Polynomial Models in Finance

Transcription

1 Polynomial Models in Finance Martin Larsson Department of Mathematics, ETH Zürich based on joint work with Damir Filipović, Anders Trolle, Tony Ware Risk Day Zurich, 11 September 2015

2 Flexibility Tractability 2/28

3 Flexibility General semimartingale models General HJM models Tractability 2/28

4 Flexibility General semimartingale models General HJM models Black-Scholes Bachelier Tractability 2/28

5 Flexibility General semimartingale models General HJM models Heston Cox-Ingersoll-Ross Black-Scholes Bachelier Tractability 2/28

6 Flexibility General semimartingale models General HJM models Multi-factor affine models Heston Cox-Ingersoll-Ross Black-Scholes Bachelier Tractability 2/28

7 Flexibility General semimartingale models General HJM models Multi-factor polynomial models Multi-factor affine models Heston Cox-Ingersoll-Ross Black-Scholes Bachelier Tractability 2/28

8 Outline Polynomial preserving processes and factor models Application: Linear-rational term structure models Application: Polynomial energy models 3/28

9 Polynomial preserving processes and factor models 4/28

10 Factor models Arbitrage-free pricing usually boils down to calculating conditional expectations: [ Price(t) = E Q e ] T r t s ds Payoff(T ) F t 5/28

11 Factor models Arbitrage-free pricing usually boils down to calculating conditional expectations: [ Price(t) = E Q e ] T r t s ds Payoff(T ) F t where ζ t = e t 0 Price(t) = 1 ζ t E P [ ζ T Payoff(T ) F t ] rs ds dq dp t is a state price density. 5/28

12 Factor models Arbitrage-free pricing usually boils down to calculating conditional expectations: [ Price(t) = E Q e ] T r t s ds Payoff(T ) F t where ζ t = e t 0 Factor model: Price(t) = 1 ζ t E P [ ζ T Payoff(T ) F t ] rs ds dq dp t is a state price density. Payoff(T ) = f (X T ) ζ t = g(x T ) where X is a factor process, f and g deterministic functions 5/28

13 Factor models Arbitrage-free pricing usually boils down to calculating conditional expectations: [ Price(t) = E Q e ] T r t s ds Payoff(T ) F t where ζ t = e t 0 Factor model: Price(t) = 1 ζ t E P [ ζ T Payoff(T ) F t ] rs ds dq dp t is a state price density. Payoff(T ) = f (X T ) ζ t = g(x T ) where X is a factor process, f and g deterministic functions Task: Find X, f, g to get tractable yet flexible class of models 5/28

14 Polynomial preserving processes Markov process X with state space E R d (Extended) generator G given by G f (x) = b(x) f (x) Tr ( a(x) 2 f (x) ) This means: df (X t ) = G f (X t ) dt + (local martingale) 6/28

15 Polynomial preserving processes Markov process X with state space E R d (Extended) generator G given by G f (x) = b(x) f (x) Tr ( a(x) 2 f (x) ) This means: df (X t ) = G f (X t ) dt + (local martingale) Definition. X is called Polynomial Preserving (PP) if G Pol n (E) Pol n (E) for all n N, where Pol n (E) = {polynomials on E of degree n}. 6/28

16 Polynomial preserving processes Examples: Geometric Brownian motion: dx t = X t (µdt + σdw t ) Ornstein-Uhlenbeck: dx t = κ(θ X t )dt + σdw t Cox-Ingersoll-Ross process: dx t = κ(θ X t )dt + σ X t dw t Etc. In fact, every affine process is polynomial preserving Lemma. X is (PP) if and only if b i Pol 1 (E) and a ij Pol 2 (E) 7/28

17 Polynomial preserving processes By definition of (PP), G restricts to an operator G Poln(E) on the finite-dimensional vector space Pol n (E) 8/28

18 Polynomial preserving processes By definition of (PP), G restricts to an operator G Poln(E) on the finite-dimensional vector space Pol n (E) Let h 1,..., h N be a basis for Pol n (E), write H(x) = (h 1 (x),..., h N (x)) 8/28

19 Polynomial preserving processes By definition of (PP), G restricts to an operator G Poln(E) on the finite-dimensional vector space Pol n (E) Let h 1,..., h N be a basis for Pol n (E), write Coordinate representations: H(x) = (h 1 (x),..., h N (x)) p(x) = H(x) p G p(x) = H(x) G p p R N G R N N 8/28

20 Polynomial preserving processes By definition of (PP), G restricts to an operator G Poln(E) on the finite-dimensional vector space Pol n (E) Let h 1,..., h N be a basis for Pol n (E), write Coordinate representations: H(x) = (h 1 (x),..., h N (x)) p(x) = H(x) p G p(x) = H(x) G p p R N G R N N Key consequence: E [p(x T ) F t ] = e (T t)g p (X t ) = H(X t ) e (T t)g p 8/28

21 Polynomial preserving processes By definition of (PP), G restricts to an operator G Poln(E) on the finite-dimensional vector space Pol n (E) Let h 1,..., h N be a basis for Pol n (E), write Coordinate representations: H(x) = (h 1 (x),..., h N (x)) p(x) = H(x) p G p(x) = H(x) G p p R N G R N N Key consequence: E [p(x T ) F t ] = e (T t)g p (X t ) = H(X t ) e (T t)g p This only involves a matrix exponential as opposed to solving a PDE which leads to tractable pricing models 8/28

22 Polynomial preserving processes Example: Pearson diffusions (Forman and Sørensen, 2008), E R: dx t = (β + bx t )dt + α + ax t + AXt 2 dw t 9/28

23 Polynomial preserving processes Example: Pearson diffusions (Forman and Sørensen, 2008), E R: dx t = (β + bx t )dt + α + ax t + AXt 2 dw t Matrix representation of G Poln(R) with respect to 1, x, x 2,...: 0 β 2 α b 2 ( ) β + a α 2 0. G = ( ) b + A 2 3 ( β a... 2) ( ) b + 3 A... 2 n(n 1) α ( ) n β + (n 1) a n ( ) b + (n 1) A 2 9/28

24 Literature PP processes: Wong (1964); Mazet (1997); Zhou (2003); Forman and Sørensen (2008); Cuchiero, Keller-Ressel, Teichmann (2012); Filipović, Gourier, Mancini (2013); Bakry, Orevkov, Zani (2014); Filipović, L. (2015); Filipović, L., Trolle (2014); etc. State price density models: Constantinides (1992); Flesaker and Hughston (1996); Rogers (1997); Rutkowski (1997); Carr, Gabaix, Wu (2010); Macrina (2013); Crépey, Macrina, Nguyen, Skovmand (2015), etc. 10/28

25 Application: Linear-rational term structure models with D. Filipović and A. Trolle 11/28

26 Linear-rational term structure models PP factor process and linear state price density: dx t = κ(θ X t )dt + dm t ζ t = e αt (1 + ψ X t ) for some martingale M, κ R d d, θ R d, α R, ψ R d 12/28

27 Linear-rational term structure models PP factor process and linear state price density: dx t = κ(θ X t )dt + dm t ζ t = e αt (1 + ψ X t ) for some martingale M, κ R d d, θ R d, α R, ψ R d Conditional expectation of X T : E[X T F t ] = θ + e κ(t t) (X t θ) 12/28

28 Linear-rational term structure models PP factor process and linear state price density: dx t = κ(θ X t )dt + dm t ζ t = e αt (1 + ψ X t ) for some martingale M, κ R d d, θ R d, α R, ψ R d Conditional expectation of X T : E[X T F t ] = θ + e κ(t t) (X t θ) Linear-rational bond prices and short rate: P(t, T ) = F (T t, X t ) = e α(t t) 1 + ψ X t + ψ e κ(t t) (X t θ) 1 + ψ X t r t = log P(t, T ) T =t = α ψ κ(θ X t ) 1 + ψ X t 12/28

29 Highlights (i) Bond prices in closed form 13/28

30 Highlights (i) Bond prices in closed form (ii) Tractable swaption pricing 13/28

31 Highlights (i) Bond prices in closed form (ii) Tractable swaption pricing (iii) Unspanned stochastic volatility (USV): Empirical fact: volatility risk cannot be hedged using bonds (Collin-Dufresne & Goldstein (02), Heidari & Wu (03), etc.) 13/28

32 Highlights (i) Bond prices in closed form (ii) Tractable swaption pricing (iii) Unspanned stochastic volatility (USV): Empirical fact: volatility risk cannot be hedged using bonds (Collin-Dufresne & Goldstein (02), Heidari & Wu (03), etc.) (iv) Extensive empirical analysis, estimation using 15 years of weekly swaps and swaptions data 13/28

33 Interest rate swaps and swaptions Swap: Bond portfolio, so valuation is immediate 14/28

34 Interest rate swaps and swaptions Swap: Bond portfolio, so valuation is immediate Swaption: Option to enter the swap at T Underlying at time T (here T T 1 < < T n, c i R): C T = n i=1 for some linear functions q i (x) c i P(T, T i ) = 1 ζ T n c i q i (X T ) i=1 14/28

35 Interest rate swaps and swaptions Swap: Bond portfolio, so valuation is immediate Swaption: Option to enter the swap at T Underlying at time T (here T T 1 < < T n, c i R): C T = n i=1 for some linear functions q i (x) c i P(T, T i ) = 1 ζ T n c i q i (X T ) Option price at time t T 1 E [ [ ζ T C + T ζ F ] 1 ( n ) ] + t = E i=1 t ζ c iq i (X T ) Ft t i=1 14/28

36 Interest rate swaps and swaptions Swap: Bond portfolio, so valuation is immediate Swaption: Option to enter the swap at T Underlying at time T (here T T 1 < < T n, c i R): C T = n i=1 for some linear functions q i (x) c i P(T, T i ) = 1 ζ T n c i q i (X T ) Option price at time t T 1 E [ [ ζ T C + T ζ F ] 1 ( n ) ] + t = E i=1 t ζ c iq i (X T ) Ft t Compute E[q(X t ) + F t ] for q(x) linear: Transform methods i=1 14/28

37 Interest rate swaps and swaptions Swap: Bond portfolio, so valuation is immediate Swaption: Option to enter the swap at T Underlying at time T (here T T 1 < < T n, c i R): C T = n i=1 for some linear functions q i (x) c i P(T, T i ) = 1 ζ T n c i q i (X T ) Option price at time t T 1 E [ [ ζ T C + T ζ F ] 1 ( n ) ] + t = E i=1 t ζ c iq i (X T ) Ft t Compute E[q(X t ) + F t ] for q(x) linear: Transform methods Contrast with affine term structure models! i=1 14/28

38 Linear-rational models: Empirics Linear-rational square root (LRSQ) model: E = R d + σ 1 X1t 0 dx t = κ(θ X t )dt +... dw t 0 σ d Xdt ζ t = e αt (1 + 1 X t ) 15/28

39 Linear-rational models: Empirics Linear-rational square root (LRSQ) model: E = R d + σ 1 X1t 0 dx t = κ(θ X t )dt +... dw t 0 σ d Xdt ζ t = e αt (1 + 1 X t ) LRSQ(m, n): Constrained to have m term structure factors and n USV factors (m n, m + n = d) Number of parameters: m 2 + 2m + 2n 15/28

40 Data and estimation approach Panel data set of swaps and ATM swaptions Swap maturities: 1Y, 2Y, 3Y, 5Y, 7Y, 10Y Swaptions on 1Y, 2Y, 3Y, 5Y, 7Y, 10Y forward starting swaps with option expiries 3M, 1Y, 2Y, 5Y 866 weekly observations, Jan 29, 1997 Aug 28, 2013 Estimation approach: Quasi-maximum likelihood in conjunction with the unscented Kalman Filter 0.08 Panel A1: Swap data 250 Panel B1: Swaption data Jan97 Jan01 Jan05 Jan09 Jan13 Panel A2: Swap fit, LRSQ(3,3) 0 Jan97 Jan01 Jan05 Jan09 Jan13 Panel B2: Swaption fit, LRSQ(3,3) 16/28

41 Fit to data 0.08 Panel A1: Swap data 250 Panel B1: Swaption data Jan97 Jan01 Jan05 Jan09 Jan13 0 Jan97 Jan01 Jan05 Jan09 Jan Panel A2: Swap fit, LRSQ(3,3) Panel B2: Swaption fit, LRSQ(3,3) Jan97 Jan01 Jan05 Jan09 Jan13 Panel A3: Swap RMSE, LRSQ(3,3) Jan97 Jan01 Jan05 Jan09 Jan13 0 Jan97 Jan01 Jan05 Jan09 Jan13 Panel B3: Swaption RMSE, LRSQ(3,3) Jan97 Jan01 Jan05 Jan09 Jan13 17/28

42 Comparison of model specifications Specification Swaps Swaptions All 3 mths 1 yr 2 yrs 5 yrs LRSQ(3,1) LRSQ(3,2) LRSQ(3,3) LRSQ(3,2)-LRSQ(3,1) 3.28 ( 8.95) LRSQ(3,3)-LRSQ(3,2) 0.12 ( 0.78) ( 2.18) 0.58 ( 2.52) ( 0.74) 0.67 ( 1.82) 0.42 ( 1.04) 0.72 ( 2.97) ( 3.66) 0.11 ( 0.46) ( 2.55) 0.49 ( 2.06) Table 3: Comparison of model specifications. The table reports meansfigure: of timeaverage series ofrmse the root-mean-squared (bps) pricing errors (RMSE) of swap rates and normal implied swaption volatilities. For swaptions, results are reported for the entire volatility surface as well as for the volatility term structures LRSQ(3, at the 1) four andoption LRSQ(3.2) maturities both in the have sample reasonable (3 months, fit 1 year, 2 years, and 5 years). Units are basis points. t-statistics, corrected for heteroscedasticity and serial.. correlation. but LRSQ(3, up to 26 3) lags is the (i.e. preferred 6 months) using model the method of Newey and West (1987), are in parentheses.,, and denote significance at the 10%, 5%, and Captures level-dependence in swaption implied vol at low rates 1% level, respectively. The sample period consists of 866 weekly observations from January Upper 29, bounds 1997 to August on short 28, rate: LRSQ(3,1) LRSQ(3,2) LRSQ(3,3) /28

43 Application: Polynomial energy models with D. Filipović and A. Ware 19/28

44 Energy futures and options on futures Spot price S t 20/28

45 Energy futures and options on futures Spot price S t Time-t price of futures contract maturing at T : F (t, T ) = E Q [S T F t ] 20/28

46 Energy futures and options on futures Spot price S t Time-t price of futures contract maturing at T : F (t, T ) = E Q [S T F t ] Time-t price of futures contract with delivery period [T, T ]: [ ] T F (t, T, T ) = E Q S u du F t T 20/28

47 Energy futures and options on futures Spot price S t Time-t price of futures contract maturing at T : F (t, T ) = E Q [S T F t ] Time-t price of futures contract with delivery period [T, T ]: [ ] T F (t, T, T ) = E Q S u du F t Time-t price of European put option expiring at T with strike K on futures contract maturing at T > T : e r(t t) E Q [ (K F (T, T )) + F t ] T 20/28

48 Energy futures and options on futures Spot price S t Time-t price of futures contract maturing at T : F (t, T ) = E Q [S T F t ] Time-t price of futures contract with delivery period [T, T ]: [ ] T F (t, T, T ) = E Q S u du F t Time-t price of European put option expiring at T with strike K on futures contract maturing at T > T : e r(t t) E Q [ (K F (T, T )) + F t ] T Time-t price of calendar spread option expiring at T with dates T, T > T and cost K: e r(t t) E Q [ (F (T, T ) F (T, T ) K) + F t ] 20/28

49 Polynomial energy model PP factor process X t, state space E R d G matrix representation of G on Pol n with respect to basis H(x) = (h 1 (x),..., h N (x)) 21/28

50 Polynomial energy model PP factor process X t, state space E R d G matrix representation of G on Pol n with respect to basis H(x) = (h 1 (x),..., h N (x)) Energy spot price model: specify S t = Λ(t) p S (X t ) for some and some seasonality function p S Pol n (E) with p S (x) > 0 on E Λ(t) = Λ(0) e t 0 λ(s)ds More generally, take convex combinations of such specifications. 21/28

51 Polynomial futures prices Time-t price of futures contract maturing at T : F (t, T, X t ) = E Q [Λ(T ) p S (X T ) F t ] is a polynomial in X t of degree n = Λ(T ) H(X t ) e (T t)g p S 22/28

52 Polynomial futures prices Time-t price of futures contract with delivery period [T, T ]: [ ] T F (t, T, T, X t ) = E Q S u du F t T 23/28

53 Polynomial futures prices Time-t price of futures contract with delivery period [T, T ]: [ ] T F (t, T, T, X t ) = E Q S u du F t = T T T E Q [S u F t ] du 23/28

54 Polynomial futures prices Time-t price of futures contract with delivery period [T, T ]: [ ] T F (t, T, T, X t ) = E Q S u du F t = T T T = H(X t ) T E Q [S u F t ] du T Λ(u)e (u t)g p S du 23/28

55 Polynomial futures prices Time-t price of futures contract with delivery period [T, T ]: [ ] T F (t, T, T, X t ) = E Q S u du F t = T T T = H(X t ) T E Q [S u F t ] du T Λ(u)e (u t)g p S du Polynomial in X t with semi-explicit coordinate representation. 23/28

56 Polynomial futures prices Time-t price of futures contract with delivery period [T, T ]: [ ] T F (t, T, T, X t ) = E Q S u du F t = T T T = H(X t ) T E Q [S u F t ] du T Λ(u)e (u t)g p S du Polynomial in X t with semi-explicit coordinate representation. Example: Λ(u) = cos(ct) for some constant c. Closed form: T T Λ(u)e (u t)g du = Re (e ( ict (G + ic) 1 e (T t)(g+ic) (T e t)(g+ic))) 23/28

57 Option pricing European put: e r(t t) [ E Q (K F (T, T, X T )) + ] F t... on an underlying with delivery period: e r(t t) [ E Q (K F (T, T, T, X T )) + ] F t Calendar spread option: e r(t t) [ E Q (F (T, T, X T ) F (T, T, X T ) K) + ] F t 24/28

58 Option pricing European put: e r(t t) [ E Q (K F (T, T, X T )) + ] F t... on an underlying with delivery period: e r(t t) [ E Q (K F (T, T, T, X T )) + ] F t Calendar spread option: e r(t t) [ E Q (F (T, T, X T ) F (T, T, X T ) K) + ] F t All these are of the form: E Q [ (polynomial(xt )) + F t ] 24/28

59 Option pricing European put: e r(t t) [ E Q (K F (T, T, X T )) + ] F t... on an underlying with delivery period: e r(t t) [ E Q (K F (T, T, T, X T )) + ] F t Calendar spread option: e r(t t) [ E Q (F (T, T, X T ) F (T, T, X T ) K) + ] F t All these are of the form: E Q [ (polynomial(xt )) + F t ] Consequence: All these options can be tractably priced 24/28

60 Illustration: One-factor Jacobi model Model: Jacobi factor process: dx t = κ(θ X t ) dt + σ X t (1 X t ) dw t Spot S t = p S (X t ) for p S (x) increasing polynomial 25/28

61 Illustration: One-factor Jacobi model Model: Jacobi factor process: dx t = κ(θ X t ) dt + σ X t (1 X t ) dw t Spot S t = p S (X t ) for p S (x) increasing polynomial Data: Electricity spot prices in 2012 from the Alberta Electricity Systems Operators (AESO) 25/28

62 Illustration: One-factor Jacobi model Model: Jacobi factor process: dx t = κ(θ X t ) dt + σ X t (1 X t ) dw t Spot S t = p S (X t ) for p S (x) increasing polynomial Data: Electricity spot prices in 2012 from the Alberta Electricity Systems Operators (AESO) Estimation: Maximum likelihood using that the transition density p t (x, y) = x a 1 (1 x) b 1 n=0 k n J n (x; a, b)j n (y; a, b)e λnt is known. 25/28

63 Illustration: One-factor Jacobi model Results for degree-5 polynomial map p S (x): 26/28

64 Illustration: One-factor Jacobi model Results for degree-5 polynomial map p S (x): 26/28

65 Conclusion (PP) processes can be used to build flexible and tractable models Linear-rational term structure models: Easily accommodates unspanned factors affecting volatility and risk premia Admits efficient pricing of swaptions Extensive empirical analysis Polynomial energy models: Modeling framework for commodities with arbitrary number of factors Futures prices in closed form Calendar spread options, basket options, etc. are as easy to price as plain vanillas Incorporating delivery period yields only minor increase in complexity (and no increase if seasonality function is additive) 27/28

66 Thank you! 28/28