Cashsettled swaptions How wrong are we?


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1 Cashsettled swaptions How wrong are we? Marc Henrard Quantitative Research  OpenGamma 7th Fixed Income Conference, Berlin, 7 October 2011
2 ... Based on: Cashsettled swaptions: How wrong are we? Available at SSRN Working Paper series, , November 2010.
3 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
4 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
5 Introduction Cashsettled swaptions are the most liquid swaptions in the interbank market for EUR and GBP. The standard pricing formulas used are not properly justified (they require untested approximations). The standard pricing formulas with standard smile description leads to arbitrage: see F. Mercurio, Cashsettled swaptions and noarbitrage, Risk, Local coherence between Market pricing formula for cashsettled and physical delivery swaption has not been analyzed.
6 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
7 Description The analyses are done for receiver swaptions. Swap start date t 0, fixed leg payment dates (t i ) 1 i n, coupon rate K and the accrual fraction (δ i ) 1 i n. Floating leg payment dates ( t i ) 1 i ñ. Swaption expiry date: θ t 0. The cash annuity for a swap rate S is (for a frequency m) C(S) = n i=1 1 m (1 + 1 m S)i. The cashsettled swaption payment is (in t 0 ) C(S)(K S) +.
8 Swap rate We work in a multicurve setup. The discounting curve is denoted P D (s, t) and the forward curve is denoted P j (s, t) where j is the Libor tenor. The physical annuity (also called PVBP or level) is A t = n δ i P D (t, t i ). i=1 The swap rate in t is ñ ( ) i=1 PD (t, t i ) P j (t, t i ) P S t = j (t, t i+1 ) 1. A t
9 Market formulas (physical delivery) The value at expiry of the delivery swaption is The generic price is A θ (K S θ ) +. N 0 E N [N 1 θ A θ (K S θ ) + ] With the numeraire A t, S t is a martingale and under lognormal model ds t = σs t dw t, the price can be computed explicitly A 0 Black(K, S 0, σ).
10 Market formulas (cashsettled) The value at expiry of the cashsettle swaption is P D (θ, t 0 )C(S θ )(K S θ ) +. With the numeraire P D (t, t 0 )C(S t ),the price becomes P D (0, t 0 )C(S 0 ) E C [(K S θ ) + ]. But S θ is not a martingale anymore. The market standard formula is to substitute C by A as numeraire and approximate the price by P(0, t 0 )C(S 0 ) E C [(K S θ ) + ] P D (0, t 0 )C(S 0 ) E A [(K S θ ) + ] = P D (0, t 0 )C(S 0 ) Black(K, S 0, σ).
11 Market formulas (cashsettled) The value at expiry of the cashsettle swaption is P D (θ, t 0 )C(S θ )(K S θ ) +. With the numeraire P D (t, t 0 )C(S t ),the price becomes P D (0, t 0 )C(S 0 ) E C [(K S θ ) + ]. But S θ is not a martingale anymore. The market standard formula is to substitute C by A as numeraire and approximate the price by P(0, t 0 )C(S 0 ) E C [(K S θ ) + ] P D (0, t 0 )C(S 0 ) E A [(K S θ ) + ] = P D (0, t 0 )C(S 0 ) Black(K, S 0, σ).
12 Market formulas (cashsettled) The value at expiry of the cashsettle swaption is P D (θ, t 0 )C(S θ )(K S θ ) +. With the numeraire P D (t, t 0 )C(S t ),the price becomes P D (0, t 0 )C(S 0 ) E C [(K S θ ) + ]. But S θ is not a martingale anymore. The market standard formula is to substitute C by A as numeraire and approximate the price by P(0, t 0 )C(S 0 ) E C [(K S θ ) + ] P D (0, t 0 )C(S 0 ) E A [(K S θ ) + ] = P D (0, t 0 )C(S 0 ) Black(K, S 0, σ).
13 Market formulas (cashsettled) The price is A 0 E A [ A 1 θ PD (θ, t 0 )C(S θ )(K S θ ) +] Consider that the annuity/annuity ratio P D (θ, t 0 )C(S θ ) A θ has a low variance and it can be replaced by its initial value (initial freeze technique) A 0 E A [ A 1 θ PD (θ, t 0 )C(S θ )(K S θ ) +] A 0 E A [ A 1 0 PD (0, t 0 )C(S 0 )(K S θ ) +] = P D (0, t 0 )C(S 0 ) E A [ (K S θ ) +].
14 Market formulas (cashsettled) The price is A 0 E A [ A 1 θ PD (θ, t 0 )C(S θ )(K S θ ) +] Consider that the annuity/annuity ratio P D (θ, t 0 )C(S θ ) A θ has a low variance and it can be replaced by its initial value (initial freeze technique) A 0 E A [ A 1 θ PD (θ, t 0 )C(S θ )(K S θ ) +] A 0 E A [ A 1 0 PD (0, t 0 )C(S 0 )(K S θ ) +] = P D (0, t 0 )C(S 0 ) E A [ (K S θ ) +].
15 Mercurio s result Mercurio presented no arbitrage conditions for cashsettled swaptions. One relates to the density deduced from option prices: C(S 0 ) x 2 Black(x, S 0, σ) C(x) dx = 1. If not a digital cashsettled option with strike 0 has a value different from its constant payoff Black implied volatility
16 Cashsettle and physical swaptions Difference between cashsettled and physical delivery swaptions for different curve environments O/N rate (%) Slope (%) 3 4
17 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
18 Data The tests are done for 5Yx10Y swaptions. For all the model used, explicit formula exists for the delivery swaptions. The market data are from 30Sep The smile is obtained with a Hagan et al. SABR approach. When a calibration is required, it is done on physical delivery swaption. At least the swaption of same maturity and strike is in the calibration basket (and calibrated perfectly).
19 Linear Terminal Swap Rate model This is not a term structure model but an approximation technique to treat the swap rate as the only state variable for the yield curve. Applied in our case, the quantity P D (θ, t 0 )A 1 θ is approximated by a swap rate function α(s θ ). It is an approximation technique more sophisticated than the initial freeze approach. It is presented in Andersen, L. and Piterbarg, V. Interest Rate Modeling, Section The cashsettled swaptions price is A 0 E A [ α(s θ )C(S θ )(K S θ ) +]. In the simple linear TSR, function is α(s) = α 1 S + α 2 and the coefficients are 1 α 2 = n i=1 δ and α 1 = 1 ( P D ) (0, t p ) α 2. i S 0 A 0
20 Linear Terminal Swap Rate model In this framework, the cashsettled swaption are priced by a replication formula + ) A 0 (k(k) Swpt(S 0, K) + (k (x)(x K) + 2k (x)) Swpt(S 0, x)dx K with k(x) = (α 1 x + α 2 )C(x). Note that the linear TSR is such that α(s 0 ) = P D (0, t 0 )A 1 0 (the approximation is exact in 0 and ATM).
21 Linear Terminal Swap Rate model Strike Mrkt DC TSR Diff. Vega Payer Receiver
22 HullWhite: mean reversion Strike Mean reversion 0.1% 1% 2% 5% 10% Payer Receiver
23 HullWhite: mean reversion Strike Mean reversion 0.1% 1% 2% 5% 10% Payer Receiver
24 HullWhite: delta Tenor Delta DSC Delta FWD Mrkt HW Diff. Mrkt HW Diff. 5Y Y Y Y Y Y Y Y Y Y Y Total Strike: ATM+1.5% Notional: 100m.
25 HullWhite: curves Differences for different curve shapes O/N rate (%) Slope (%) 3 4
26 G2++: correlation Calibration on one swaption, imposed mean reversions (1% and 30%), imposed volatility ratio (4). Pricing: numerical integration. Strike Correlation 90% 45% 0% 45% 90% Payer Receiver
27 G2++: term structure of volatility Calibration to 10Y and 1Y swaptions. Missing numbers due to not perfect calibrations. Strike Correlation 90% 45% 0% 45% 90% Payer Receiver
28 LMM: multifactor Calibration to one swaptions (tenor 10Y); displacement is 10%. Pricing by MonteCarlo. Strike Angle 0 π/4 π/2 3π/4 π Payer Receiver
29 LMM: multifactor Calibration to swaptions with tenor 1Y to 10Y (10 calibrating instruments); displacement is 10%. Pricing by MonteCarlo. Strike Angle 0 π/4 π/2 3π/4 π Payer Receiver
30 LMM: skew Calibration to swaptions with tenor 10Y (one calibrating instrument); angle is π/2. Strike Displacement Payer Receiver
31 Multimodels: summary Market data as of 30Sep Strike Price difference (bps) All figures computed with OpenGamma open source OGAnalytics library.
32 Multimodels: summary Market data as of 30Sep Strike Price difference (bps) All figures computed with OpenGamma open source OGAnalytics library.
33 Multimodels: summary Market data as of 30Sep HullWhite TSR G2++ LMM 7.98 Strike Price difference (bps) All figures computed with OpenGamma open source OGAnalytics library.
34 Multimodels: summary Market data as of 23Apr Strike Hull White G2++ LMM Price difference
35 Multimodels: summary Market data as of 30Apr Strike Hull White G2++ LMM Price difference
36 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
37 Conclusion The standard market formula is arbitrable with Black or SABR smile (Mercurio result). We presented multimodels analyses of cashsettled swaption prices and delta. The model prices can be very far away from standard market formula prices. Within one model, the price range from noncalibrated parameters can be large. Risk uncertainty between different models for cashsettled swaption is not large but usually applies to nonnetting positions. The cashsettled swaption are liquid but not as simple as they may look!
38 Conclusion The standard market formula is arbitrable with Black or SABR smile (Mercurio result). We presented multimodels analyses of cashsettled swaption prices and delta. The model prices can be very far away from standard market formula prices. Within one model, the price range from noncalibrated parameters can be large. Risk uncertainty between different models for cashsettled swaption is not large but usually applies to nonnetting positions. The cashsettled swaption are liquid but not as simple as they may look!
39 Cash settled swaptions 1 Introduction 2 Cash settle swaptions description and market formulas 3 Multimodels analysis 4 Conclusion 5 Appendix: approximation in HullWhite model
40 Approximation in HullWhite model In the case of the HullWhite model, it is possible to obtain an efficient approximated formula for the cashsettled swaptions. All variable can be written as function of a normally distributed random variable X. Exercise boundary: S θ (X ) < K which is equivalent to X < κ where κ is defined by S θ (κ) = K. The rate S θ is roughly linear in X and the annuity is also roughly linear. The result is more parabola shape than a straight line. A third order approximation is required to obtain a precise enough formula. The payoff expansion around a reference point X 0 is C(S θ (X ))(K S θ (X )) U 0 +U 1 (X X 0 )+ 1 2 U 2(X X 0 ) ! U 3(X X 0 ) 3.
41 Approximation in HullWhite model Theorem In the extended Vasicek model, the price of a cashsettled receiver swaption is given to the third order by [ ( P(0, t 0 ) E exp α 0 X 1 ) 2 α2 0 (U 0 + U 1 (X X 0 )+ 1 2 U 2(X X 0 ) )] 3! U 3(X X 0 ) 3 ( U 0 U 1 α U 2(1 + α0) 2 1 ) 3! U 3( α α 0 ) N( κ) ( + U U 2( 2 α 0 + κ) + 1 ) 3! U 3( 3 α κ α 0 κ 2 2) 1 exp ( 12 ) κ2. 2π where κ is given above, U i are the payoff expansion coefficients, κ = κ + α 0 and α 0 = α 0 + X 0.
42 Approximation in HullWhite model Strike Int. Appr. 2 IntAp. 2 Approx. 3 IntAp. 3 Payer Receiver
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