Pricing basket options with an eye on swaptions

Pricing basket options with an eye on swaptions Alexandre d Aspremont ORFE Part of thesis supervised by Nicole El Karoui. Data from BNP-Paribas, London. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 1

Introduction Why baskets, swaptions, calibration? Interest rate derivatives trading Focus on structured products activity Discuss stability, speed and robustness few stable methods for model calibration and risk-management How do we extract correlation information from market option prices? A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 2

Activity Figure 1: OTC activity in interest rate options. Source: Bank for International Settlements. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 3

Structured Interest Rate Products Structured derivatives desks act as risk brokers buy/sell tailor made products from/to their clients hedge the resulting risk using simple options in the market manage the residual risk on the entire portfolio P&L comes from a mix of flow and arbitrage... A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 4

Derivatives Production Cycle market data model calibration pricing & hedging risk-management A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 5

Derivatives Production Cycle, Trouble... market data: illiquidity, Balkanization of the data sources model calibration: inverse problem, numerically hard pricing & hedging: American option pricing in dim. 2 risk-management: all of the above... A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 6

Objective However, it works Numerical difficulties creates P&L hikes, poor risk description,... Our objective here: improve stability, robustness Focus first on calibration Using new cone programming techniques to calibrate models and manage portfolio risk A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 7

Outline Pricing baskets Application to swaptions Cone programming, a brief introduction (next talk) IR model calibration (next talk) Risk-management (next talk) A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 8

Pricing baskets Everything you ever wanted to know about basket options without ever daring to ask is in Carmona & Durrleman (2003), in SIREV. What this means for today: Either something you don t want to know or something you didn t know you wanted to know Let me know... A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 9

Multivariate Black-Scholes We now look at the problem of pricing a basket in a generic Black & Scholes (1973) model with n assets F i s such that: df i s/f i s = σ i sdw s where σ i R n and dw s is a n dimensional B.M. We study the dynamics of a basket of forwards F ω s = n i=1 w if i s We look for an approximation to the price of a basket call: ( n ) + E w i FT i K i=1 A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 10

Multivariate Black-Scholes We can write the dynamics of the basket as: df ω u F ω u = ( n i=1 ω i,uσ i u) dwu d ω i,s ω i,s = ( n j=1 ω j,s ( σ i s σ j s ) ) ( dw s + ) n j=1 ω j,sσsds j where we have used: ω i,s = ω i F i s n i=1 ω if i s We notice that 0 ω i,s 1 with n j=1 ω i,s = 1. We also set: σ i s = σ i s σ ω s with σ ω s = n j=1 ω i,t σ j s note that σ ω s = n j=1 ω i,tσ j s is F t measurable. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 11

Multivariate Black-Scholes We can develop these dynamics around small values of n j=1 ω j,s σ j s. For some ε > 0, we write: df ω,ε s = F ω,ε s d ω ε i,s = ωε i,s ( σs ω + ε ) n j=1 ω j,s σ s j dw s ( σ s i ε ) ( n j=1 ωε j,s σj s dw s + σs ω ds + ε ) n j=1 ω j,s σ sds j As in Fournié, Lebuchoux & Touzi (1997) and Lebuchoux & Musiela (1999) we compute: [ ] C ε = E (F ω,ε T k) + (Ft ω, ω t ) and approximate it around ε = 0 by: C ε = C 0 + C (1) ε + o(ε) Both C 0 and C (1) (as well as C (2),... ) can be computed explicitly. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 12

Price Approximation: order zero The order zero term can be computed directly as the solution to the limit BS PDE: C 0 s + σω s 2 x 2 2 C 0 2 = 0 x 2 C 0 = (x K) + for s = T and we get C 0 as a Black & Scholes (1973) price with variance σ ω s 2 : C 0 = BS(T, F ω t, V T ) = F ω t N(h(V T )) κn (h(v T ) V T ) with h (V T ) = ( ) ) F ω ln( t κ + 1 2 V T et V T = VT T t σ ω s 2 ds A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 13

Price Approximation: order one As in Fournié et al. (1997), we then look at the PDE satisfied by C ε and differentiate it with respect to ε. The PDE associated with the multivariate BS dynamics is: { L ε 0 C ε = 0 C ε = (x k) + en s = T A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 14

Price Approximation: order one where 2 L ε 0 = Cε s + n σω s + ε y j σ s j x 2 2 C ε j=1 2 x 2 n + σ s, j σs ω n + ε j y k σ s σs ω, σ s k ε 2 + + j=1 j=1 k=1 2 n n σj s ε y k σ s k y 2 j 2 C ε 2 y 2 k=1 j n σ s, j σs ω n + ε j y k σ s σs ω, σ s k ε 2 j=1 k=1 n 2 y k σ s k xy 2 C ε j x y j k=1 n 2 y k σ s k C y ε j y j k=1 A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 15

Price Approximation: order one Take the limit in ε = 0 (mod. regularity conditions,... ): { ( n L 0 0C (1) + j=1 y j j σ s, σs ω C ε = 0 en s = T ) x 2 2 C 0 x 2 = 0 We then compute C (1) using the Feynmann-Kac representation: C (1) = F ω t E [ exp ( s t T t n ( j ω j,t σ s, σs ω s exp 1 ) j σ 2 u σu ω 2 du j=1 ( σ ω u + σ j u) dwu ) Vs,T n t ( ln F ω t K + s t σω udw u 1 2 V t,s + 1 2 V s,t Vs,T )] ds which can be computed explicitly. The same technique produces C (2),... A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 16

Price Approximation: summary The price of a basket call: ( n ) + E w i FT i K i=1 is approximated by a regular call price C = BS(w T F t, K, T, V T ) with V T = T t Tr(Ω t X s )ds where and Tr(Ω t X s ) = n i,j=1 Ω t,i,jx s,i,j Ω t = ŵ t ŵ T t = n i,j=1 ŵi,tŵ j,t σ it s σ j s with ŵ i,t = w if i t w T F t A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 17

Price Approximation: summary We can get a better approximation of the price by using instead: C 0 is given by the BS formula above: We get C (1) as: C (1) = w T F t T N C ε = C 0 + C (1) C 0 = BS(w T F t, K, T, V T ) t ln wt F t K n j σ ŵ s, σs w j,t exp j=1 + s t V 1/2 T σ j u, σ w u V 1/2 T ( s ) j 2 σ u, σu w du t du + 1 2 V T ds A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 18

Hedging Interpretation Suppose we are hedging the option with the approximate vol. σs ω and, as in El Karoui, Jeanblanc-Picqué & Shreve (1998), we track the hedging error: e T = 1 2 T n ω i,s σs i σs ω 2 (Fs ω ) 2 2 C 0 (Fs ω, V t,t ) 2 x 2 ds t i=1 At the first order in σ j s, we get: e (1) T = T t n i σ s, σs ω i=1 ωi,s F ω s n(h(v s,t, Fs ω )) ds V 1/2 s,t We finally have: C (1) = E [ e (1) T ] A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 19

Outline Pricing baskets Application to swaptions Cone programming, a brief introduction (next talk) IR model calibration (next talk) Risk-management (next talk) A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 20

Swaps The swap rate is the rate that equals the PV of a fixed and a floating leg: swap(t,t 0, T n ) = floating B(t, T0 ) B(t, T floating n+1 ) level(t, T fixed 0, Tn fixed ) where level(t, T fixed 0, T fixed n ) = n+1 i=1 coverage(t fixed i 1, T fixed )B(t, T fixed i i ) A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 21

Swaps The swap rate can be expressed a basket of forward rates: swap(t, T 0, T n ) = n w i (t)k(t, T i ) i=0 where K(t, T i ) are the forward rates with maturities T i, with the weights w i (t) given by w i (t) = float coverage(ti, T float float i+1 )B(t, T level(t, T fixed 0, Tn fixed ) i+1 ) Empirically, these weights are very stable (see Rebonato (1998)). A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 22

Libor Market Model In the Libor Market Model, the zero coupon volatility is specified to make Libor rates ( ) θ+δ 1 + δl(t, θ) = exp r(t, v)dv lognormal martingales under their respective measures: dk(s, T i )/K(s, T i ) = σ(s, T i )dw QT i +δ s θ where σ(s, T i ) R n and dw QT i +δ s is a n dimensional B.M. and K(s, T i ) = L(s, T i s) This volatility definition, the forward curve today and the Heath, Jarrow & Morton (1992) arbitrage conditions fully specify the model. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 23

Pricing Swaptions We let Q LV L be the swap forward martingale probability measure given by: dq LV L N dq T δcvg(i, b)β 1 (T i+1 ) t = B(t, T)β(T) Level(t, T, T N ) Following Jamshidian (1997), we can write the price of the Swaption with strike k as a that of a call on a swap rate: Ps(t) = Level(t, T, T N )E Q LV L t i=1 ( n ) + ω i (T)K(T,T i ) k i=0 In other words, the swaption is a call on a basket of forwards. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 24

A Remark on the Gaussian HJM We can also express the price of the swaption as that of a bond put: Ps(t) = B(t, T)E Q T t 1 B(t, T N+1 ) kδ + N B(t, T i ) i=i T In the Gaussian H.J.M. model (see El Karoui & Lacoste (1992), Musiela & Rutkowski (1997) or Duffie & Kan (1996)), this expression defines the price of a swaption as that of a put on a basket of lognormal zero coupon prices. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 25

Approximations We will make two key approximations: We replace the weights w i (s) by their value today w i (t). We approximate the swap rate n i=0 w i(t)k(s, T i ) by a sum of Q LV L lognormal martingales F i s with: F i t = K(t, T i ) and df i s/f i s = σ(s, T i s)dw LV L s A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 26

Swaption pricing formula We can write the order zero price approximation for Swaptions: Swaption = Level(t, T, T N ) ( ) swap(t,t, T N )N(h) κn(h V 1/2 T ) with where h = ( ln ( swap(t,t,tn ) κ V 1/2 T ) + 1 2 V T ) V T = T t N i=1 ˆω i (t)σ(s, T i s) 2 K(t, T i ) ds and ˆω i (t) = ω i (t) swap(t,t, T N ) and dk(s, T i ) = σ(s, T i s)k(s, T i )dw Q T i+1 s. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 27

Errors Can we quantify the error: What s the contribution of the weights in the swap s volatility? What about the drift terms coming from the forwards under Q LV L? What is the precision of the basket price approximation? First two questions: wait for next talk... A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 28

Price Approximation: Precision We plot the difference between two distinct sets of swaption prices in the Libor Market Model. One is obtained by Monte-Carlo simulation using enough steps to make the 95% confidence margin of error always less than 1bp. The second set of prices is computed using the order zero approximation. The plots are based on the prices obtained by calibrating a BGM model to EURO Swaption prices on November 6 2000, using all cap volatilities and the following swaptions: 2Y into 5Y, 5Y into 5Y, 5Y into 2Y, 10Y into 5Y, 7Y into 5Y, 10Y into 2Y, 10Y into 7Y, 2Y into 2Y, 1Y into 9Y (choice based on liquidity). A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 29

Price Approximation: Precision Figure 2: Error (bp) for various ATM swaptions. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 30

Price Approximation: Precision Figure 3: Error vs. moneyness, on the 5Y into 5Y A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 31

Price Approximation: Precision Figure 4: Error vs. moneyness on the 5Y into 10Y. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 32

Price Approximation: Precision We compare again with Monte-Carlo. The model parameters are F0 i = {0.07,0.05,0.04,0.04,0.04} w i = {0.2,0.2,0.2,0.2,0.2} T = 5 years, the covariance matrix is: 11 100 0.64 0.59 0.32 0.12 0.06 0.59 1 0.67 0.28 0.13 0.32 0.67 0.64 0.29 0.14 0.12 0.28 0.29 0.36 0.11 0.06 0.13 0.14 0.11 0.16 These values correspond to a 5Y into 5Y swaption. Our goal is to measure only the error coming from the pricing formula and not from the change of measure/martingale approximation A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 33

Price Approximation: Precision Absolute Error 0.0002 0.0001 0-0.0001-0.0002 0.2 0.4 0.6 0.8 1 Moneyness in Delta Figure 5: Order zero (dashed) and order one absolute pricing error (plain), in basis points. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 34

Price Approximation: Precision Absolute Error 0.0003 0.00025 0.0002 0.00015 0.0001 0.00005 0-0.00005 0 0.2 0.4 0.6 0.8 1 Moneyness in Delta Figure 6: Order zero (dashed) and order one absolute pricing error (plain), in basis points, zero correlation. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 35

Price Approximation: Precision 0.25 Relative Error 0.2 0.15 0.1 0.05 0 0.2 0.4 0.6 0.8 Moneyness in Delta Figure 7: Order zero (dashed) and order one relative pricing error (plain), equity case. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 36

Conclusion Order zero BS like formula sufficient for ATM swaptions Equity case: use order one Change of measure between Q LV L and Q T negligible (volatilities too low). Next talk: Will discuss calibration and risk-management issues A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 37

Outline Pricing baskets Application to swaptions Cone programming, a brief introduction (next talk) IR model calibration (next talk) Risk-management (next talk) A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 38

References Black, F. & Scholes, M. (1973), The pricing of options and corporate liabilities, Journal of Political Economy 81, 637 659. Carmona, R. & Durrleman, V. (2003), Pricing and hedging spread options, SIAM Rev. 45(4), 627 685. Duffie, D. & Kan, R. (1996), A yield factor model of interest rates, Mathematical Finance 6(4). El Karoui, N., Jeanblanc-Picqué, M. & Shreve, S. E. (1998), On the robustness of the black-scholes equation, Mathematical Finance 8, 93 126. El Karoui, N. & Lacoste, V. (1992), Multifactor analysis of the term structure of interest rates, Proceedings, AFFI. Fournié, E., Lebuchoux, J. & Touzi, N. (1997), Small noise expansion and importance sampling, Asymptotic Analysis 14, 361 376. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 39

Heath, D., Jarrow, R. & Morton, A. (1992), Bond pricing and the term structure of interest rates: A new methodology, Econometrica 61(1), 77 105. Jamshidian, F. (1997), Libor and swap market models and measures, Finance and Stochastics 1(4), 293 330. Lebuchoux, J. & Musiela, M. (1999), Market models and smile effects in caps and swaptions volatilities., Working paper, Paribas Capital Markets.. Musiela, M. & Rutkowski, M. (1997), Martingale methods in financial modelling, Vol. 36 of Applications of mathematics, Springer, Berlin. Rebonato, R. (1998), Interest-Rate Options Models, Financial Engineering, Wiley. A. d Aspremont, ORFE ORF557, stochastic analysis seminar, BCF, Sep. 2004. 40