , i.e. show that we have dS(t, T1 , T3 ) = σ1.3 (t)S(t, T1 , T3 )dŴt2 , where σ1,3 (t) is a process to be determined. f) Using the Black-Scholes formula, compute an approximation of the swaption price i h r T1 IE∗ e− t rs ds P (T1 , T1 , T3 )(S(T1 , T1 , T3 ) − κ)+ Ft ˆ 2 (S(T1 , T1 , T3 ) − κ)+ | Ft , = P (t, T1 , T3 )IE at time t ∈ [0, T1 ]. You will need to approximate σ1,3 (s), s ≥ t, by “freezing” all random terms at time t. Hint. Given X a centered Gaussian random variable with variance v 2 we have v2 IE∗ (em+X − κ)+ = em+ 2 Φ(v + (m − log κ)/v) − κΦ((m − log κ)/v), where Φ denotes the Gaussian cumulative distribution function. Exercise 14.6 Consider a portfolio (ξtT , ξtS )t∈[0,T ] made of two bonds with maturities T , S, and value Vt = ξtT P (t, T ) + ξtS P (t, S), 0 ≤ t ≤ T, at time t. We assume that the portfolio is self-financing, i.e. dVt = ξtT dP (t, T ) + ξtS dP (t, S), 0 ≤ t ≤ T, (14.27) and that it hedges the claim (P (T, S) − κ)+ , so that " 427 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault h rT + Vt = IE∗ e− t rs ds (P (T, S) − κ) h + = P (t, T ) IET (P (T, S) − κ) i Ft i Ft , 0 ≤ t ≤ T. a) Show that i h rT + IE∗ e− t rs ds (P (T, S) − K) Ft h i wt wt + = P (0, T ) IET (P (T, S) − K) + ξsT dP (s, T ) + ξsS dP (s, S). 0 0 b) Show that under the self-financing condition (14.27), the discounted portrt folio price Vet = e− 0 rs ds Vt satisfies dVet = ξtT dP̃ (t, T ) + ξtS dP̃ (t, S), rt rt where P̃ (t, T ) = e− 0 rs ds P (t, T ) and P̃ (t, S) = e− 0 rs ds P (t, S) denote the discounted bond prices. c) Show that h i h i + + IET (P (T, S) − K) |Ft = IET (P (T, S) − K) w t ∂C + (Xu , T − u, v(u, T ))dXu . 0 ∂x Hint: use the martingale property and the Itô formula. d) Show that the discounted portfolio price V̂t = Vt /P (t, T ) satisfies ∂C (Xt , T − t, v(t, T ))dXt ∂x P (t, S) ∂C = (Xt , T − t, v(t, T ))(σtS − σtT )dB̂tT . P (t, T ) ∂x dV̂t = e) Show that dVt = P (t, S) ∂C (Xt , T − t, v(t, T ))(σtS − σtT )dBt + V̂t dP (t, T ). ∂x f) Show that dVet = P̃ (t, S) ∂C (Xt , T − t, v(t, T ))(σtS − σtT )dBt + V̂t dP̃ (t, T ). ∂x g) Compute the hedging strategy (ξtT , ξtS )t∈[0,T ] of the bond option. h) Show that ∂C log(x/K) + τ v 2 /2 √ (x, τ, v) = Φ . ∂x τv 428 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives Exercise 14.7 Consider a LIBOR rate L(t, T, S), t ∈ [0, T ], modeled as dL(t, T, S) = µt L(t, T, S)dt + σ(t)L(t, T, S)dWt , 0 ≤ t ≤ T , where (Wt )t∈[0,T ] is a standard Brownian motion under the risk-neutral measure P∗ , (µt )t∈[0,T ] is an adapted process, and σ(t) > 0 is a deterministic function. a) What is the dynamics of L(t, T, S) under the forward measure P̂ with numeraire Nt := P (t, S) ? b) Rewrite the price h rS i IE∗ e− t rs ds φ(L(T, T, S))Ft (14.28) at time t ∈ [0, T ] of an option with payoff function φ using the forward measure P̂. c) Write down the value of the above option price (14.28) using an integral. Exercise 14.8 Given n bonds with maturities T1 , . . . , Tn , consider the annuity Pj−1 numeraire P (t, Ti , Tj ) = k=i (Tk+1 − Tk )P (t, Tk+1 ) and the swap rate S(t, Ti , Tj ) = P (t, Ti ) − P (t, Tj ) P (t, Ti , Tj ) at time t ∈ [0, Ti ], modeled as dS(t, Ti , Tj ) = µt S(t, Ti , Tj )dt + σS(t, Ti , Tj )dWt , 0 ≤ t ≤ Ti , (14.29) where (Wt )t∈[0,Ti ] is a standard Brownian motion under the risk-neutral measure P∗ , (µt )t∈[0,T ] is an adapted process and σ > 0 is a constant. Let i h r Ti IE∗ e− t rs ds P (Ti , Ti , Tj )φ(S(Ti , Ti , Tj ))Ft (14.30) at time t ∈ [0, Ti ] of an option with payoff function φ. a) Rewrite the option price (14.30) at time t ∈ [0, Ti ] using the forward swap measure P̂i,j defined from the annuity numeraire P (t, Ti , Tj ). b) What is the dynamics of S(t, Ti , Tj ) under the forward swap measure P̂i,j ? c) Write down the value of the above option price (14.28) using a Gaussian integral. d) Apply the above to the computation at time t ∈ [0, Ti ] of the put swaption price i h r Ti IE∗ e− t rs ds P (Ti , Ti , Tj )(κ − S(Ti , Ti , Tj ))+ Ft with strike price κ, using the Black-Scholes formula. Hint. Given X a centered Gaussian random variable with variance v 2 we have " 429 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault IE[(κ − em+X )+ ] = κΦ(−(m − log κ)/v) − em+ v2 2 Φ(−v − (m − log κ)/v), where Φ denotes the Gaussian cumulative distribution function. Exercise 14.9 Consider a bond market with two bonds with maturities T1 , T2 , whose prices P (t, T1 ), P (t, T2 ) at time t are given by dP (t, T1 ) = rt dt + ζ1 (t)dBt , P (t, T1 ) dP (t, T2 ) = rt dt + ζ2 (t)dBt , P (t, T2 ) where (rt )t∈R+ is a short term interest rate process, (Bt )t∈R+ is a standard Brownian motion generating a filtration (Ft )t∈R+ , and ζ1 (t), ζ2 (t) are volatility processes. The LIBOR rate L(t, T1 , T2 ) is defined by L(t, T1 , T2 ) = P (t, T1 ) − P (t, T2 ) . P (t, T2 ) Recall that a caplet on the LIBOR market can be priced at time t ∈ [0, T1 ] as i i h r T2 h + ˆ (L(T1 , T1 , T2 ) − κ)+ Ft , IE e− t rs ds (L(T1 , T1 , T2 ) − κ) Ft = P (t, T2 )IE (14.31) under the forward measure P̂ defined by r T1 P (T1 , T2 ) dP̂ = e− 0 rs ds , dP P (0, T2 ) under which B̂t := Bt − wt 0 ζ2 (s)ds, t ∈ R+ , (14.32) is a standard Brownian motion. In the sequel we let Lt = L(t, T1 , T2 ) for simplicity of notation. a) Using Itô calculus, show that the LIBOR rate satisfies dLt = Lt σ(t)dB̂t , 0 ≤ t ≤ T1 , (14.33) where the LIBOR rate volatility is given by σ(t) = P (t, T1 )(ζ1 (t) − ζ2 (t)) . P (t, T1 ) − P (t, T2 ) b) Solve the equation (14.33) on the interval [t, T1 ], and compute LT1 from the initial condition Lt . c) Assuming that σ(t) in (14.33) is a deterministic function, show that the price 430 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives i h ˆ (LT − κ)+ Ft P (t, T2 )IE 1 of the caplet can be written as P (t, T2 )C(Lt , v(t, T1 )), where v 2 (t, T1 ) = w T1 |σ(s)|2 ds, and C(t, v(t, T1 )) is a function of Lt and v(t, T1 ). t d) Consider a portfolio (ξt1 , ξt2 )t∈[0,T1 ] made of bonds with maturities T1 , T2 and value Vt = ξt1 P (t, T1 ) + ξt2 P (t, T2 ), at time t ∈ [0, T1 ]. We assume that the portfolio is self-financing, i.e. dVt = ξt1 dP (t, T1 ) + ξt2 dP (t, T2 ), 0 ≤ t ≤ T1 , (14.34) and that it hedges the claim (LT1 − κ) , so that i i h r T1 h + ˆ (LT − κ)+ Ft , Vt = IE e− t rs ds (P (T1 , T2 )(LT1 − κ)) Ft = P (t, T2 )IE 1 + 0 ≤ t ≤ T1 . Show that we have i h r T1 + IE e− t rs ds (P (T1 , T2 )(LT1 − κ)) Ft h i wt wt ˆ (LT − κ)+ + = P (0, T2 )IE ξs1 dP (s, T1 ) + ξs2 dP (s, T1 ), 1 0 0 0 ≤ t ≤ T1 . e) Show that under the self-financing condition (14.34), the discounted portrt folio price Vet = e− 0 rs ds Vt satisfies dVet = ξt1 dP̃ (t, T1 ) + ξt2 dP̃ (t, T2 ), where P̃ (t, Tk ) = e− prices. f) Show that rt 0 rs ds P (t, Tk ), k = 1, 2, denote the discounted bond h h i w t ∂C i ˆ (LT − κ)+ Ft = IE ˆ (LT − κ)+ + (Lu , v(u, T1 ))dLu , IE 1 1 0 ∂x and that the discounted portfolio price V̂t = Vt /P (t, T2 ) satisfies dV̂t = ∂C ∂C (Lt , v(t, T1 ))dLt = Lt (Lt , v(t, T1 ))σ(t)dB̂t . ∂x ∂x Hint: use the martingale property and the Itô formula. g) Show that dVt = (P (t, T1 ) − P (t, T2 )) ∂C (Lt , v(t, T1 ))σ(t)dBt + V̂t dP (t, T2 ). ∂x h) Show that " 431 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault dVet = ∂C (Lt , v(t, T1 ))d(P̃ (t, T1 ) − P̃ (t, T2 )) ∂x ∂C + V̂t − Lt (Lt , v(t, T1 )) dP̃ (t, T2 ), ∂x and deduce the values of the hedging portfolio (ξt1 , ξt2 )t∈R+ . Exercise 14.10 Consider a bond market with tenor structure {Ti , . . . , Tj } and bonds with maturities Ti , . . . , Tj , whose prices P (t, Ti ), . . . P (t, Tj ) at time t are given by dP (t, Tk ) = rt dt + ζk (t)dBt , P (t, Tk ) k = i, . . . , j, where (rt )t∈R+ is a short term interest rate process and (Bt )t∈R+ denotes a standard Brownian motion generating a filtration (Ft )t∈R+ , and ζi (t), . . . , ζj (t) are volatility processes. The swap rate S(t, Ti , Tj ) is defined by S(t, Ti , Tj ) = where P (t, Ti , Tj ) = P (t, Ti ) − P (t, Tj ) , P (t, Ti , Tj ) j−1 X (Tk+1 − Tk )P (t, Tk+1 ) k=i is the annuity numéraire. Recall that a swaption on the LIBOR market can be priced at time t ∈ [0, Ti ] as !+ j−1 r Ti X IE∗ e− t rs ds (Tk+1 − Tk )P (Ti , Tk+1 )(S(Ti , Tk , Tk+1 ) − κ) F t k=i i h + = P (t, Ti , Tj ) IEi,j (S(Ti , Ti , Tj ) − κ) Ft , (14.35) under the forward swap measure Pi,j defined by r Ti P (Ti , Ti , Tj ) dPi,j = e− 0 rs ds , dP P (0, Ti , Tj ) 1 ≤ i < j ≤ n, under which B̂ti,j := Bt − j−1 X P (t, Tk+1 ) (Tk+1 − Tk ) ζk+1 (t)dt P (t, Ti , Tj ) (14.36) k=i 432 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives is a standard Brownian motion. Recall that the swap rate can be modeled as dS(t, Ti , Tj ) = S(t, Ti , Tj )σi,j (t)dB̂ti,j , 0 ≤ t ≤ Ti , (14.37) where the swap rate volatilities are given by j−1 X P (t, Tj ) P (t, Tl+1 ) (Tl+1 −Tl ) (ζi (t)−ζl+1 (t))+ (ζi (t)−ζj (t)) P (t, Ti , Tj ) P (t, Ti ) − P (t, Tj ) l=i (14.38) 1 ≤ i, j ≤ n, cf. e.g. Proposition 10.8 of [83]. In the sequel we denote St = S(t, Ti , Tj ) for simplicity of notation. σi,j (t) = a) Solve the equation (14.37) on the interval [t, Ti ], and compute S(Ti , Ti , Tj ) from the initial condition S(t, Ti , Tj ). b) Assuming that σi,j (t) is a deterministic function of t for 1 ≤ i, j ≤ n, show that the price (14.24) of the swaption can be written as P (t, Ti , Tj )C(St , v(t, Ti )), where v 2 (t, Ti ) = w Ti t |σi,j (s)|2 ds, and C(x, v) is a function to be specified using the Black-Scholes formula Bl(K, x, σ, r, τ ), with IE[(xem+X − K)+ ] = Φ(v + (m + log(x/K))/v) − KΦ((m + log(x/K))/v), where X is a centered Gaussian random variable with mean m = rτ −v 2 /2 and variance v 2 . c) Consider a portfolio (ξti , . . . , ξtj )t∈[0,Ti ] made of bonds with maturities Ti , . . . , Tj and value j X Vt = ξtk P (t, Tk ), k=i at time t ∈ [0, Ti ]. We assume that the portfolio is self-financing, i.e. dVt = j X ξtk dP (t, Tk ), 0 ≤ t ≤ Ti , (14.39) k=i and that it hedges the claim (S(Ti , Ti , Tj ) − κ)+ , so that !+ j−1 rT X ∗ − t i rs ds Vt = IE e (Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ) Ft k=i i h + = P (t, Ti , Tj ) IEi,j (S(Ti , Ti , Tj ) − κ) Ft , " 433 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html N. Privault 0 ≤ t ≤ Ti . Show that !+ j−1 rT X ∗ − t i rs ds (Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ) IE e Ft k=i j w h i X t + = P (0, Ti , Tj ) IEi,j (S(Ti , Ti , Tj ) − κ) + ξsk dP (s, Ti ), k=i 0 0 ≤ t ≤ Ti . d) Show that under the self-financing condition (14.39), the discounted portrt folio price Vet = e− 0 rs ds Vt satisfies dVet = j X ξtk dP̃ (t, Tk ), k=i rt where P̃ (t, Tk ) = e− 0 rs ds P (t, Tk ), k = i, . . . , j, denote the discounted bond prices. e) Show that i h + IEi,j (S(Ti , Ti , Tj ) − κ) Ft h i w t ∂C + = IEi,j (S(Ti , Ti , Tj ) − κ) + (Su , v(u, Ti ))dSu . 0 ∂x Hint: use the martingale property and the Itô formula. f) Show that the discounted portfolio price V̂t = Vt /P (t, Ti , Tj ) satisfies dV̂t = ∂C ∂C (St , v(t, Ti ))dSt = St (St , v(t, Ti ))σti,j dB̂ti,j . ∂x ∂x g) Show that dVt = (P (t, Ti ) − P (t, Tj )) ∂C (St , v(t, Ti ))σti,j dBt + V̂t dP (t, Ti , Tj ). ∂x h) Show that j−1 dVt = St ζi (t) X ∂C (St , v(t, Ti )) (Tk+1 − Tk )P (t, Tk+1 )dBt ∂x k=i j−1 X ∂C +(V̂t − St (St , v(t, Ti ))) (Tk+1 − Tk )P (t, Tk+1 )ζk+1 (t)dBt ∂x k=i + ∂C (St , v(t, Ti ))P (t, Tj )(ζi (t) − ζj (t))dBt . ∂x 434 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html " Pricing of Interest Rate Derivatives i) Show that dVet = ∂C (St , v(t, Ti ))d(P̃ (t, Ti ) − P̃ (t, Tj )) ∂x ∂C +(V̂t − St (St , v(t, Ti )))dP̃ (t, Ti , Tj ). ∂x j) Show that ∂C (x, v(t, Ti )) = Φ ∂x log(x/K) v(t, Ti ) + v(t, Ti ) 2 . k) Show that we have log(St /K) v(t, Ti ) + d(P̃ (t, Ti ) − P̃ (t, Tj )) dVet = Φ v(t, Ti ) 2 log(St /K) v(t, Ti ) −κΦ − dP̃ (t, Ti , Tj ). v(t, Ti ) 2 l) Show that the hedging strategy is given by log(St /K) v(t, Ti ) ξti = Φ + , v(t, Ti ) 2 log(St /K) v(t, Ti ) log(St /K) v(t, Ti ) + − ξtj = −Φ −κ(Tj −Tj−1 )Φ , v(t, Ti ) 2 v(t, Ti ) 2 and ξtk = −κ(Tk+1 − Tk )Φ " log(St /K) v(t, Ti ) − v(t, Ti ) 2 , i ≤ k ≤ j − 2. 435 This version: June 23, 2016 http://www.ntu.edu.sg/home/nprivault/indext.html