Semi-analytic Lattice Integration of a Markov Functional Term Structure Model

Transcription

1 Semi-analytic Lattice Integration of a Markov Functional Term Structure Model... Christ Church College University of Oxford A thesis submitted for the degree of Master of Science in Mathematical Finance Hilary 9

2 Abstract One common use of Markov functional models is to approximate LIBOR market models and to avoid complications the terminal forward measure is typically used. If this method is applied to long term structures (ten or more years), the distribution of the early LIBORs in the term structure has a very large tail, which is normally not completely captured by common numerical techniques (either Monte Carlo or grid-based methods). A numerical method that is frequently applied to Markov functional models is known as the semi-analytic lattice integrator (Sali) tree. This thesis examines the implications of the long tails on the Sali tree. Adequate boundary conditions and grid sizes are derived in order to capture the effect of the long tails. It turns out that this method either exhibits stability problems or demands for a relatively small lattice spacing. The reason for this is examined in detail and several variations of the Sali tree to avoid this effect are suggested and analysed. Furthermore the optimisation of the grid parameters is considered in order to reduce the necessary computation time.

3 Contents Introduction Preliminaries 3. Markov Functional Models Definition Sali-Trees Cubic Splines Stability of Cubic Splines Markov Black-Derman-Toy Model 7 3. The Model Analytical Calibration Numerical Approach Approaches to Improve Convergence 8 4. Other Spline Types Akima Interpolation Tension Splines Definition Selection of Tension Factors Results Splitting off the Asymptotic Behaviour Optimising the Grid 9 5. Grid Size Lattice Spacing i

4 6 Conclusion Summary Outlook A Analytical solution of the MBDT-Model 39 A. Calibration A. Estimation of the Tails A.. Non-Asymptotic Contributions A.. Integral over the Tail B Tension Splines 4 C Expectation Values of Splines 44 C. Cubic Splines C. Tension Splines D Combined Approaches to Improve Convergence 46 Bibliography 49 ii

5 List of Figures 3. Analytic L L i for different tenors and levels of volatility I t (x) analytic according to eq. (3.) comparison of a i and å i Effect of h Oscillations of the cubic spline approximation Effect of h with Akima interpolation Effect of h with splines under tension Tension factor σ Effect of h on L L with spilt asymptotics f for n = and ψ = Sali L L i for different tenors and levels of volatility Effect of different tail extrapolations on L L with spilt asymptotics 3 5. Grid size needed for calibration in figure Sali L L i for different tenors and levels of volatility with optimised grid D. Effect of h on L L with spilt asymptotics and Akima splines D. Effect of h on L L with spilt asymptotics and splines under tension 48 iii

6 Chapter Introduction Markov functional models, suggested by Hunt, Kennedy and Pelsser [, 9] as tools for pricing exotic derivatives, have the characteristic property that the discount bond prices are at any time a function of some low dimensional Markov process. With LIBOR market models they share the easy calibration to market prices, but due to the low dimension of the random process they allow for a much more efficient implementation. So one common use of Markov functional models is to approximate LIBOR market models. For simplicity, Markov functional models are typically formulated in the terminal forward measure. If this method is applied to long term structures (ten years or longer), the distribution of the early LIBORs in the term structure has a very large tail, which is typically not captured completely by common numerical methods. Either the grid or tree is too small or a vast number of Monte Carlo steps would be necessary to capture these contributions. A method that is often used to implement Markov functional models is the semianalytic lattice integrator (Sali) tree [6]: The backward integration is done on a grid using exact formulae to integrate piecewise defined functions against the propagation kernel of the driving stochastic process, interpolating the resulting function at the prior time. The goal of this thesis is to analyse the effect of the large tails observed for long term structures on the accuracy of the Sali tree. The determination of an adequate grid-size will be examined as well as a proper treatment of the semi-infinite intervals beyond the grid. This thesis is structured as follows: In chapter basic concepts are introduced. Markov functional models are defined and the Sali tree will be outlined. Special attention is payed to cubic splines that are normally used as basic functions.

7 This is followed by the definition of a model that is still analytically solvable but complex enough to show the main characteristics of a realistic Markov functional model in chapter 3. The analytical calibration of the model is presented as well as a Sali approach. In contrast to the standard Sali tree, the contributions of the semi-infinite intervals beyond the grid are taken into account in order to keep the effects of the long tails and to derive realistic boundary conditions for the cubic spline interpolation on the finite grid. In chapter 4 three variations of the Sali approach are introduced with the goal to achieve better results at larger lattice spacing. Two are based on different spline types, the third is based on splitting off the exponential behaviour that is responsible for the long tails and apply the Sali approach to well behaved functions. The purpose of chapter 5 is to optimise the grid in order to get precise results with a minimum of computation time while chapter 6 summarises the results.

8 Chapter Preliminaries. Markov Functional Models In practice, exotic derivatives are priced by calibrating a model to the market prices of liquid simple derivatives and then using this model to price the exotic derivative. As such the role of the model could be described as an extrapolation tool. As the model should describe prices in an efficient market it must be arbitrage free. For a practical applications two other features are important. The model should be well-calibrated, i.e. correctly price a large class of liquid instruments without over-fitting allow for an efficient implementation Hunt, Kennedy and Pelser [, 9] suggested a Markov functional model for this purpose, where the randomness comes through a low dimensional Markov process and the interest rates are a functional of this random process. In a single currency economy a Markov functional model can be described as follows: Let D tt be the value at time t of a zero coupon bond maturing at time T with D TT =. The underlying assets should be a finite number of these bonds with T T = {T i i =,...,n}. This is enough for the present purpose, but the treatment can be generalised to an infinite number of underlyings. Let F t be the filtration representing the information available at time t. Any trading strategy in the market should be self-financing. The value of a portfolio generated by such a trading strategy is called a price process and any price process that is positive almost surely is called a numeraire. Given a numeraire N we assume there is a measure N equivalent to the natural measure P such that the process (D tt /N t ) T T is an {F t } martingale. 3

9 We assume that any derivative can be replicated by a self financing portfolio. If we further assume that there is a time limit, at which the value of a derivative is determined purely by the asset prices, the value of this derivative at any earlier time is given by V t = N t E N [V /N F t ] = N t E N [V T /N T F t ] (.) for t T... Definition Let X t be a time inhomogeneous Markov process under N and the boundary function S : [, ] [, ] with S S a real function defining the boundary times up to which the following assumptions on the discount bonds are valid. We assume that the prices of the pure discount bonds are a function of the random process: D ts = D ts (x t ) for t S (.) the same should apply to the numeraire N: N t = N t (x t ) for t (.3) Then a Markov functional interest rate model is completely determined by. the law of the process X under N. the functional form of the numeraire N t (X t ) for t 3. because of equation (.) we do not need the form of D ts (x t ) for all times. It is sufficient to know the functional at the boundary t = S. The calibration of the model demands the determination of these three elements.. Sali-Trees A numerical method which is well suited for the treatment of Markov functional models is the semi-analytic lattice integrator (Sali) tree [3, 6]. Let T = {t,t...t n } be a set of discrete time steps with t i < t j for i < j. Assume that the conditional pdf f(x t x t ) is available for all t,t T, t < t. 4

10 The calculation of the expectation value of a quantity V at a time step t i from the distribution at t i+ V i (x i ) = E i [V i+ x i ] = f(x i+ x i )V i+ (x i+ )dx i+, (.4) where x i is a short notation for x ti, is done in two steps:. The integral is evaluated at time t i for a finite number of grid points x t,k, k =...n i. So the method relies on the fact that the function V i+ is such that the integral (.4) can be evaluated either analytically or by efficient numerical integration. This is further discussed in the next paragraph.. Then the function V i (x i ) is approximated by fitting a spline or another piecewise polynomial function to the grid {x i,k } k=...ni, leading to a function Ṽi(x i ). To do the next time-step, we have to integrate Ṽi(x i )f(x i x i ). As Ṽi is piecewise a linear combination of the base functions b k (x), this can be done if the integral xi,j+ x i,j b k (x i )f(x i x i )dx i can be evaluated efficiently. This is certainly the case if X t is a Brownian motion and cubic splines or any other set of piecewise polynomial interpolation functions are used. The integrals relevant for this case are given in Appendix C.. To get the iteration started, the payoff V T is approximated by the base functions b k (x) as well. In general a much smaller number of grid points is needed compared with conventional trees to get a similar precision. A notable strength of the Sali-tree comes with payoff functions that are non-continuous either themselves or in their first derivative. It is possible to define domains of integration where the payoff is well-behaved. This major advantage of the Sali-tree is described in [6]. As it is not needed for the calibration of the model described in the next chapter this point will not be elaborated further..3 Cubic Splines The following section summarises properties of cubic splines that are relevant for this thesis. A detailed introduction can be found in most textbooks about numerical mathematics such as [4] Let = {a = x < x <... < x n = b} be a partition of the interval (a,b) and f : (a,b) R a real function. The cubic spline for f on is a function S : (a,b) R with the following properties: 5

11 S (x i ) = f(x i ) for all i {,...,n} S C (a,b), i.e. f is twice continuously differentiable On each subinterval (x i,x i+ ) S corresponds to a polynomial of order three. To make the cubic spline unique, two further equations are needed which are usually chosen to be a condition imposed on the spline s derivatives at the boundaries. Common boundary conditions are either vanishing second derivative or a fixed first derivative. The first case is usually referred to as a natural spline. Determining the parameters of the polynomials boils down to a linear equation. The implementation used in this work is taken from [5]..3. Stability of Cubic Splines Cubic splines are guaranteed to converge towards the original function with diminishing distance between the grid points. To be more precise: Let f C 4 (a,b), assume an L R exists, so that f (4) (x) L x [a,b]. Given a sequence of grids m = {a = x (m) < x (m) <... < x (m) n m = b} with maximal lattice spacing m = max(x (m) i+ x(m) i ) (.5) i and sup m,i m x (m) i+ x(m) i K (.6) for some number K R, then for i {,,, 3} constants C i independent of m exist, so that for all x [a,b] f (i) (x) S (i) m (x) C i LK m 4 i. (.7) This ensures that S m will eventually converge to f. As the derivatives of f enter the right hand side of (.7) via L, the lattice spacing necessary to obtain a decent precision might be prohibitively small. Note that (.6) is always fulfilled as long as equidistant grids are used. Hall and Mayer [8] were able to prove the following estimates for c i = C i K: c = 5/384, c = /4, c = (K + K )/, where c and c are optimal. 6

12 Chapter 3 Markov Black-Derman-Toy Model 3. The Model To demonstrate the effects of the long tails we choose a model that is still analytically tractable but shows the main features of a Markov functional model for a term structure of LIBOR rates. It can be considered as the Markov functional version of the Black-Derman-Toy model [7]. In this work it will be referred to as Markov Black-Derman-Toy model or MBDT model. Let L i be the LIBOR-rate from time T i to T i+ and δ i the accrual factor for that period. In the terminal measure, i.e. using the last discount bond D Ti,T n as a numeraire, we assume L i = L i + ( exp q i ψ(t i ) T i g(t)dw t q i ψ(t i ) ) T i g(t) dt q i (3.) where ψ and g are positive, real functions of time, < q i, Li positive, real numbers and W t is a standard Brownian motion under the terminal measure. We consider only the lognormal case, i.e. q i =. The integral W Gi = T i g(t)dw t is a normal variable with variance G i = T i g(t) dt and can thus be considered as a time changed Brownian motion. So (3.) with q i = depends only on G i and ψ i = ψ(t i ): L i = L i exp ( ψ i WGi ) ψi G i = L i E i ( ψ i WGi ) (3.) where E is Doléan s exponential of a contiuous martingale X t : E t (X) = exp(x t var(x t)). (3.3) 7

13 and we use the shorthand E i (X) = E ti (X). Note that E s [E t (X)] = E s (X) for all s t. We use the short form D i,j = D Ti,T j ( W Gi ) with i < j for the discount bonds and D i,j = D i,j D i,n for the numeraire adjusted discount bonds in the terminal measure. 3. Analytical Calibration Assume that ψ i and G i are given. So for the model calibration we only need to determine the L i from the market values at time t =, i.e. D,i. With we arrive at D i,i+ D i,n = D i,i+ = ( + δ i L i (T i )) (3.4) D j,i = D j,i D j,n [ ] = E j D i,n [ ] = E j Di,i+ ( + δ i Li E i ( ψ i WG )) (3.5) (3.6) where j < i and E i [ ] = E[ F Ti ] is the expectation value given the filtration at time T i. For j = in particular we get D,i = D,i+ + δ i Li E [E i ( ψ i WG ) D i,i+ ] (3.7) which allows to calculate L i from the initial prices of the zero bonds D,i and D,i+ and the distribution of D i,i+ ( W Gi ). Starting with D j,n = j n (3.8) we can determine L i by induction. In Appendix A. it is shown that D i,i+ has the form D i,i+ = n i j= X i.j E i ( Y i,j WG ) (3.9) with constants X i,j and Y i,j given by (A.3,A.4) that depend only on those L j with j > i. We can thus get L i using (3.7): L i = D,i D,i+ δ i E [E i ( ψ i WG ) D i,i+ ] (3.) From (3.6) with i = j we get the functional form of the numeraire D i,n. Together with the definition of the Markov process W and the form of the discount bonds at 8

14 the boundary D n,n = the definition of the Markov functional model is complete according to section... Below results for L are shown with the following parameters: The value of the discount bonds at time T are chosen to get a flat initial LIBOR rate of L = 5% and different tenor structures with yearly resets and annual compounding are used. For simplicity we choose G i = T i. Figure 3. shows the convexity adjustment L L for different time independent levels of the volatility ψ Figure 3.: L L with L calculated analytically according to (3.), a flat initial LIBOR rate of L = 5%, a tenor of (from top left), and 3 years and different values of the volatility ψ, which is assumed to be time-independent 9

15 Now we have a closer look on the functional form of Di,i+ (x t ). As can be seen from equation (3.9) it is the sum of terms that each are proportional to exp( Y i,j x). The expectation values of these terms are ( E t [exp( Y i,j x)] = exp ( )) (x xt ) + Y i,j x dx π(gi+ G t ) G i+ G t ( ) = π(gi+ G t ) exp Y i,j(g i+ G t ) Y i,j x t exp ( (G i+ G t ) ( ) ) x xt + Y i,j dx. (3.) G i+ G t For a given x t the main contribution to the integral is at x = x t Y i,j (G i+ G i ) where, according to (A.4) Y i,j could get as large as n i+ ψ i, leading to contributions far from the central value x =. This behaviour is most striking in the denominator of (3.), where t =. Figure (3.) shows the integrand I i (x) = E i ( ψ i WGi ) D i,i+ n(x;g i ). (3.) For e.g. i = 4 there are significant contributions at x, six times the standard deviation from the central value. The observation described above will be of interest as soon as we use numerical methods to calibrate the model. Trees, finite differences and Monte Carlo methods all have some kind of finite cut-off for x. The first two explicitly by the grid size, the later implicitly, as a finite number of runs will lead to a negligible probability several standard deviations from the central value. For a Monte Carlo integration that uses a uniform sampling of the distribution W Gi over a region including six standard deviations from the central value, about Monte Carlo steps would be necessary. The effect for Monte Carlo simulation has already been investigated by Merrill Lynch Quantitative Risk Management []. 3.3 Numerical Approach The expense to calculate the L i numerically using the above analytic solution is of order n, which starts to become prohibitively large for longer tenors. So even in this case of an analytically solvable model a numerical approach like the Sali-tree is necessary. As there are no discontinuities involved in D, we do not have to care about different domains of integration. Let x i,k be the grid for the underlying stochastic process W G at time-step i and D i,j the Sali approximation for D i,j. Then the Sali step i (i ) is:

16 ,8, ,4, ,4,3, , Figure 3.: I t (x) (eq. 3.) for a tenor of years, for several t and ψ =.5 (above) and ψ =.3 (below). The other parameters are chosen as in fig. 3.. While we see only a slight asymmetry for ψ =.5, the contributions far from the central value are significant for the larger volatility.

17 . Starting with D i,i+ calculate L i using (3.).. For each x i,k evaluate ( D i,i ) k = E [ ( + δ i Li exp( ψ i x i ψ i G i /)) D i,i+ WGi = x i,k ] (3.3) 3. From these ( D i,i ) k determine D i,i by fitting the splines. Here we consider cubic splines that are continuous in their second derivative as described in section.3. Apart from the question which splines to use, the main decisions necessary for applying this method are the boundary conditions and the placement of grid points. For the time being we assume equally spaced grid points ( x, x + h,..., x) with h = x/n, but we have a closer look at the boundary conditions: As the single summands in (3.9) grow exponentially for x <, natural boundary conditions, i.e. vanishing second derivative at the boundaries, are clearly inappropriate. Instead, we try to get a reasonable estimate for the first derivative at the boundaries. As the method that is developed here should not be limited to the simple case of a model that is in principle solvable analytically, we will not use detailed knowledge about the analytical form of D i,n (x i ). Instead, estimates for the asymptotical behaviour of D i,n (x i ) for x i and x i are needed. For x i from (3.) we get L j a.s. j > i. So D i,n (x i ) for x i As L j is monotonous in x i, this is also true for D i,n (x i ) and we may assume a zero first derivative at the upper boundary. To determine the boundary conditions for x i, define j = x j x i for j > i. From (3.4) and (3.) we get for x D j,j+ (x j ) = ( + δ i L i (x i + j )) exp(ψ j (x i + j )) (3.4) and therefore E[D j,j+ W Gi = x i ] exp(ψ j x i ) (3.5) With (3.5) D i,i+ [ ] = E WGi = x i D i+,n ( ) n exp x i ψ j j=i+ (3.6)

18 So we assume that D i,i+ (x i ) has the asymptotic form D i,i+ (x i ) b i exp( a i x i ) for x i (3.7) and the boundary condition at x i = x is D i,i+ = a i b i exp( a i x i ). To determine a i and b i we could either set a i to å i = n j=i+ ψ j and determine b i from the first grid point b i = exp(a i x)( D i,i+ ) or we could determine both a i and b i from the first two grid points: a i = ln(( D i,i+ ) /( D i,i+ ) )/h (3.8) b i = ( D i,i+ ) exp(a i x). (3.9) For a correct estimation of the asymptotic behaviour it is important to choose x large enough so that the asymptotic behaviour dominates all other terms for x > x. An estimation of D i,i+ (x i )/(b i exp( a i x i )) for x > x would be helpful but in most practical cases unrealistic to achieve. Instead, the difference å i a i will be used as a consistency check to see if x is chosen large enough so that the asymptotic behaviour can be assumed for x < x. x will be chosen so that å i a i stays within a fixed interval ( δ a,δ a ) for all i. In this special case of an analytically solvable model, the role of the asymptotic behaviour can be seen by comparing it to the analytic solution (3.9). The exponential with maximal coefficient is singled out as the leading term for x. It depends on the factors X i,j what value of x is necessary for this exponential to dominate the others. For the derivation of an upper limit to the non-asymptotic terms see appendix A.. The result justifies the use of å i a i as an indicator for an adequate grid size. The boundary effects on the first grid points have to be taken into account. When taking the expectation value (3.3) it is insufficient to calculate the integral from x to x as this will lead to wrong results for x k close to ± x and to wrong boundary conditions. At each further time-step these errors will cause deviations closer to the centre of the grid. Instead, the integral for the semi-infinite intervals (, x) and 3

19 ( x, ) will be approximated by the integral over the asymptotic behaviour: ( D i,i ) k = ( + δ i Li exp( ψ i x ψ i G i /)) D i,i+ (x)n(x x k ;G i G i )dx b i = + + x N j= x N j= + b i e a i x n(x x k ;G i G i )dx (x i ) j+ (x i ) j ( + δ i Li e ψ ix ψ i G i/) Di,i+ (x)n(x x k ;G i G i )dx n(x x k ;G i G i )dx (3.) (x i ) j+ (x i ) j ( + δ i Li e ψ ix ψ i G i/) Di,i+ (x)n(x x k ;G i G i )dx e a i x k +(G i G i )a / ( + erf ( ( )) + x x k erf (Gi G i ) ( )) x x k + a(g i G i ) (Gi G i ). (3.) As D i,i+ is a polynomial of third order in each of the intervals (x i,j,x i,j+ ), the integral can be solved analytically. To verify, whether x is large enough for the asymptotic behaviour described above to be a good approximation, we compare the theoretical value å i to the value calculated from the to first grid points using (3.8). Figure 3.3 shows the numerical value for a i compared to the theoretical value å i for n =, h =.5, ψ {.5,.3} and several values of x as a function of the time-step i. As we can see, the asymptotic behaviour described above is a good approximation for these parameters if x > 5. For ψ =.3 the effect of the contributions far from the central value is clearly visible, as even with x = 3 the asymptotic behaviour can be observed for large i, but large deviations from the asymptotic behaviour can be observed, as soon as the additional peak shown in figure 3. becomes significant. In the present and the next chapter numerical examples are with a tenor of and ψ (.5,.3). Figure 3.3 illustrates that a grid size given by x = 6 is fully appropriate. In chapter 5. the choice of an optimal grid size will be investigated further. 4

20 a i 3 6,5 analytical a i 5 4 a 3 4 5,5 3,5 5 5 i 5 5 i Figure 3.3: å i and a i according to (3.8) for n =, h =.5, different grid sizes x and ψ =.5 (left) and ψ =.3 (right) After determining the boundary conditions and the grid size, the spacing between the grid points must be set. For the time being we stick with equally spaced points and vary h. Figure 3.4 illustrates that h has to be chosen relatively small to get reliable results. For any larger grid spacing the numerical solution is good up to some point i. For any j < i L j is practically zero. The reason for this lies in a well known problem with cubic splines (see e.g. [4, ]): Cubic splines are a global interpolation method, which means that every grid point affects the parameters for the spline in every single interval. This can lead to oscillations in the whole domain. For a series of lattices m on a finite interval [ x, x] with lattice spacing m, a function f C (4) [ x, x] and corresponding cubic spline S m the convergence theorem for cubic splines (see section.3.) states that S m does converge uniformly to f. But this convergence is influenced by the fourth derivative of f: f(x) S m (x) c L m 4 (3.) with f (4) (x) L x [a,b] and c = 5/384. In our case we have a series of functions f i that grow exponentially. At the lower boundary f i (x) b i exp( a i x). So L f (4) i ( x) f i ( x)a 4 i and as soon as c (a i h) 4 > the error might even be larger than the function value itself. Figure 3.5 shows D i,i+ (x) for i =,, 9, 8. It can clearly be seen that first even with well behaved grid points the oscillations set in and at a later step these oscillations lead to implausible grid points. In principle this problem could be solved by choosing h small enough, but this would lead to excessive need of computing time. The following chapter will show alternative approaches. 5

21 analytic i analytic i Figure 3.4: The effect of h on the quality of the Sali approach to L L i with n =, x = 6, ψ =.3 (above) and ψ =. (below) 6

22 x x i = i = x i = x i = 8 Figure 3.5: Di,i+ (x) for i =,, 9, 8 for n =, x = 6, ψ =. and h =. The dotted line connects the values at the grid points, the solid line shows the cubic spline defined by these points. It can clearly be seen that the oscillations of the spline that set in at i = lead to inconsistent values at the grid points for smaller i. 7

23 Chapter 4 Approaches to Improve Convergence To handle the oscillations that were observed for the spline approximation, basically two approaches can be used. Either the method used to calculate the splines can be varied or the function that is approximated can be changed. Both ideas are further investigated in the following sections. 4. Other Spline Types Though cubic splines are much less prone to over-oscillation than e.g. fitting of a polynomial, the phenomenon is well known in the literature (see e.g. [4, ]). This is in part due to the cubic spline s lack of locality. A local change in the input data will modify the curve even far away from this point. Several other versions of splines have been presented to handle this problem. We will use Akima interpolation [] and tension splines [3, 4, 4]. Let again be = {a = x < x <... < x N = b} a partition of the interval (a,b) and y...y N R. As for cubic splines we search a function y : (a,b) R with y(x i ) = y i i N, but the additional conditions that make y unique differ for each type of splines. 4.. Akima Interpolation Like cubic splines, Akima interpolation is based on cubic polynomials. The condition of a continuous second derivative is abandoned and instead the first derivative at a given grid point is estimated using the neighbouring points. With q i = y i y i x i x i for i {,...N} (4.) 8

24 the first derivative at x i is estimated as y i = q i q i+ q i+ + q i+ q i+ q i+ q i+ q i+ + q i+ q i+ for i {3,...N } (4.) If q i+ = q i+ and q i+ = q i+ then y i can not be derived from (4.). In this case y i = (q i + q i+ )/. The derivative at the first two points and the last two points has to be chosen by other means. For D j,j+ (x j ) we use the asymptotic behaviour described in section 3.3 and assume D j,j+(x j,i ) = { aj b j exp( a j x j,i for i {, } for i {N,N } (4.3) with a j and b j from (3.8) and (3.9). Considering an interval (x,x ) with y,y,y,y given, the polynomial can be expressed as y(x) = p + p (x x ) + p (x x ) + p 3 (x x ) 3 (4.4) where p = y (4.5) p = y (4.6) p = [3(y y )/(x x ) y y ]/(x x ) (4.7) p 3 = [y + y (y y )/(x x )]/(x x ). (4.8) (4.9) The integrations needed to determine the expectation values of the polynomials can again be done using appendix C. Figure 4. shows L L i calculated using a Sali tree with Akima interpolation. Compared to figure 3.4 it shows no abrupt transition to the over-oscillating behaviour and an overall better convergence. 4.. Tension Splines Splines under tension, first suggested by Schweikert in 966 [3], are a common tool for shape preserving interpolation, i.e. for interpolation that should avoid the spurious oscillations observed for cubic splines. The name comes from the fact that the curve can be interpreted as a very light and flexible bar, that is not only constrained to run through certain points, but is also pulled at the ends. The strength of this tension is described by a tension factor σ. 9

25 analytic i analytic i Figure 4.: The effect of h on the quality of the Sali approach with Akima interpolation instead of cubic splines. The graphs show L L i with n =, x = 6, ψ =.3 (above) and ψ =. (below)

26 4... Definition For σ = no tension is applied, which should lead to normal cubic splines. For σ the tension should minimise the length of the curve, leading to straight lines connecting the given function values, thus avoiding spurious oscillations but loosing smoothness and any non-spurious extrema. The derivation of the formulation of splines under tension used here can be found in [4]. It is summarised in appendix B. The basic assumptions are a continuous second derivative of y and piecewise linearity of the term y (x) σ y(x) in each interval (x i,x i+ ). This leads to the following form of the interpolation: for x (x i,x i+ ), where y(x) = y() i+ sinh(σ (x x i )) + y () i sinh(σ (x i+ x)) sinh(σ (x i+ x i )) + (y i+ y () i+ )(x x i) + (y i y () i )(x i+ x) x i+ x i (4.) σ = σ xn x n (4.) is used instead of σ and the parameters y () i are determined by a system of linear equations very similar to cubic splines. Boundary conditions are needed either for y (x j ) or y (x j ) with j {,N}. The same conditions as in section 3.3 are used. An algorithm to determine y () i is given in [5]. The use of σ instead of σ is to avoid scaling effects when the grid size is changed. The integrations that are needed to determine the expectation values of the tension splines can still be done analytically. The results are given in appendix C Selection of Tension Factors Special care has to be taken when choosing the tension factor σ. If it is too small, the same problems as with cubic splines will be observed. If it is chosen too large, smoothness will be lost and extrema will be underestimated. If additional conditions are know about the function, like convexity or bounds on a derivative, it is in principle possible to determine the set of σ for which the constraint is satisfied []. In this work no such conditions are used as they would limit the applicability of the results to more complex models for which no analytical solution is known. So another way has to be found to rule out values of σ that lead to unstable results. To avoid spurious oscillations as observed in figure 3.5, σ will be chosen so that within

27 any two adjacent intervals (x i,x i ), (x i,x i+ ) there is at most one inflexion point. The derivatives of y in I i = (x i,x i+ ) are: [ ] y (x) = σ y () i+ cosh(σ (x x i )) y () i cosh(σ (x i+ x)) / sinh(σ h i ) [ ] + y i+ y i y () i+ + y() i /h i (4.) [ ] y (x) = σ y () i+ sinh(σ (x x i )) + y () i sinh(σ (x i+ x)) / sinh(σ h i )(4.3) with h i = x i+ x i. As all sinh terms in y are positive for x i < x < x i+, y has no zeros if y () i+ y() i >. Now assume y () i+ y() i <. As y (x j ) = σ y () j, the second derivative has at least one zero in I i. y (x) = y () i+ sinh(σ (x x i )) = y () i sinh(σ (x + x)). (4.4) As the left hand side is zero for x = x i and strictly increasing in I i and the right hand side is zero for x = x i+ and strictly deceasing in I i, exactly one x I i exists with y (x) =. With the same reasoning we see that if y () i =, there are no further inflexion points in (x i,x i+ ), though y might be constantly zero, this is still at most one inflexion point. To summarise: If y () i+ y() i > there is no inflexion point in I i, with y () i+ y() i < there is exactly one inflexion point in I i and with y i = there is at most one inflexion point in (x i,x i+ ). When the calibration of the splines is started, σ is set to a positive, but small value. When two inflexion points are observed within two adjacent intervals, σ is increased by a fixed amount σ until no two such intervals exist. Though y (x) x I i, this convergence is not uniform. So it can not be guaranteed that the condition can be fulfilled for any σ <. Due to this fact σ will only be increased up to a maximum value σ max. Then σ max will be used as tension factor. The performance can be optimised by choosing σ relatively large and then optimising σ by nested intervals Results Fig. 4. shows the result of the model calibration for n = and ψ {.,.3}. Though the results are much better than for the simple cubic splines, the splines under tension show no improvement compared to Akima interpolation. Figure 4.3 shows the values of σ p determined by the algorithm described above for the same cases. The value of σ max = is not reached, so the condition of no

28 analytic i analytic i Figure 4.: The effect of h on the quality of the Sali approach with splines under tension instead of cubic splines. The graphs show L L i with n =, x = 6, ψ =.3 (above) and ψ =. (below) 3

29 oscillations is always fulfilled. The deviation of the Sali result using splines under tension is not due to oscillations, but due to a loss of smoothness caused by the high tension factor. σ σ δ x = δ x = δ x =.5 δ x =.5 3 δ x = δ x = δ x =.5 δ x = i 5 5 i Figure 4.3: Tension factor σ as a function of the time-step i for n =, x = 6, ψ =.3 (above) and ψ =. (below) 4. Splitting off the Asymptotic Behaviour In the last chapter we saw that cubic splines are unsuitable to handle exponential growth. The basic idea discussed in the present section is to keep the cubic splines and split of the exponential asymptotic behaviour instead. Consider the two terms that have to be integrated during a Sali step (3., 3.3). Using we define f s,i for s {, } by g s,i (x i ) = ( + b (s) i exp( a i x i )) s {, } (4.5) g,i (x i ) f,i (x i ) = exp( ψ i x i ψ i G i /) D i,i+ (x i ) (4.6) g,i (x i ) f,i (x i ) = ( + δ i Li exp( ψ i x i ψ i G i /)) D i,i+ (x i ) (4.7) where the b (s) i are chosen so that f s,i( x) =. If b (s) i the asymptotic behaviour will not be split off, a normal Sali-step will be performed instead. Using the asymptotic behaviour of D discussed in Section 3.3, we get: f s,i(x i ) for x i. (4.8) Natural boundary conditions, i.e. f s,i(x i ) = at x = ± x will be used for f s,i. The exponential term that has been split off can be handled analytically: E j [exp( ψ i x i ψ i G i /) D i,i+ (x i )] = E j [( + b () i exp( a i x i )) f,i (x i )] 4

30 = = = + b () i + b () ( + b () i exp( a i x i )) f,i (x i ) n(x i ;G i G j ) dx i f,i (x i ) n(x i ;G i G j ) dx i exp( a i x i ) f,i (x i ) n(x i ;G i G j ) dx i f,i (x i ) n(x i ;G i G j ) dx i (4.9) i e a i (G i G j ) f,i (x i a i (G i G j )) n(x i ;G i G j ) dx i and = E j [( + δ i Li exp( ψ i x i ψ i G i /)) D i,i+ (x i ) x j ] + b () f,i (x i ) n(x i x j ;G i G j ) dx i (4.) i e a i (G i G j ) f,i (x i a i (G i G j )) n(x i x j ;G i G j ) dx i From here on we can proceed like in section 3.3 with the sole exception that the cubic spline interpolation is used on f n,i which should lead to a much more stable behaviour. The tails x > x and x < x are less problematic in this case, as f s,i (x) for x < x and f s,i (x) s for x > x. Nevertheless they will be taken into account and f s,i (x) will be assumed to be constant outside ( x, x). Figure 4.4 shows L i for different values of h and the same parameters as in figure 3.4 with the only difference that the asymptotics are split off here. As it can clearly be seen the stability is improved significantly. The deviations observed for h = are caused by oscillations as those observed in figure 3.5. This can be seen in figure 4.5. Figure 4.6 shows the result of the Sali-approach using cubic splines with split-off exponential behaviour for the same set of tenors and volatilities ψ as in figure 3. using the analytical solution. With h = the analytical solution can be reproduced in most cases except for long tenor and very high volatility. To get an impression whether x = 6 is large enough so that the asymptotic behaviour is a good approximation for 5

31 analytic i analytic i Figure 4.4: The effect of h on the quality of the Sali approach with split off asymptotic behaviour. The graphs show L L i with n =, x = 6, ψ =.3 (above) and ψ =. (below) 6

32 x Figure 4.5: f,i according to (4.7) for different time-steps i (n =, ψ =., h = ). all x < x, the limit to the non-asymptotic terms c (A.4) is monitored. For all sets of tenor and volatility considered and every time step it is smaller than.. The combination of the methods described above, splitting off the exponential behaviour and using a spline type that is less prone to overoscillations does show an improvement compared to the simple use of the other spline types but is not as good as using the normal cubic splines after splitting off the exponential. As long as no overoscillations occur, the simple cubic splines seems to be best suited among the examined spline types. For completeness, figures showing these results are given in appendix D. When applying this method to other models it is not necessary to identify the exact asymptotic behaviour. It suffices to identify a function g s,i (x) so that f s,i (x) is well behaved in the sense that it shows less spurious oscillations than the original function g s,i (x) f s,i (x) and the integrations involved can still be performed in an efficient manner. 7

33 Figure 4.6: L L with L calculated using a Sali tree with split off asymptotic behaviour, a tenor of (from top left), and 3 years and different values of ψ with x = 6 and h = (symbols). The lines show the analytic results from figure 3. for comparison 8

34 Chapter 5 Optimising the Grid Up to now no attention has been paid to the structure of the underlying grid. In principle, three aspects can be considered to optimise the grid : The size of the grid given by the maximal value x. If this is chosen too small, essential contributions far from equilibrium might get lost. If x is chosen too large, computation time will be higher than necessary. In addition this may increase stability problems at the lower end of the grid. The value of the lattice spacing h. For each interpolation technique used up to now, the effect of changing h has been demonstrated (fig. 3.4, 4., 4. and 4.4). The conflict between performance and stability should be solved such that h is small enough to get an accurate result but within this restriction as large as possible. The concept of equidistant points can be abandoned in favour of a lattice that is adapted to the structure of the function. The convergence theorem for cubic splines clearly favours a homogeneous grid, as this case allows for a minimal factor K (.6) compared with other grids with the same size and number of grid points. So within this work only homogeneous grids will be considered. The optimisation should be done with the goal to minimise computation time while keeping an acceptable level of precision. As the integration (3.) has to be done for each single grid point and the integration implies a sum over all grid points, the computing time is of order O(N ) as long as the number of grid points is the same for each time-step. But the Sali method does not rely on a constant grid for all time steps. So all of the above adjustments can be done during the calibration for each single time-step. 9

35 The optimisation will be done on the basis of cubic splines with split-off asymptotic behaviour from section 4., as these led to the most reliable results. 5. Grid Size According to equation (4.9) two integrations are done over f,i. The first with a normal distribution centred at x =, the other centred at x = a i (G i G j ), both with variance G i G j. So it must be guaranteed that a reasonable vicinity of both points is covered by the grid. In the following a range of ±3 G i G j is considered to be sufficient. The validity of this assumption is confirmed a posteriori using the bound on non-asymptotic terms derived in appendix A... As in the first part of a Sali-step the integration (4.9) is done with j =, the grid should at least cover the interval G i = (x,x N ) = ( a i G i 3 G i, 3 G ) i. (5.) The last step i = earns special attention. All other values of x are integrated over, averaging out small oscillations, but for i = the limit of zero variance is reached and the two integrals in equation (4.9) evaluate as f, (). To obtain maximum precision for this single point, the grid is chosen so that x = is a grid point for all time-steps. So x and x N are chosen as with minimal m,m N N. x = m h < a i G i 3 G i (5.) x N = m N h > 3 G i (5.3) It is not guaranteed that D i,i+ already shows the asymptotic behaviour at x = a i G i 3 G i. So neither a i nor b i can be determined from the first two grid points. å i from section 3.3 will be used. b(s) i equations (4.6,4.7) it can be seen that b() i = exp will be determined as follows: From ( ) G iψi b i (5.4) b() i = δ i Li b() i (5.5) where b i defined in equation (3.7) can be determined from b () i by using equation (4.): As f,i (x) for x, both integrals in equation (4.) converge to for 3

36 x j. So b i = b () i expressions for b() i b() i b (s) i = exp are derived: exp(a i(g i G j )/). From this and b N = the following ( ) G n iψi δ j Lj exp ( ψjg j + ) ψ j(g j G n ) j=i+ (5.6) = δ i Li b() i. (5.7) The above considerations are valid for the integration over f,i in equation (4.9). For equation (4.) the situation is basically different as the Gaussians are centred around x = x j and x = a i (G i G j ) + x j, where x j can be any grid-point. So vicinities of these points can not be covered by any finite grid and the tails still have to be taken into account. As f s,i (x ) can be far from, the tail x < x has to be handled in a different manner than in section 4.. As f s,i (x) for x, assuming f s,i (x) = f s,i (x ) for x < x clearly overestimates the tail, though f s,i (x) = clearly underestimates it. As an estimation of the functional form for x < x an extrapolation is used that fits with the known values for x x and x : An exponential decay if f s,i (x ) < f s,i (x ) and a Gaussian f s,i (x) = + c exp(d x) (5.8) f s,i (x) = + c ( d(x m) ) (5.9) if f s,i (x ) > f s,i (x ). The parameters are chosen so that the extrapolation fits with values at the first two or three grid points respectively leading to for the exponential decay and for the Gaussian. c = log(y ) log(y ) x x (5.) d = (y ) exp(ax ) (5.) log(y ) log(y ) log(y m = ) log(y ) (x x ) x + x ( ) (5.) log(y ) log(y ) (x log(y ) log(y ) x ) + x x d = log(y ) log(y ) (x c) (x c) (5.3) c = (y ) exp(b(x c) ) (5.4) 3

37 The above estimation of the tails can still lead to inconsistent results if d < in equation (5.9). In this case either an other estimate for the tail is needed or the grid has to be chosen larger. It must be emphasised that the importance of the tails does not come from equation (4.9), where the tails are far from the central value of the Gaussian, but from equation (4.), where the Gaussian is centred around any grid-value. Choosing a more or less arbitrary functional form for x < x is in principle not different from choosing a cubic polynomial for each interval inside the grid. The different functional form comes from the fact that the known behaviour for x has to be taken into account. For consistency reasons the same approach should be used on the upper interval (x N, ) though the numerical impact is negligible.,5 no tail f(x)=f(x ) for x<x f(x)= for x<x exponential analytical solution,5 5 5 i Figure 5.: Effect of different tail extrapolations on the quality of the Sali-approach with spilt asymptotics. The graphs show L L i with n =, ψ =. and h =. It is obvious that the exponential tails defined in equations ( ) are necessary to reproduce the analytical results. Figure 5. illustrates the effect of the different approaches to the interval (,x ). All estimations of a constant tail clearly over- or underestimate the integral, while the exponential extrapolation reproduces the analytical solution with high accuracy. 3

38 The number of grid points that were used for each single step is displayed in figure 5. The total number of integrations done is 368 compared to 4678 integrations needed in the previous chapters with a fixed x = x N = 6. Note that the driving factor on the grid size is basically different from the previous chapters. While in the previous chapters it was important to choose x large enough so that the asymptotic behaviour takes over, after splitting off the exponential behaviour the approximated function is well behaved so that the grid can be limited to a much smaller interval. This approach relies on the fact that the coefficients of the asymptotic exponential can be determined analytically (i.e. å i and bi ) as they are needed in (4.9) and (4.). Otherwise a larger grid would be needed to determine them numerically. 5 N i Figure 5.: The Grid size needed for the calibration in figure 5. as a function of the time-step i. As both å i and b i are known analytically, the estimation of the influence of nonasymptotic terms used previously can be applied in a straightforward manner to get a boundary on the integral over the lower tail x < x as shown in appendix A... The integration is done for f,i with σ = G i, x = a i G i 3σ and both µ = and µ = a i G i, so according to (A.7) the error from the lower tail is bounded by Θ x = c [ ( 3 + ψ Gi erfc exp ) + erfc ( ) Gi ψ a i G i ( 3 + (ai + ψ) ) ] G i (5.5) with time-independent volatility ψ. If ψ i varies with i, ψ has to be replaced by min j i ψ i. The ratio between Θ x and the integral (4.9) will be monitored to verify the adequacy of the choice of the grid size. This estimate is rather rough as it assumes an approximation f,i for x < x, whereas the functional form introduced earlier is supposed to be much more precise. On the other hand this approximation can be used to verify the functional form used as an approximation for the tail. 33

39 5. Lattice Spacing The basic idea for an automatic selection of the lattice spacing h is the same as for the selection of the tension factor σ for splines under tension in section 4... If oscillations are observed, h is reduced by a constant factor w and the time-step is repeated. In more detail, after performing a time-step i with lattice-spacing h, D i,i is split into an asymptotic part and a non-asymptotic part as defined in equation (4.6). The non-asymptotic part f,i is checked for oscillations as in the case of splines under tension in section 4... In the case of cubic splines it is trivial that a change of the sign of y (x i ) indicates a point of inflexion. If within any two adjacent intervals two points of inflexion are observed, the calculation of D i,i is repeated with lattice spacing h/w. Figure 5.3 shows the result of the Sali approach using cubic splines with splitoff exponential behaviour and optimised grid size given by equations (5.,5.3) and a lattice spacing determined by the algorithm described in the present section with starting value h = and w =.5. In the case n = 3 and ψ =. the estimation of the tail (5.9) fails which is indicated by an upside-down Gaussian with d <. Here it is necessary to revert to a larger grid size. The relative effect of the lower tail, Θ x divided by (4.9) can get as large as 3% for the last non-trivial step i =, but otherwise stays well below.%. Comparing figure 5.3 to the matching results for a fixed grid in figure 4.6 we see that the adjustment of the lattice spacing leads to a good convergence even with very high values of ψ. Table 5. shows the number of integrations needed for the calibration with fixed and optimised grid respectively. As each integration contains a sum over all grid points, the total number of summands is given as well, indicating the computation cost. For short tenor the improvement is significant. For longer tenor a smaller lattice spacing than used previously for the fixed grid is necessary to avoid the errors for high volatility. This leads to a higher computational cost. The actual computation time on an average desktop system (Athlon64 X 4+) is given in table 5.. For long tenor the computation time of the analytical solution gets out of hand, making it necessary to use the SALI approach even for this simple model with a known analytical solution. 34