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The Swaption Cube

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The Swaption Cube The Swaption Cube Anders B. Trolle Ecole Polytechnique F´ed´eralede Lausanne and Swiss Finance Institute Princeton-Lausanne workshop, May 13, 2011 Introduction I A standard European swaption is an option to enter into a fixed versus floating forward starting interest rate swap at a predetermined rate on the fixed leg I Understanding the swaption market is important from a practical perspective... I By some measures (such as the notional amount of outstanding options) the world's largest derivatives market I Many standard fixed income securities, such as fixed rate mortgage-backed securities and callable agency securities, embed swaption-like options I Many exotic interest rate derivatives and structured products also embed swaptions I Many large corporations are active in the swaption market either directly or indirectly (through the issuance and swapping of callable debt) I and from an academic perspective... I Swaptions contain valuable information about interest rate distributions Many empirical studies on the swaption market. This paper differs from these in two important ways I All existing studies are limited to only using data on at-the-money (ATM) swaptions. In contrast, we analyze a proprietary data set consisting of swaption cubes, which sheds new light on the swaption market. I Existing studies are mostly concerned with the pricing and hedging of swaptions using reduced-form models. Key objective of the paper is to understand the fundamental drivers of prices and risk premia in the swaption market. I Vast literature on interest derivatives I Paper is related to a growing literature linking the term structure of interest rates to macro factors (Ang and Piazzesi (2003), Gallmeyer, Hollifield, and Zin (2005), Ang, Piazzesi, and Wei (2006), Smith and Taylor (2009), Bekaert, Cho, and Moreno (2010), Bikbov and Chernov (), Chun (2010), and Joslin, Priebsch, and Singleton (2010)) I This literature is mainly based on Gaussian models and, consequently, primarily concerns itself with the determinants of the conditional mean of interest rates I We complement this literature by studying the determinants of the conditional volatility and skewness of interest rates, which are critical for derivatives prices I While existing macro-term structure papers focus on USD market, we consider both USD and EUR markets to see if differences in monetary policy objectives has effect on the relative importance of various macro drivers Agenda I The swaption cube data I A model independent analysis of the swaption cube I A dynamic term structure model for swap rates I Fundamental drivers of the swap rate distributions The swaption cube data The swaption cube is an object that shows how swaption prices vary along three dimensions: I The maturities of the underlying swaps (2, 5, 10, 20, and 30 years) I The expiries of the options (1, 3, 6, 9 months and 1, 2, 5, and 10 years) I The option strikes (15 different degrees of moneyness measured as the strike minus the at-the-money level of the swaption ( 400, 300, 200, 150, 100, 50, 25, and 0 basis points))± ± ± ± ± ± ± I Consider both USD and EUR markets, which are by far the most liquid markets. I For the USD market, the data is from December 19, 2001 to January 27, 2010 (419 weeks, total of 172,658 quotes). For the EUR market, the data is from June 6, 2001 to January 27, 2010 (449 weeks, total of 172,500 quotes) I Data is from ICAP plc., which is the largest inter-dealer broker in the interest rate derivatives market and as such provides the most accurate quotations. A model independent analysis of the swaption cube I For a given option expiry and swap maturity, conditional moments of the swap rate distribution (under the appropriate pricing measure) at a time horizon equal to the option expiry can be inferred from the implied volatility smile I We analyze how conditional moments vary with option expiry, swap maturity, and across time in order to establish a set of robust stylized facts regarding the swaption cube I Consider a fixed versus floating interest rate swap for the period Tm to Tn with a fixed rate of K. At every time Tj in a pre-specified set of dates Tm+1; :::; Tn, the fixed leg pays τj−1K, where τj−1 is the year-fraction between times Tj−1 and Tj . I The value of the swap at time t < Tm (assuming a notional of one) is given by Vm;n(t) = P(t; Tm) P(t; Tn) KAm;n(t); − − where n X Am;n(t) = τj−1P(t; Tj ) j=m+1 and P(t; T ) denotes the time-t price of a zero-coupon bond maturing at time T . I The time-t forward swap rate, Sm;n(t), is the rate on the fixed leg that makes the present value of the swap equal to zero, and is given by P(t; Tm) P(t; Tn) Sm;n(t) = − : Am;n(t) I A payer swaption is an option to enter into an interest rate swap, paying the fixed leg at a predetermined rate and receiving the floating leg. Let m;n(t; K) denote the time-t value of a European P payer swaption expiring at Tm with strike K on a swap for the period Tm to Tn. At expiration, the swaption has a payoff of + + Vm;n(Tm) = (1 P(Tm; Tn) KAm;n(Tm)) − − + = Am;n(Tm)(Sm;n(Tm) K) : − I At time t < Tm, its price is given by h R Tm i Q − r(s)ds + m;n(t; K) = E e t Am;n(Tm)(Sm;n(Tm) K) P t − h +i = Am;n(t)E A (Sm;n(Tm) K) ; t − where Q denotes expectation under the risk-neutral measure and A denotes expectation under the annuity measure associated with using Am;n(t) as numeraire. I From Bakshi and Madan (2000), Carr and Madan (2001), and Bakshi, Kapadia, and Madan (2003) it follows that for any fixed Z, we can write any twice continuously differentiable function of Sm;n(Tm), g(Sm;n(Tm)), as 0 g(Sm;n(Tm)) = g(Z) + g (Z)(Sm;n(Tm) − Z) Z 1 Z Z 00 + 00 + + g (K)(Sm;n(Tm) − K) dK + g (K)(K − Sm;n(Tm)) dK: Z 0 I Taking expectations under the annuity measure and setting Z = Sm;n(t), we obtain an expression in terms of prices of out-of-the-money receiver and payer swaptions A Et [g(Sm;n(Tm))] = g(Sm;n(t)) Z 1 Z Sm;n(t) 1 00 00 + g (K)Pm;n(t; K)dK + g (K)Rm;n(t; K)dK : Am;n(t) Sm;n(t) 0 We can use this result to compute conditional moments of the swap rate distribution at a time horizon equal to the expiry of the option. I Conditional mean µt E A [Sm;n(Tm)] = Sm;n(t): ≡ t I Conditional variance h i A 2 Var (Sm;n(Tm)) = E A (Sm;n(Tm) µt ) = t t − 2 Z 1 Z Sm;n(t) m;n(t; K)dK + m;n(t; K)dK Am;n(t) Sm;n(t) P 0 R I Conditional skewness h i A 3 Et Sm;n(Tm) − µt SkewA S (T ) = = t m;n m 3=2 VartA Sm;n(t; Tm) 6 R 1 R Sm;n(t) K − Sm;n(t) Pm;n(t; K)dK + K − Sm;n(t) Rm;n(t; K)dK Am;n(t) Sm;n(t) 0 3=2 VartA Sm;n(t; Tm) I Conditional kurtosis h i A 4 Et Sm;n(Tm) − µt A S (T ) = = Kurtt m;n m 2 VartA Sm;n(t; Tm) 12 R 1 2 R Sm;n(t) 2 K − Sm;n(t) Pm;n(t; K)dK + K − Sm;n(t) Rm;n(t; K)dK Am;n(t) Sm;n(t) 0 2 VartA Sm;n(t; Tm) Tenor Option expiry 1mth 3mths 6mths 9mths 1yr 2yrs 5yrs 10yrs Panel A: USD market 2 yrs 110.1 112.0 114.4 117.4 120.7 123.6 116.9 98.2 (35.5) (31.5) (27.9) (26.9) (26.6) (24.6) (18.0) (12.0) 5 yrs 122.4 122.2 121.8 121.1 121.3 120.4 111.7 93.2 (37.4) (33.5) (29.4) (27.4) (26.3) (23.2) (16.5) (10.4) 10 yrs 115.9 115.8 115.4 114.7 114.4 113.3 104.7 86.8 (36.6) (32.7) (28.7) (26.5) (25.0) (22.0) (15.0) (9.2) 20 yrs 106.9 105.7 104.0 102.7 101.8 99.5 89.9 74.1 (36.1) (31.4) (26.7) (24.0) (21.8) (18.6) (12.3) (7.9) 30 yrs 103.5 101.7 99.9 98.6 97.5 95.1 85.4 69.6 (37.6) (31.5) (26.3) (23.4) (20.9) (17.1) (10.8) (6.3) Panel B: EUR market 2 yrs 80.7 81.2 81.6 81.3 80.8 80.6 78.0 72.1 (29.2) (24.6) (20.2) (17.2) (15.3) (12.9) (8.7) (6.3) 5 yrs 83.6 82.4 80.9 79.6 78.6 77.1 74.2 69.1 (25.3) (21.1) (17.1) (14.8) (13.4) (11.5) (8.5) (7.2) 10 yrs 74.7 74.7 74.3 73.7 73.3 73.4 71.9 67.1 (23.1) (20.5) (17.6) (15.9) (15.0) (13.6) (9.9) (7.6) 20 yrs 72.3 72.0 71.2 70.4 70.0 69.3 67.2 61.9 (30.0) (26.4) (21.8) (19.3) (18.0) (15.4) (10.9) (8.2) 30 yrs 71.6 71.2 70.3 69.3 68.7 67.9 65.3 59.8 (36.8) (32.5) (26.7) (23.0) (20.8) (17.6) (12.9) (9.2) Notes: The table shows average conditional volatilities (annualized and in basis points) of the future swap rate distributions under the annuity measure A.
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