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**Chapter 3 The Swap Market Model**

**Swaps**

Deﬁnition 3.1 A swap is an agreement between two parties to exchange cash-ﬂows in the future, at some agreed dates.The most common type of swap is a ”plain vanilla” interest rate swap. Here company B agrees to pay company A cash ﬂows equalto interest at a pre-determined ﬁxed rate on a notional principal (it is not exchanged but used only for the calculation of interest payments) for a number of years. At the same time company A agrees to pay company B cash-ﬂows equal to interest at a ﬂoating rate,which, in many interest rate swap agreements, is the LIBOR (or the JIBAR in the South African market[Chapter 4]). Exchanging thesame amount makes no sense, hence the principal is not exchanged.

*Valuation of Interest rate swaps: Market Practice*

When swaps and other over-the-counter derivatives are valued, thecash-ﬂows are usually discounted using LIBOR zero-coupon interest rates. This is because LIBOR is the cost of funds for a ﬁnancial institution.

Relationship of swaps to bonds A swap is the same as an agreement in which

1. Company B has lent company A a certain amount (not principal)at the x-month LIBOR rate.

2. Company A has lent company B the same amount at a ﬁxed rate per annum. The value of the money to B is therefore the diﬀerence between the values of the two bonds. Deﬁne Bfix – time 0 value of ﬁxed-rate bond underlying the swap, Bfloat – time 0 value of ﬂoating-rate bond underlying the swap. Then V, the value of the swap to company B is Vswap = Bfloat −Bfix. (3.1)If all interest and principal are realized at the end of the period, say n years, then the rate involved is called an n-year zero rate, also known as zero-coupon rate or n-year spot rate. Deﬁne:ti: time when ith payments are exchanged, i = 1,…,n,L: notional principal in swap agreement, Li = Li(0) = L(0,ti): LIBOR zero- rate for a maturity ti, K: ﬁxed payment made on each payment date.

Then,Bfix =n X i=1 Ke−Liti + Le−Lntn. (3.2)For the ﬂoating rate bond, immediately after a payment date, we have Bfloat = L because this is now identical to a newly issued ﬂoating rate bond. But, immediately before the next payment date, we have Bfloat = L plus ﬂoating rate payment, say K∗, to be paidon the next payment date. Today’s swap value is its value before tomorrow’s payment discounted at the LIBOR rate L1 for time t1,i.e, Bfloat = (L + K∗)e−L1t1 (3.3) where K∗ is the ﬂoating-rate payment already known.

Substituting the two equations into Vswap we get Vswap =−Ãn X i=1 Ke−Liti + Le−Lntn!+ (L + K∗)e−L1t1. (3.4) The value of the swap to A will be negative. K∗ is, in precise form, calculated taking into account the accrual day-count convention (out of 365 or 360 days).TERMINOLOGY

The set of ﬂoating rate payments is called the ﬂoating leg while that of ﬁxed rate payments is called the ﬁxed leg.Receiver swap: in this case the holder of a receiver swap receivesthe ﬁxed leg and pays the ﬂoating leg.

Payer swap: the holder of this one pays the ﬁxed leg and receivesthe ﬂoating leg.

*General Theory of Swaps*

Now consider resettlements dates T0,T1,…,TN; αi = Ti −Ti−1. Deﬁnition 3.2 The payments in a Tn ×(TN −Tn) payer swap are as follows: – Payments will be made and received at Tn+1,Tn+2,…,TN. – For every elementary period [Ti,Ti+1],i = 1,…,N−1, the LIBOR rate Li+1(Ti) is set at time Ti and the ﬂoating leg αi+1Li+1(Ti) is received at Ti+1. We assume a notional principal of L≡1. – For the same period the ﬁxed leg αi+1K is paid at Ti+1, where Kis a ﬁxed rate (swap rate). This K is not quite the same as the one on page 63. If an amount of αi+1Li+1(Ti) =p(Ti,Ti)−p(Ti,Ti+1) p(Ti,Ti+1) is received at time Ti+1, then αi+1Li+1(Ti)p(t,Ti+1) is received at time Ti. But this is just p(Ti,Ti) − p(Ti,Ti+1). If payoﬀ of this contract at time Ti is p(Ti,Ti) − p(Ti,Ti+1), then because of noarbitrage, the value of the ﬂoating payment at time t is given by the expression p(t,Ti)−p(t,Ti+1). (3.5) Hence, the total value of the ﬂoating side at time t for t≤Tn is p(t,Tn)−p(t,Tn+1) + p(t,Tn+1)−p(t,Tn+2) + ···+ p(t,TN−1)−p(t,TN)=N−1 X i=n [p(t,Ti)−p(t,Ti+1)]= p(t,Tn)−p(t,TN) = pn(t)−pN(t).

The total value on the ﬁxed side is N−1 X i=n p(t,Ti+1)αi+1K = KN Xi =n+1αip(t,Ti)= KN Xi =n+1

αipi(t).The net value PSN n (t,K) of the Tn×(TN −Tn) payer swap at time t < Tn is Bfloat −Bfix = p(t,Tn)−p(t,TN)−K N Xi =n+1 αipi(t)i.e PSN n (t,K) = pn(t)−pN(t)−K N Xi =n+1αipi(t). (3.6)

But pn(t)−pN(t)−KN Xi =n+1αipi(t) = 0⇔K = pn(t)−pN(t) PN i=n+1 αipi(t)

Deﬁnition 3.3 The par or forward swap rate RN n (t) of the Tn × (TN −Tn) swap is the value of K forwhich PSN n (t,K) = 0, i.e,K = RN n (t,K) =pn(t)−pN(t) PN i=n+1 αipi(t). (3.7)

Deﬁnition 3.4 For each pair n,k with n < k, the process in the denominator of the above equation, Sk n(t), deﬁned by Sk n(t) = Sk(t,Tn) =k Xi =n+1 αipi(t) (3.8) is known as the accrual factor or as the present value of a basis point.

Note that Snk(t) represents the value at time t of a portfolio of bonds with diﬀerent maturities. It is clear that RN n (t) =pn(t)−pN(t) SN n (t), 0≤t≤Tn. (3.9)

Hence, the arbitrage-free price of a payer swap with swap rate K is PSN n (t,K) = pn(t)−pN(t)−K N Xi =n+1 αipi(t) = pn(t)−pN(t)−KSN n (t) = RN n (t,K) N Xi =n+1 αipi(t)−KSN n (t) = RN n (t,K)SN n (t)−KSN n (t) = hRN n (t,K)−KiSN n (t) = hRN n (t)−KiSN n (t) (3.10)Equally, the price of a receiver swap is given by RSN n (t) =hK −RN n (t)iSN n (t).

** Swaptions: Deﬁnition and Market Practice**

Deﬁnition 3.5 A Tn ×(TN −Tn) payer swaption with strike K is a contract which at the exercise date Tn gives the holder the right but not the obligation to enter into a Tn×(TN −Tn) swap with ﬁxed swap rate K. We see from the deﬁnition that a payer swaption is a contingentTn-claim that pays XN n = maxhPSN n (Tn,K),0i = maxh³RN n (Tn,K)−K´SN n (Tn),0i = SN n (Tn)maxh(RN n (Tn)−K),0i(3.11)

which is a call option on RN n with strike K. Hence,

Deﬁnition 3.6 (Black’s formula for swaptions) The Black-76 formula for a Tn ×(TN −Tn) payer swaption with strike K is deﬁned as PSN n (t) = SN n (t)nRN n (t)N[d1]−KN[d2]o, (3.12) where

d1 =1 σn,N√Tn −t »lnÃRN n (t) K !+ 1 2σ2n,N(Tn −t)#d2 = d1 −σn,NqTn −t.

The constant σn,N is known as the Black volatility. Given a market price for the swaption, the Black volatility implied by the Black for mula is referred to as the implied Black volatility.

The task at hand is to build an arbitrage-free model with the property that the theoretical prices derived within the model has the structure of the Black formula in the above deﬁnition.

**The Swap Market Models**

Lemma 3.7 Denote the martingale measure for the numeraire Sk n(t)by Qk n. Then the forward swap rate Rk n is a Qk n – martingale. Proof 3.8 We are required to prove that E³Rk n(s)|Ft´= Rk n(t). EQk n³Rk n(s)|Ft´ = EQk n »pn(s)−pk(s) Sk n(s) |Ft#,0≤t≤s = pn(t)−pk(t) Sk n(t) = Rk n(t) since Rk n is the value of a self-ﬁnancing portfolio ( a long Tn bond and a short Tk bond), divided by the value of the self-ﬁnancing portfolioSk n(t).

Deﬁnition 3.9 Consider resettlement dates T0,…,TN, with 0 ≤ n < k ≤N. Furthermore, consider a deterministic function of time σn,k(t). A swap market model with volatilities σn,k(t) is speciﬁed by assuming that the par swap rates have dynamics of the form dRk n(t) = Rk n(t)σn,k(t)dWk n(t), (3.13) where Wk n(t) is Wiener under Qk n.

* Pricing Swaptions in the Swap Market Model*

The swap market model price of a Tn ×(TN −Tn) swaption is PSNN n (t) = SN n (t)En,NhmaxhRN n (Tn)−K,0i|Fti,0≤t≤Tn. (3.14) Since the equation in (3.13) describing the dynamics of RN n is a

GBM, then RN n (Tn) = RN n (t)eRTn tσn,N(t)dWk n(s)−1 2RTn t ||σn,N(s)||2ds. (3.15) And, since σn,N is deterministic, then conditional onFt, the process RN n (Tn) is log-normal, that is we can write RN n (Tn) = RN n (t)eY N n (t,Tn), (3.16) where Y N n (t,Tn) is normally distributed with expected value mN n (t,Tn) =− 1 2ZTn t ||σn,N(s)||2ds, (3.17)

and variance vN n2(t,Tn) =ZTn t ||σn,N(s)||2ds. (3.18) These results give rise to the following swaption pricing formula.

Proposition 3.10 In the swap market model, the Tn ×(TN −Tn) payer swaption price with strike K is given by PSNN n (t) = SN n (t)nRN n (t)N[d1]−KN[d2]o, (3.19)

where d1 =1vN n √Tn −t »lnÃRN n (t) K !+ 1 2vN n2#d2 = d1 −σn,N which is of the form of the Black- formula in Deﬁnition 3.6.Equation (3.19) shows that the numeraire for pricing swaptions is

SN n (t), whereas the numeraire for pricing caplets (ﬂoorlets) is αipi(t). Proof The pay-oﬀ of a payer swaption is given by PSNN n (t) = SN n (t)En,Nhmax[RN n (Tn)−K,0]|Fti. (3.20) But since RN n (t) is a GMB,RN n (Tn) = RN n (t)exp(ZTn t σn,N(t)dWk n(s)−1 2ZTn t ||σn,N(s)||2ds)= RN n (t)eY N n (t,Tn), (3.21)by the deterministic character of σn,N and the log-normality of RN n (Tn). Here Y N n (t,Tn)∼N(µ,σ) = N³−1 2RTn t ||σn,N(s)||2ds,(RTn t ||σn,N(s)||2ds)1/2´. Now lettingmN n (t,Tn) = −

1 2ZTn t ||σn,N(s)||2ds, vN n 2(t,Tn) = ZTn t ||σn,N(s)||2ds, the value of the payer swaption is given byPSNN n (t) = SN n (t)En,N maxh[RN n (Tn)−K,0]+i. (3.22) WriteRTn t σn,N(t)dWN n (t) as vN n x where X ∼N(0,1). Then PSNN n (t) = SN n (t) √2π Z∞ −∞ RN n (t)exp vN n x− vN n 2 2 −K + e−x2/2dx (3.23) and RN n (t)exp vN n x−PN n 2 2 −K = 0 74 ⇔exp N X n x−PN n 2 2 =K RN n Solve the following equation for x:exp vN n x− vN n 2 2 =K RN n vN n x−vN n22 = lnÃK RN n! x = a = ln³K RN n´+ vN n 2 2 vN n. (3.24) Thus PSNN n =SN n (t) √2π Z∞ −∞ RN n (t)exp vN n x−vN n2 2 −K +e−x2/2dx=SN n (t) √2π Z∞ aRN n (t)evN n xe−vN n2 2 e−x2/2dx− SN n (t) √2π Z∞ aKe−x2/2dx. Let II = −SN n (t) √2π Z∞ aKe−x2/2dx = −SN n (t)K » 1 √2πZ∞ ae−x2/2dx#= −SN n (t)K(1−N(a)). And let I = SN n (t) √2π Z∞ a RN n (t)evN n xe−vN n2 2 e−x2/2dx = SN n (t)RN n (t)e−vN n 2 2 √2π Z∞ a evN n x−x2/2dx.But

vN n x−x2/2 = −1 2hx2 −2vN n xi = − 1 2hx2 −2vN n x + (−vN n )2 −(−vN n )2i = − 1 2³x−vN n´2 + vN n 2 2 . I = SN n (t)RN n (t)e−vN n 2 2 √2π Z∞ a e−1 2(x−vN n )2+vN n 2 2 dx= SN n (t)RN n (t)·1√ 2π Z∞ ae−1 2(x−vN n )2dx. (3.25) Let y = x − vN n . Then dy = dx and a − vN n ≤ y < ∞ as a≤x <∞. Hence I =SN n (t)RN n (t)e−vN n22 √2π Z∞ a−vN ne−y2 2 evN n2 2 dy= SN n (t)RN n (t) » 1 √2πZ∞ a−vN ne−y2 2 dy# = SN n (t)RN n (t)h1−N(a−vN n )i = SN n (t)RN n (t)N³−(a−vN n )´.

PSNN n (t) = I + II= SN n (t)RN n (t)N³−(a−vN n )´+ SN n (t)K(1−N(a)) = SN n (t)hRN n (t)N³−(a−vN n )´+ K(1−N(a))i. Now if we let d1 = −(a−vN n ) and d2 = −a = ln(RN n /K)−vN2 n vN n , we get that PSNN n = SN n (t)hRN n (t)N(d1)−KN(d2)i.¦ (3.26) 76In the same manner we obtain that the price of a receiver swaption is given by RSNN n (t) = SN n (t)hKN(−d2)−RN n (t)N(−d1)i. (3.27)

**Contents
**

**1 Probability Measures**

1.1 Risk Neutral Probability Measures:- Discrete Case

1.2 ForwardRisk NeutralProbabilityMeasures- Discrete Case

**2 LIBOR Market Models**

2.1 Deﬁning the LIBOR rate

2.2 The LIBOR market model: Risk Neutral Valuation

2.3 Floors: Deﬁnition and Market Practice

2.4 Terminal Measure Dynamics and Existence of LIBOR market model 2.5 Interest Rate Collars: Market Practice

**3 The Swap Market Model**

3.1 Swaps

3.2 Swaptions: Deﬁnition and Market Practice

3.3 The Swap Market Models

3.4 Compatibility of LIBOR and Swap market models

**4 South African Market**

4.1 Historical background

4.2 Interest rate caps and ﬂoors: The South African context

4.3 The SAFEX-JIBAR market model

**5 Computational Analytics**

5.1Concluding Remarks

5.2 SAFEX-JIBAR Swap and Swaptions models

5.3 Calibration and Other Issues

**6 New directions in interest rate theory**

**7 Conclusion**

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A THEORETICAL AND EMPIRICAL ANALYSIS OF THE LIBOR MARKET MODEL AND ITS APPLICATION IN THE SOUTH AFRICAN SAFEX JIBAR MARKET