Calculating Forward Rate Curves Based on Smoothly Weighted Cubic Splines

Calculating Forward Rate Curves Based on Smoothly Weighted Cubic Splines February 8, 2017 Adnan Berberovic Albin Göransson Jesper Otterholm Måns Skytt Peter Szederjesi

Contents 1 Background 1 2 Operational analysis 2 2.1 Goals................................................. 2 2.2 Limitations.............................................. 2 3 Mathematical model 3 3.1 Analytical solution.......................................... 3 3.1.1 Identifying the basic variables................................ 6 3.1.2 Linearising g e f....................................... 7 3.1.3 Regularisation........................................ 9 4 Evaluation 4.1 Procedure............................................... 4.1.1 LSExp Vs. Spline-approximation of LSExp and f..................... 4.1.2 Principal component analysis PCA............................ 4.2 Results and analysis......................................... References 18 i

1 Background This project focuses on calculating forward rate curves with an approximation of the method LSExp Least Square, Exponential Weighting, as described in [1]. By approximating LSExp with smoothly weighted cubic splines for the yield curve and a least square tted third degree polynomial for the weight function, an analytical simplication of the regularisation is derived which makes it possible to avoid numerical approximations. The yield curve is t to quoted OIS-instruments from the Swedish market. The general optimisation model in [1] is given by min f,x,z e,z b s.t. hf + 1 2 zt e E e z e + 1 2 zt b E bz b g e f + F e z e = b e g l g b f + F b z b g u f f l f F, 1 where hf is a regularisation of forward rate curve which implies that the yield curve gets a desired property. The two remaining terms in the objective function penalises pricing errors for exact prices, z e, respectively prices limited to a price gap, z b. The functions g e f and g b f transforms forward rates to market prices for instruments that require unique prices and instruments whose prices are within a price gap respectively, and b e, b l and b u are the observed market prices. F e and F b are diagonal matrices that decides which instruments are allowed to deviate from market prices, and E e and E b are diagonal matrices with penalty for eventual deviations. f l gives a lower bound for the yield curve and the condition f F allowed the model to limit the curve to a given class of functions. Blomvall and Ndengo suggets the regularisation hf = 1 2 Tn for the roughness in the forward rate curve [1]. T 0 w 1 tf t 2 dt + 1 2 Tn T 0 w 2 tf t 2 dt. 2 The weight functions in the regularisation, w 1 t and w 2 t, intends to decide how much of the derivatives that contributes to the regularisation hf. In the method LSExp w 1 t is set to 0 and w 2 t is set to { { w 2 exp t th t = 365 ln δ} om t t h, 3 1 om t > t h where δ is the rate of information decay and is suggested in [1] to be set to δ = 2, and t h decides when to stop weighting the derivatives and is suggested to be set to t h = 730 days. Let T k = { T0 k, T 1 k,..., T m} k, with 0 = T k 0 T1 k... T m, k be the set of maturities for all contracts that the optimisation takes into consideration. The relation between the spot rate for time point t i, r ti, and the quoted xed leg for an OIS with maturity Tj k, is then given by F j = T k j 1 e r T j k Nj. 4 i=1 tf i,j e rt i,j t i,j In 4 t f i,j is the time since the last settlement, t i,j is the time until interest payment i from the time of valuation and N j is the number of payments, all for an OIS with maturity Tj k. The discount factor is 1

calculated with the spot rate for the corresponding time horizon, r ti, which itself can be calculated from the continuous forward rate ft through where r ti t 0 is the interest until time point t i observed at time point t 0. r ti t 0 = 1 ti ftdt 5 t i t 0 t 0 2 Operational analysis Under the assumption that the project should be performed with the given resources and the intended time span, some limitations has to be made. Additionally the goal with the project is also discussed as well as the pros and cons with the results. 2.1 Goals The purpose of the project is to analyse whether or not a spline approximation of the method LSExp yields reasonable yield curves that can be used for pricing interest rate derivatives, risk management of these and eventually as an interpolation tool for OTC-traded OIS. Whether or not the results are reasonable is analysed in two parts. First o, the yield curves principal components are analysed, and the second part compares the yield curves as well as the principal components from this method, with the corresponding results yielded from the LSExp method. This is done in order to see how well the spline approximation behaves qualitatively as well as how reasonable the result is. 2.2 Limitations In the mathematical model of the optimisation problem in 1 the requirement that takes the price gap in consideration will be ignored, and the problem will then only be solved using the equality condition, where mid prices are used as exact prices. Additionally, the inequality condition with a lower limit will also be ignored, in order to allow for any negative forward rate. As a consequence of this, the penalty term for the price dierence in price gaps will be eliminated from the objective function. Furthermore, g e f is a non linear function, which contributes to a computationally intensive and complex optimisation problem. For this reason g e f will be linearised in order to yield an analytical solution given an operating point. The matrix F e will be required to be non singular. The model is built under the assumption that all nodes knot points as well as end points for the splines of the forward rate structure, f i, to coincide with the maturities for the OIS contracts, T k. Therefore, the set T s T k with Ti s T s for i = 0,..., n, where n = T s, and 0 = T0 s < T 1 s <... < T n s is dened, which denotes all nodes that build the splines, where the nodes in this case coincide with the maturities of the contracts. Furthermore, w 2 t in 3 is approximated with a third degree polynomial for t t h if t h 730; if t h > 730 more splines might be needed, but these cases are not researched. The time t h which denotes when a weighting of derivatives in the regularisation should cease does not need to coincide with any maturity of the contracts. No comparison between other models for estimation of forward rate curves will be performed, with the exception of LSExp as mentioned initially. 2

3 Mathematical model Under the limitations discussed in section 2.2, 1 is simplied as min f,z e s.t. hf + 1 2 zt e E e z e g e f + F e z e = b e f F, 6 where b e is set to mid prices, F e is the identitiy matrix, since pricing errors are allowed for all instruments, and E e = pi, where the design parameter p decides how much pricing errors are penalised. The forward rate structure will be approximated with splines, f i t = a f i t T s i 3 + b f i t T s i 2 + c f i t T s i + d f i i I 7 in the intervals [Ti s, T i+1 s ], where I is the index set with all pieces of the yield curve that is built by all the splines, where the elements are ordered integers 1 such that they correspond to the contracts that is used to decide the knot points for the splines i.e d.v.s I = {0, 1,..., n 1} if the forward rate curve is built by n + 1 quoted instruments, which means that F will be the set of all smoothly weighted cubic splines. For cubic splines the following has to hold f i Ti+1 s = f i+1ti+1 s f i T i+1 s = f i+1 T i+1 s, i I \ {n 1} 8 f i T i+1 s = f i+1 T i+1 s in order to guarantee continuity and smoothness. The regularisation hf in 2 is solved analytically. The weight function w 2 t in 3 is approximated with a cubic spline for t t h and is given by w 2 t = { p w t om t t h, 1 om t > t h, where p w t = a w t 3 + b w t 2 + c w t + d w. For choices of t h and δ in conjunction with [1], it is enough that p w is a cubic spline, with respect to the limitations. But for other choices of parameters it might be more suitable to use more splines, tentatively the same knot points could be used as for the forward rate curve f in order to make the maths easier. 9 3.1 Analytical solution The choice to use splines for ft causes the regularisation in 2 to become quadratic in the spline coecients a 0,..., d 0,..., a n,..., d n. By linearising g e f and substituting for z in the objective function, the second term becomes quadratic in the coecients as well, which results in an optimisation problem with a quadratic objective function and the linear constraints 8, which are solved with respect to osets in the spline coecients, a 0,..., d 0,..., a n,..., d n. The yielded optimum yields a new operation point for the linearisation, where the problem is solved again. 3

Let f = a 0,..., d 0,..., a n,..., d n T denote the vector with the spline coecients that build f. By setting f = f + f, g e f is linearised according to g e f; T s g e f; T s + g e f; T s T f, 10 where g e f; T s T is the jacobian of g e. Since the constraints in 8 constitutes an underdetermined system of equations, the variables is split up into basic as well as nonbasic variables. In order to make the notation easier, the permutation matrix P is used, which allowed the denitions [ ] f f = P f = B 11 f N and x = P f = [ xb x N ], where index B and N indicates basic as well as nonbasic variables. With the linearisation in 10, the rst constraint in 6 can be rewritten as z e = Fe b e g e f; T s g e f; T s T x = Ax a, 13 [ where A = [A B A N ] = F g e f; T s T B F g e f; ] T s T N with g e f; T s = P g e f; T s, and a = F g e f; T s b e. The constraints in 8 can trivially be expressed as B f + x = b 14 for some B = [B B B N ] and b, and the optimisation model in 6 is then rewritten as where denes the quadratic form of the regularisation. 1 min x,z e 2 f + x T H f + x + 1 2 zt e E e z e s.t. z e = Ax a B f + x = b, [ ] HBB H H = BN H NB From x B is expressed in terms of x N according to Substituting this in 13 yields H NN B f + x = B B f B + x B + BN f N + x N = b B B x B = b B N f N + x N f B x B = B B b BN f N B B f B B B B Nx N. 15 16 17 z e = Ax a = A B x b + A N x N a = A B B B b BN fn B B fb B B B Nx N + AN x N a = 18 A N A B B B B N xn + A B B B b BN fn B B fb a 4

and elimination of z e in the objective function in 15 results in ze T Ez e = Ax a T E Ax a = x T N AN A B B B B T N E AN A B B B B N xn + 2 A B B T B b BN f N B B f B a E AN A B B B B N xn + A B B T B b BN fn B B fb a E AB B B b BN fn B B fb a for the penalty term. The regularisation is expanded according to 19 f + x T H f + x = x T Hx + 2 f T Hx + f T H f, 20 where x T Hx = x T BH BB x B + x T NH NB x B + x T BH BN x N + x T NH NN x N and = / HNB T / = H BN = x T B H BB x B + 2x T BH BN x N + x T NH NN x N 17 = B B b BN fn B B fb B B B T Nx N HBB B B b BN f N B B f B B B B T Nx N HBN x N + 2 B B + x T NH NN x N = x T N + 2 B B B + B B B B N B B T HBB B B B N 2 B B B T N HBN + H NN T b BN f N B B f B HBN b BN f N B B f B T HBB B B B N x N b BN fn B B fb T HBB B B b BN fn B B fb B B B Nx N x N b BN fn B B fb 2 f T Hx = 2 f B T H BB x B + 2 f N T H NB x B + 2 f B T H BN x N + 2 f N T H NN x N = 2 f B T H BB B B b BN f N B B f B B B B Nx N + 2 f N T H NB B B b BN f N B B f B B B B Nx N + 2 f B T H BN x N + 2 f N T H NN x N = 2 f B T HBN H BB B B B N + f N T HNN H NB B B B N xn + 2 f B T H BB B B b BN fn B B fb + 2 f N T H NB B B b BN f N B B f B. By combining 19, 20, 21 and 22 the objective function in 15 is rewritten as 21 22 Φ = 1 2 x N Hx N + h T x N + d, 23 where H = B B B T N HBB B + A N A B B B B N B B N 2 B B B T N HBN + H NN 24 T E AN A B B B B N 5

and h = B T B b BN f N B B f B HBN B T B b BN fn B B fb HBB B + f B T HBN H BB B B B N HNN H NB B + f T N + A B B B B B N B B N b BN fn B B fb a T E AN A B B B B N, and where d is independent of x N. The optimisation problem can now, starting from the initial guess f, be solved by nding the solution to the linear system of equations 25 Φ = Hx N + h = 0. 26 The set of spline coecients that dene ft, f is yielded by [ ] f = f + P T xb, 27 x N and is used as the starting point in the next iteration of the solution process: f k+1 = f k. What remains is to nd B, b, g e f and H. 3.1.1 Identifying the basic variables The requirement for continuity and smoothness in the knot points between the splines 8 is rewritten to get a zero sum. With f i and f i + 1 substituted with the spline expression in 7 and inverted row order, the linear system of equations for knot point Ti+1 s yields 3a i Ti+1 s T i s + b i b i+1 = 0 3a i Ti+1 s T i s2 + 2b i Ti+1 s T i s + c i c i+1 = 0 28 a i Ti+1 s T i s3 + 2b i Ti+1 s T i s2 + c i Ti+1 s T i s + d i d i+1 = 0 where only the contribution from the constant term in the right hand side in 8 is kept, the rest of the terms becomes zero. Let β i R 3 4n, where n is the number of splines yield 28 through β i f = 0. The requirements for all the splines F, is then given by β f + f = b 29 where b = 0 and β = β 0 β 1. β n 2 R3n 4n. 30 If f B = b 1, c 1, d 1, b 2, c 2, d 2,..., b n, c n, d n T, row operations in β i yields the modied matrix 6

B i = β i,1 β i,1 β i,2 2T i+1 T i β i,1 β i,2, i = 1, 2,..., n 2, 31 β i,3 T i+1 T i 2 β i,1 T i+1 T i β i,2 β i,3 where β i,j indicates row j in matrix β i. From B i B = β 0 B 1. B n 2 R3n 4n, 32 is yielded which denes a linear system of equations that guarantees continuity and smoothness in the knot points of the splines. The structure in B is such that all basic variables, f B, is expressed in only nonbasic variables f N. consequence of this is that B can be split up in B B and B N with a permutation according to A B = BP T = [ B B B N ] 33 where B B = I R 3n 3n och B N R 3n n+3. The choice of basic and nonbasic variables is based on the structure for β which allows the easy calculation of B. The permutation matrix P is used widely to order the spline coecients starting with basic variables and ending with nonbasic variables. Permutating the original constraint 29 yields the rewritten constraint B f + x = B B f B + x B + B N f N + x N = b 34 which is split into two terms: basic and nonbasic variables, with b = 0. This yields 14 which is used for eliminating variables in the objective function. 3.1.2 Linearising g e f The rst constraint in the simplied optimisation model, g e f + F e z e = b e, needs to have a dened function g e. The function takes the entire forward rate structure and transforms it to the corresponding rate for the xed leg for an OIS, in unity with earlier notation. g e,j f = F j 4 = T k j 1 e r T j k Nj i=1 tf i,j e rt i,j t i,j = Γ jf Π j f, 35 When the spot rate is calculated from the forward rate structure, all the splines contribute to a sum of integrals, where each spline f i t comprises a part of the entire forward rate structure ft. Then the 7

following relation between the forward rate structure and the spot rate is yielded, rt k j 0 5 = 1 Tj k ftdt = 1 k T j 0 7 = 1 k T j i {i:t s i T j k } k T j i {i:ti s T j k } τi T i f i tdt [ a f i 4 τ i T s i 4 + bf i 3 τ i T s i 3 + cf i 2 τ i T s i 2 + d f i τ i T s i ] 36 where τ i = min { T s i+1, T j k}. The partitioning of the forward rate curve in dierent cubic splines entails a sum of all active splines up until the maturity of the contract T k j. In order to express the jacobian g e f; T s T, rst the numerator, Γ j f, and denominator, Π j f, in 35, is split apart, whereupon partial derivatives of these with respect to each spline coecient is calculated according to and Γ j f a i Γ j f b i Γ j f c i Γ j f d i = e r T k j = e r T k j = e r T k j = e r T k j T k j 1 T k j 1 T k j 1 T k j 4 τ i Ti s4, om Tj k T i s, 0 annars 3 τ i Ti s3, om Tj k T i s, 0 annars 2 τ i Ti s2, om Tj k T i s, 0 annars τ i T s i, om T k j T s i, 0 annars 37 Π j f a i Π j f b i Π j f c i Π j f d i = k {k:t k i,j T i s} t i,je r t k t i,j i,j 1 4 τ i T s = k {k:t k i,j T i s} t i,je r t k t i,j i,j 1 3 τ i T s = k {k:t k i,j T i s} t i,je r t k t i,j i,j 1 2 τ i T s i 4 i 3 i 2 = k {k:t k i,j T i s} t i,je r t k t i,j i,j τ i Ti s4, 38 where N j is the number of cash ows under the term of the contract, t i,j is the time until cash ow i, t i,j is the time since the previous cash ow, and where now τ i = min { Ti+1 s, t i,j}. For contracts with maturities longer than one year t i,j = 1, under the assumption that cash ows occur yearly for all OIS. Finally all the elements in the jacobian is calculated with the quotient rule for derivatives according to where g e f; T s = g j a i = g 1 a 0. g 1 d n g m a 0.... g m d n, 39 Γ j f a i Π j f Γ j f Π jf a i Π j f 2. 40 Note that the elements in the nal matrix are transposed with respect to a jacobian. 8

3.1.3 Regularisation In order to nd an explicit expression for the regularisation, the weight function w 2 t in 9 is least square tted to a cubic polynomial, and the forward rate curve is modelled with cubic splines f i t according to p w t = a w t 3 + b w t 2 + c w t + d w f i t = a f i t T i s 3 + b f i t T i s 2 + c f i t T i s + d f i f it = 3a f i t T i s 2 + 2b f i t T i s + c f i f i t = 6a f i t T i s + 2b f i f i t 2 = 36a f i 2 t Ti s 2 + 24a f i bf i t T i s + 4b f i 2. 41 p w tf i t2 is determined and split into four terms where the distributive property has been used to break out the coecients for p w t p w tf i t 2 = q a i t + q b i t + q c i t + q d i t q a i t = a w 36t 3 t T s i 2 a f i 2 + 24t 3 t T s i a f i bf i + 4t3 b f i 2 q b i t = b w 36t 2 t T s i 2 a f i 2 + 24t 2 t T s i a f i bf i + 4t2 b f i 2 q c i t = c w 36tt T s i 2 a f i 2 + 24tt T s i a f i bf i + 4tbf i 2 q d i t = d w 36t T s i 2 a f i 2 + 24t T s i a f i bf i + 4bf i 2 42 which together with 3 and hf = 1 2 n T s i+1 i=0 T s i w 2 tf i t 2 dt 43 yields n 2hf = i=0 T s i+1 T s i ι w 2 tf i t 2 dt = i=0 + T s i+1 T s i th p w tf i t 2 dt + p w tf ι t 2 dt Tι s T s ι+1 t h f ι t 2 dt + n i=ι+1 T s i+1 T s i f i t 2 dt 44 where ι = max{i : T i t h }, since the second derivative in the regularisation stops being weighted with w 2 t when t > t h. This is split into two sub problems: and f T i Q 1 i, t 2 f i = p w tf i t 2 dt 45 f T i Q 2 i, t 2 f i = f i t 2 dt 46 9

where f i = a i, b i, c i, d i T are the coecients for spline i that constructs the forward rate curve. For equation 45 the following holds p w tf i t 2 dt = = qi a t + qi b t + qi c t + qi d t dt q a i tdt + q b i tdt + q c i tdt + q d i tdt = f T i Q a i, t 2 f i + f T i Q b i, t 2 f i + f T i Q c i, t 2 f i + f T i Q d i, t 2 f i 47 = f T i Q a i, t 2 + Q b i, t 2 + Q c i, t 2 + Q d i, t 2 f i = f T i Q 1 i, t 2 f i which are calculated termwise. All four integrals in 47 are quadratic forms which is shown below. [ qi a tdt = a w 36a f 1 i 2 6 t6 2 5 T i s t 5 + 1 4 T i s 2 t 4 + 24a f 1 i bf i 5 t5 1 ] 4 T i s t 4 + 4b f 1 t2 i 2 4 t4 t 1 = a w a f i 2 6t 6 2 t 6 1 72 5 t5 2 t 5 1Ti s + 9t 4 2 t 4 1Ti s 2 + a f i bf i 24 5 t5 2 t 5 1 6t 4 2 t 4 1Ti s + b f i 2 t 4 2 t 4 1 48 which by observation is a quadratic form, which is why the ansatz in the two last rows in 47 are made. Similar calculations are performed for the remaining integrals: t2 qi b tdt = b w a f 36 i 2 5 t5 2 t 5 1 18t 4 2 t 4 1Ti s + t 3 2 t 3 1Ti s 2 q c i tdt = c w q d i tdt = d w + a f i bf i 6t 4 2 t 4 1 8t 3 2 t 3 1Ti s + b f 4 i 2 t 3 3 2 t 3 1 a f i 2 9t 4 2 t 4 1 24t 3 2 t 3 1T s i + 18t 2 2 t 2 1T s i 2 + a f i bf i 8t 3 2 t 3 1 t 2 2 t 2 1Ti s + b f i 2 2 t 2 2 t 2 1 a f i 2 t 3 2 t 3 1 36t 2 2 t 2 1T s i + 36t 2 T s i 2 + a f i bf i t 2 2 t 2 1 24t 2 Ti s + b f i 2 4 t 2. 49 10

The expression above is written on a quadratic form with the following matrices 6t 6 2 t6 1 72 5 t5 2 t5 1 T i s + 9t4 2 t4 1 T i s2 5 t5 2 t5 1 3t4 2 t4 1 T s i 0 0 Q a i, t 2 = a w 5 t5 2 t5 1 3t4 2 t4 1 T i s t 4 2 t4 1 0 0 0 0 0 0 0 0 0 0 36 5 t5 2 t5 1 18t4 2 t4 1 T i s + t3 2 t3 1 T i s2 3t 4 2 t4 1 4t3 2 t3 1 T s i 0 0 Q b i, t 2 = b w 3t 4 2 t4 1 4t3 2 t3 1 T i s 4 3 t3 2 t3 1 0 0 0 0 0 0 0 0 0 0 9t 4 2 t4 1 24t3 2 t3 1 T i s + 18t2 2 t2 1 T i s2 4t 3 2 t3 1 6t2 2 t2 1 T s i 0 0 Q c i, t 2 = c w 4t 3 2 t3 1 6t2 2 t2 1 T i s 2t 2 2 t2 1 0 0 0 0 0 0 0 0 0 0 t 3 2 t3 1 36t2 2 t2 1 T i s + 36t 2 Ti s2 6t 5 2 t5 1 t 2 T s i 0 0 Q d i, t 2 = d w 6t 2 2 t2 1 t 2 Ti s 4t 2 0 0 0 0 0 0 0 0 0 0 Q 1 i, t 2 = Q a i, t 2 + Q b i, t 2 + Q c i, t 2 + Q d i, t 2 50 where zeros has been added in order to satisfy the form of f i by expanding to 4 4 matrices. It remains to calculate Q 2 i, t 2 in order to nally yield the regularisation hf in matrix form. According to 46 and 41 we have f i t 2 dt = a f i 2 t 3 2 t 3 1 36t 2 2 t 2 1T s i + 36t 2 T s i 2 + a f i bf i t2 2 t 2 1 24t 2 T s i + b f i 2 4t 2 51 which yields t 3 2 t3 1 36t2 2 t2 1 T i s + 36t 2 Ti s2 6t 2 2 t2 1 t 2 T s i 0 0 Q 2 i, t 2 = 6t 2 2 t2 1 t 2 Ti s 4t 2 0 0 0 0 0 0 52 0 0 0 0 and the regularisation is then hf = 1 2 ι i=0 f T i Q 1 i T s i, T s i+1f i + f T ι Q 1 ι T s ι, t h f ι + f T ι Q 2 ι t h, T s ι+1f ι + n i=ι+1 f T i Q 2 i T s i, T s i+1f i 53 which on matrix form is written as H = P diag Q 1 0T0 s, T1 s,..., Q 1 ιtι, s Tι s, Q 1 ι Tι s, t h + Q 2 ι t h, T ι+1, Q 2 ι+1tι+1, s Tι+2, s..., Q 2 ntn, s Tn s P T 54 where diaga 1,..., A n creates a block diagonal of matrices. 11

4 Evaluation The result of the produced forward rate curve with the simplied model will be analysed with the forward rate curves' principal components in order to evaluate how reasonable they are. Additionally, a comparison of the results from this model and the results from LSExp will be made. Depending on what the forward rate curve will be used for, dierent types of evaluations could be of interest. One such example is to use the forward rate curve as a tool for interpolation or extrapolation for pricing OIS with maturities that are not quoted on the market. In order to evaluate this, it could be of additional interest to price OIS with such maturities, and compare the prices yielded with observed market quotes. 4.1 Procedure In order to perform the evaluation, a framework has been implemented in order to analyse the results given the methods below which has been chosen to evaluate the yielded forward rate curves. 4.1.1 LSExp Vs. Spline-approximation of LSExp and f A pre existing implementation of LSExp was used in order to compare the results. LSExp yields reasonable forward rate curves [1], and in order to evaluate the results from the project, the yielded forward rate curves will be compared to these by examining the structure as well as principal components. The methodology is discussed in section 4.1.2. 4.1.2 Principal component analysis PCA The most common systematic risks are shift, twist and buttery; these risk factors have the greatest impact on changes in interest rates for longer maturities [1]. The risk factors that aect the forward rate curve in the short end is usually explained by principal components with less impact on the total variance. This is due to the fact that there exists a lot of information on forward rates within a short period in the future, but after that period there is almost no information on how the forward rates will move, which is why changes in these rates are mainly aected by systematic risks. In order to perform a PCA on forward rate curves, the daily changes in interest rates has to be calculated for each day, for each maturity, even the maturities that a contract does not exist for in order to yield a large vector that is seemingly continuous. Then the covariance matrix C is calculated for these changes in interest rates over time. Since covariance matrices are always positively semidenite as well as symmetric, an eigenvaluedecomposition can be performed according to C = UΛU T, where U contains eigenvectors u k to C, with corresponding eigenvalue λ k in the diagonal matrix Λ. Since the eigenvectors for C are orthogonal given that all the eigenvalues are unique, these can be used to study independent changes in the forward rate curve. Each eigenvector u k explains one type of change, and its impact on the total variation in the λ forward rate curve is described by the eigenvalue according to k i λ i. 4.2 Results and analysis Below is presented the maturities for the contracts that are included in the models, as well as the dierent parameter congurations for spline knowpoints and penalties for pricing errors. A unique conguration of

knot points is indexed with a number under U in table 2, in order to keep track on the dierent data les during calculation of all the data that is required for the evaluation. Table 1: All maturities T k. 0 1 52 1 2 3 6 Maturities 9 1 2 3 4 5 6 7 8 9 10 15 20 25 30 Table 2: Parameter conguration used for the evaluation. U Spline knot points p 3 6 5 0 1 2 3 4 5 6 7 8 9 10 15 20 1000000 6 1 0 1 2 5 8 10 20 30 1000 6 2 0 1 2 3 4 5 6 7 8 9 10 15 20 25 30 1000 6 1 0 1 2 5 8 10 20 1000 2 0 6 3 0 1 2 3 1 2 3 4 5 6 7 8 9 10 15 20 1000 6 9 1 2 3 4 5 6 7 8 9 10 15 20 1000 4 0 1 5 10 30 1000 6 1 0 1 2 5 8 10 20 30 100 6 2 0 1 2 3 4 5 6 7 8 9 10 15 20 100 5 0 3 6 1 2 3 4 5 6 7 8 9 10 15 20 1 With a penalty parameter of p = 100 the constructed forward rate curves are considerably more curved in the long end compared to those with p = 1000. By examining the dierences between priced OIS with the constructed forward rate curves and the quoted prices, it can be conrmed that p = 100 causes a much greater pricing error than p = 1000. By examining the pricing error over time, it was observed that contracts with very short maturities under certain days were suddenly priced entirely dierently from market quotes, even for p = 1000. This pricing error was amplied when knot points for 3/ and 6/ was not taken into the consideration. The gures below initially presents some priced OIS with constructed forward rate curves in relation to market quotes, and thereafter forward rate curves with dierent parameter congurations are presented. a p = 1, 20 years, Spline conguration 5. b p = 1000, 20 years, Spline conguration 1. 13

Figure 1: Forward rate curves. a p = 1, 20 years, Spline conguration 5. b p = 100, 30 years, Spline conguration 1. Figure 2: Forward rate curves. a p = 1000, 30 years, Spline conguration 2. b p = 1000, 20 years, Spline conguration 3. Figure 3: Forward rate curves. 14

a p = 1000, 20 years, Spline conguration 2. b p = 1000000, 20 years, Spline conguration 5. Figure 4: Forward rate curves. LSExp with a penalty for pricing error at p = 1000 gives generates very unstable forward rate curves, which could depend on the fact that the model discretises the forward rate curve in a huge amount of points, where the pricing error is penalised in each discretised point. In the approximative model, the degrees of freedom are considerably fewer, and the pricing error is penalised in fewer points. With p = 1000000 in the approximative model a similar behaviour is yielded in the forward rate curve as for p = 1000 in LSExp. With p = 1 the forward rate curve as well as the principal components become very unstable, with no clear shift, twist nor buttery. In the gures below dierent congurations of principal components are presented for dierent parameter congurations. Figure 5: Comparison between Shift, Twist and Buttery between LSExp1E0 and LSExpSplines1E0. 15

Figure 6: Comparison between Shift, Twist and Buttery between LSExp1E0 and LSExpSplines1E2. Figure 7: Comparison between Shift, Twist and Buttery between LSExp1E0 and LSExpSplines1E3. Figure 8: Comparison between Shift, Twist and Buttery between LSExp1E0 and LSExpSplines1E3. 16

Figure 9: Comparison between Shift, Twist and Buttery between LSExp1E0 and LSExpSplines1E3. Figure 10: Comparison between Shift, Twist and Buttery between LSExp1E0 and LSExpSplines1E6. With p = 1 the forward rate curves are much smoother, but at the same time the rst principal component is very straight. In all evaluations, the rst principal component derails a lot, which could be the cause of that the forward rate curves constructed do not have any requirement to be dampened in the last knot point. If a requirement that does not dampen the rst derivative in the last knot point causes no "nancial" shift in the forward rate curve to be observed. This could be solved by adding a condition for the last knot point similarly to Deventer & Imai 1997. The model is therefore very sensitive to parameter congurations, but for spline congurations without the early knot points a/ and 2/ in combination with a penalty parameter of p = 1000 the model is considered stable enough to construct reliable forward rate curves over time. These congurations are also considered to give a good enough balance between pricing errors and smoothness in the forward rate curve. A weakness that was identied in the model is that the rst principal component does not show a real parallell shift, but instead derails towards longer maturities, probably due to no condition for the rst derivative in the last knot point. 17

References [1] J. Blomvall and M. Ndengo, Accurate measurements of yield curves and their systematic risk factors, 2016. 18