Callable Swaps, Snowballs and Videogames

Transcription

1 Callable Swaps, Snowballs and Videogames Claudio Albanese Presented at Stanford University, October 2007

2 History of short rates (fund rates) for US dollar, the Euro and the Japanese Yen. 1

3 Brief (and incomplete) history of interest rate derivatives. Black model. Vasicek-Hull-White, CIR model and more general affine models (Duffie, Singleton, Cheyette, etc.) Market models: Heath-Jarrow-Morton and Brace-Gatarek-Musiela SABR (Hagan et al.) 2

4 Common themes. All models have some degree of analytic solvabilty thought necessary for calibration. Lattice models are implemented using implicit differentiation schemes and smoothing techniques. Measure change techniques are ubiquitous. The stochastic process for the drift of rates is not model explicitly. The stochastic process for the drift of rates is not model explicitly. Engineering implementations involve large clusters. 3

5 Real world, derivatives and building the missing link. Processes used for interest rate derivatives deviate substantially from the observed real world process and don t possess these qualitative features. See also the review paper by Q. Dai and K. Singleton (2003), Term Structure Dynamics in Theory and Reality. Review of Financial Studies 16, Consistent pricing of long dated exotic derivatives requires is very sensitive to model risk. A better agreement with the real world process is necessary. Punchline: A suitable mathematical and engineering implementation framework can be built by implementing operator methods on massively parallel GPU architectures. 4

6 Stochastic calculus Arbitrage free pricing Sigma algebra axioms Dynamics described by SDEs Pade approximants Double precision Exponential propagation of errors at high frequencies Diffusions and sparse matrices Analytic solvability CPUs and CPU clusters Calibrated market models with drift restrictions Measure changes Stochastic integrals Montecarlo methods Operator methods Arbitrage free pricing Convergence estimates on simplicial sequences Dynamics described by Markov generators Fast exponentiation Single precision Homogeneization Jump processes and full matrices Reducibility to manipulations of matrices small enough to fit in memory CPU/GPU pairs and GPU clusters Calibrated econometric models with explicit drift modeling Operator manipulations possibly without probabilistic interpretation Abelian processes Dynamic conditioning

7 Arbitrage free pricing Arbitrage free pricing is the leading pricing framework adopted in the industry. According to the fundamental theorem of arbitrage-free pricing, if one finds prices by risk neutral valuation of future payoffs and if the chosen pricing measure is mutually absolutely continuous with respect to the statistical measure, then there are no arbitrage opportunities. When using operator methods, there is no benefit in considering numeraire assets other than the money market account. Utility pricing and indifference pricing have been considered in the stochastic control literature, not (yet) with operator methods. 5

8 Measure theory based on topological spaces and operator algebras The theory of integration can be rooted on sigma algebras and measure spaces. It can also be rooted on the theory of locally compact, Hausdorff topological spaces whereby one defines integrals as linear bounded functionals over the linear space of continuous functions and then completes with respect to various norms to obtain L p spaces. The latter approach leads to operator algebras, i.e. C algebras. The second approach has the advantage from a computational viewpoint to be understandable constructively. Constructive mathematics is a branch of mathematics whereby one limits logic constructs so that any object needs to be explicitly constructed prior to applying an existential qualifier to it. Since Finance is a very applied and computational science, a fully constructive approach to Mathematical Finance seems appropriate. 6

9 Dynamics described by Markov generators Consider a finite state space Λ. A Markov generator is given by a time-dependent matrix of elements L(x, x ; t) indexed by x, x Λ that allows one to define a Markov process and construct transition probability kernles U(x, t; x, t ) as the solutions of the backward differential equation: d dt U(x, t; x, t ) + y L(x, y; t)u(y, t; x, t ) = 0 (1) with final time condition U(x, t; x, t) = δ xx. 7

10 Jump processes and full matrices Specifying an SDE is equivalent to specifying a Markov generator. When discretized, the generator corresponding to a diffusion process is a tri-diagonal matrix. However, when working with generators directly on GPU platforms, sparsity patterns do not yield any appreciable advantage and one can freely consider full matrices. The single-name propagator satisfying equation (1) is given by the so called path-ordered exponential ( t2 ) U(t 1, t 2 ) = P exp t 1 L(s)ds. (2) There are two useful methods to express a path-ordered exponential: Feynman path integrals and Dyson expansions. 8

11 Feynman path-integrals The Feynman path-integral expansion is given by P exp ( t2 t 1 L(s)ds ) where δt N = t 2 t 1 N. An application is given below. ( = lim I+δtN L(t 1 ) )( I+δt N L(t 1 +δt 2 ) )... ( I+δt N L(t N ) ) N (3) 9

12 Dyson s formula Dyson s formula is given by where P exp ( t2 t 1 L(s)ds ) = n=0 ( 1 t2 n! P t 1 L(s)ds ) n. (4) P ( t2 t 1 ) n L(s)ds = n! t2 t 1 ds 1 t2 t2 ds 2... ds n L(s 1 )L(s 2 )...L(s n ). (5) s 1 s n 1 An application is given below. 10

13 Fast exponentiation The Feynman path integral representation is interesting as this formula can be implemented numerically very efficiently on GPU architectures by the method of fast exponentiation. The method works as follows. Assume that the dynamic generators L(t) are piecewise constant as a function of time. Suppose L(t) = L i in the time interval [t i, t i +( t) i ]. Assume δt be chosen so small that the following two conditions hold: (FE1) min y Λ (1 + δtl i(y, y)) 1/2 ( t) (FE2) log i 2 = n N. δt This condition leads to intervals δt of the order of one hour of calendar time and this is indeed the choice we make. To compute e ( t) il i (x, y), we first define the elementary propagator u δt (x, y) = δ xy + δtl i (y, y) (6) and then evaluate in sequence u 2δt = u δt u δt, u 4δt = u 2δt u 2δt,... u 2 n δt = u 2 n 1 δt u 2 n 1 δt. 11

14 Internal smoothing, sensitivities and floating point errors Matrix multiplication is accomplished numerically by invoking either the single precision routine sgemm or the double precision routine dgemm. For practical purposes discussed in pricing theory, single precision is almost always sufficient. In fact, the algorithm of fast-exponentiation as described above with a δt satisfying the above bound has selfsmoothing properties which lessen the impact of floating point errors to be far less than what one would naivley expect with a back-of-the envelope worst case estimate. 12

15 A new notion of solvability The traditional notion of analytic solvability involves reducing the calculation of certain quantities to the evaluation of special functions (usually a variant of hypergeometric functions) by means of Taylor expansion or a Pade approximant (Black-Scholes, CEV, CIR, HW, Duffie s affine models, Heston model, Albanese-Kuznetsov-Lawi classification schemes, etc..). A second notion of solvability is based on asymptotic expansions (Hagan s SABR, Pieterbarg s SV-BGM expansions, Papanicolao et al. volatility models). A new notion of solvability that arises with operator methods is the reducibility of the problem to the multiplication of matrices of size small enough that all the required buffers fit in the CPU and GPU memory spaces. This definition changes quite radically the flavour of mathematical work and also moves the boundary of solvable models to include a much larger number of processes and payoffs. 13

16 CPUs and CPU clusters The traditional hardware frameworks for computational finance are given by single core CPUs over which one executes single threaded code As an alternative, Montecarlo algorithms run on CPU clusters with slow interconnects over which one spawns multiple processes typically with a job queing PVM type algorithm. 14

17 CPU/GPU pairs and GPU clusters Operator methods perform best on massively parallel multi-core architectures. A current example would be a Tesla GPU with 16 singleinstruction-multiple-data SIMD processors, each with 32 data registers and one instruction register, with 1.5GB shared fast-access memory. Such GPUs are linked to the CPU by a 100 Mhz bus on a PCI-E connection and GPU-CPU data transfer. Although this transfer rate is orders of magnitude faster than the typical inter-node communication speed in a cluster, the transfer on a bus is a possible bottleneck which needs to be avoided by executing GPU side calculations as much as possible while keeping data buffers resident in GPU memory. As a next level, GPU clusters allow one to execute loosely couple GPU jobs in parallel while being coordinated by a CPU. 15

18 Execution times in seconds for various portfolios under various configurations. Task GPU-O GPU-D Host-O Initialization Calibration to term structure of rates European swaptions Portfolio of CMS spread range accruals: 9 callable swaps and 3 callable snowballs ATM European swaptions ATM European swaptions and ATM Bermuda swaptions GPU-O: using the GPU with host side optimized code. GPU-D: using the GPU with host side debug code. Host-O: using the host only with optimized code and Intel MKL libraries. 16

19 Ratio: column 3 versus column 1.

20 Economic models without drift restrictions Market models are characterized by a large number of dynamically constrained processes. They are well suited for Montecarlo simulations but not to the application of operator methods. The models that are best suited to operator methods are specified through a minimalistic filtration without drift restrictions. These tend to be models with an intuitive economic content and explanatory power such as short rate interest rate models, credit equity models specified as defaultable equity models, stochastic skew FX models with stochastic drift, etc.. 17

21 Measure Changes A mathematical framework is largely characterized by its morphisms, the operations we allow ourselves to carry out on the objects at our disposal. In the traditional measure theoretic approach to finance, measure changes play a pivotal role as they allow one to map stochastic processes into stochastic processes. With operator methods, numeraire changes correspond to a transformation of generators of the form L G (t) = 1 G(t) L(t)G(t) + 1 G(t) G(t) t where G(t) is a diagonal operator corresponding to the new numeraire. (7) 18

22 Although this mapping allows one to rederive all the classic results of continuous time finance from analytic solvability to quanto-options and drift restrictions, this is not particularly useful to take full advantage of the formalism.

23 Operator morphisms The most useful operator manipulations are Path exponentiation Operator deformation and differentiation Operator lifting for path-dependent processes Block diagonalizations for Abelian processes Kernel splitting for dynamic conditioning 19

24 Time dependent adjustment function λ(t) to fit the term structure of rates. Notice that this function is very close to 1. This is achieved by calibrating the drift of the monetary policy process. 20

25 Zero curves in the deflation regime. 21

26 Zero curves in the regime with drift -75 bp/year. 22

27 Zero curves in the regime with drift -25 bp/year. 23

28 Zero curves in the regime with drift +25 bp/year. 24

29 Zero curves in the regime with drift +100 bp/year. 25

30 Backbone of 2y-10Y correlation, i.e. correlation between daily returns of the 2Y swap rate versus the 10Y swap rate as a function of the short rate. 26

31 Backbone of 1y-20Y correlation, i.e. correlation between daily returns of the 1Y swap rate versus the 20Y swap rate as a function of the short rate. 27

32 Projected occupancy probabilities of monetary policy regimes. 28

33 Implied volatilities for at-the-money European swaptions of tenor 2Y compared to market data. 29

34 Implied volatilities for at-the-money European swaptions of tenor 5Y compared to market data. 30

35 Implied volatilities for at-the-money European swaptions of tenor 20Y compared to market data. 31

36 Implied volatility skews for European swaptions of tenor 2Y compared to market data. 32

37 Implied volatility skews for European swaptions of tenor 5Y compared to market data. 33

38 Implied volatility skews for European swaptions of tenor 20Y compared to market data. Only implied volatilites for extreme strikes 16% over the forward failed to compute, as the graph shoes. 34

39 Implied volatility backbone, i.e. the scatterplot of the implied at the money volatility of 2Y into 2Y European swaptions versus the corresponding forward rate. 35

40 Implied volatility backbone, i.e. the scatterplot of the implied at the money volatility of 4Y into 5Y European swaptions versus the corresponding forward rate. 36

41 Implied volatility backbone, i.e. the scatterplot of the implied at the money volatility of 10Y into 20Y European swaptions versus the corresponding forward rate. 37

42 Bermuda premium backbone, i.e. the Bermuda premium of 4Y into 2Y at-the-money swaptions plotted against the corresponding forward rate. 38

43 Bermuda premium backbone, i.e. the Bermuda premium of 5Y into 10Y at-the-money swaptions plotted against the corresponding forward rate. 39

44 2Y into 5Y convexity backbone, i.e. the convexity correction for of 2Y into 5Y European constant-maturity-swaps (CMS) with respect to European swaptions. 40

45 3Y into 2Y convexity backbone, i.e. the convexity correction for of 3Y into 2Y European constant-maturity-swaps (CMS) with respect to European swaptions. 41

46 10Y into 20Y convexity backbone, i.e. the convexity correction for of 10Y into 20Y European constant-maturity-swaps (CMS) with respect to European swaptions. 42

47 Pricing functions of callable CMS spread range accruals. 43

48 Pricing functions of callable CMS spread range accruals. 44

49 Pricing functions of callable CMS spread range accruals. 45

50 Pricing functions of callable CMS spread range accruals. 46

51 Pricing functions of callable CMS spread range accruals. 47

52 Pricing functions of callable CMS spread range accruals. 48

53 Pricing functions of callable CMS spread range accruals. 49

54 Pricing functions of callable snowball CMS spread range accruals. 50

55 Pricing functions of callable snowball CMS spread range accruals. 51

56 Pricing functions of callable snowball CMS spread range accruals. 52

57 Conclusions Operator methods are an emerging mathematical framework for finance and econometrics which is suitable for semi-parametric and non parametric modeling. We showed an example concerning long dated fixed income derivatives but I worked out several others concerning credit, equity and energy derivatives. Just google my name. The practical engineering applications of operator methods rely on fast implementations of matrix-matrix multiplication algorithms, which can nowadays be nest achieved on massively parallel GPUs optimized for single precision floating point arithmetics. 53