Asian SWIFT method Efficient wavelet-based valuation of arithmetic Asian options ´ Alvaro Leitao, Luis Ortiz-Gracia and Emma I. Wagner CMMSE 2018 - Rota July 10, 2018 ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 1 / 24
Motivation Arithmetic Asian options are still attractive in financial markets, but it numerical treatment is rather challenging. The valuation methods relying on Fourier inversion are highly appreciated, particularly for calibration purposes, since they are extremely fast, very accurate and easy to implement. Lack of robustness in the existing methods (number of terms in the expansion, numerical quadratures, truncation, etc.). The use of wavelets for other option problems (Europeans, early-exercise, etc.) has resulted in significant improvements in this sense. In the context of arithmetic Asian options, SWIFT provides extra benefits. ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 2 / 24
Outline Problem formulation 1 The SWIFT method 2 SWIFT for Asian options 3 Numerical results 4 Conclusions 5 ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 3 / 24
Problem formulation In Asian derivatives, the option payoff function relies on some average of the underlying values at a prescribed monitoring dates. Thus, the final value is less volatile and the option price cheaper. Consider N + 1 monitoring dates t i ∈ [0 , T ] , i = 0 , . . . , N . Where T is the maturity and ∆ t := t i +1 − t i , ∀ i (equal-spaced). Assume the initial state of the price process to be known, S (0) = S 0 . 1 � N Let averaged price be defined as A N := i =0 S ( t i ), the payoff of N +1 the European-style Asian call option is v ( S , T ) = ( A N − K ) + . The risk-neutral option valuation formula, � v ( x , t ) = e − r ( T − t ) E [ v ( y , T ) | x ] = e − r ( T − t ) v ( y , T ) f ( y | x ) d y , R with r the risk-free rate, T the maturity, f ( y | x ) the transitional density, typically unknown, and v ( y , T ) the payoff function. ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 4 / 24
The SWIFT method A structure for wavelets in L 2 ( R ) is called a multi-resolution analysis . We start with a family of closed nested subspaces in L 2 ( R ), � � V m = L 2 ( R ) , . . . ⊂ V − 1 ⊂ V 0 ⊂ V 1 ⊂ . . . , V m = { 0 } , m ∈ Z m ∈ Z where f ( x ) ∈ V m ⇐ ⇒ f (2 x ) ∈ V m +1 . Then, it exists a function ϕ ∈ V 0 generating an orthonormal basis, denoted by { ϕ m , k } k ∈ Z , for each V m , ϕ m , k ( x ) = 2 m / 2 ϕ (2 m x − k ). The function ϕ is called the scaling function or father wavelet . For any f ∈ L 2 ( R ), a projection map of L 2 ( R ) onto V m , denoted by P m : L 2 ( R ) → V m , is defined by means of � P m f ( x ) = c m , k ϕ m , k ( x ) , with c m , k = � f , ϕ m , k � . k ∈ Z ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 5 / 24
The SWIFT method In this work, we employ Shannon wavelets. A set of Shannon scaling functions ϕ m , k in the subspace V m is defined as, ϕ m , k ( x ) = 2 m / 2 sin( π (2 m x − k )) = 2 m / 2 ϕ (2 m x − k ) , k ∈ Z , π (2 m x − k ) where ϕ ( z ) = sinc ( z ), with sinc the cardinal sine function. Given a function f ∈ L 2 ( R ), we will consider its expansion in terms of Shannon scaling functions at the level of resolution m . Our aim is to recover the coefficients c m , k of this approximation from the Fourier transform of the function f , denoted by ˆ f , defined as � ˆ e − i ξ x f ( x ) d x , f ( ξ ) = R where i is the imaginary unit. ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 6 / 24
The SWIFT method Following wavelets theory, a function f ∈ L 2 ( R ) can be approximated at the level of resolution m by, � f ( x ) ≈ P m f ( x ) = c m , k ϕ m , k ( x ) , k ∈ Z where P m f converges to f in L 2 ( R ), i.e. � f − P m f � 2 → 0 , when m → + ∞ . The infinite series is well-approximated (see Lemma 1 of [2]) by a finite summation, k 2 � P m f ( x ) ≈ f m ( x ) := c m , k ϕ m , k ( x ) , k = k 1 for certain accurately chosen values k 1 and k 2 . ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 7 / 24
The SWIFT method Computation of the coefficients c m , k : by definition, � � ϕ m , k ( x ) d x = 2 m / 2 f ( x ) ϕ (2 m x − k ) d x . c m , k = � f , ϕ m , k � = f ( x ) ¯ R R Using the classical Vieta’s formula truncated with 2 J − 1 terms, the cosine product-to-sum identity and the definition of the characteristic function, the coefficients, c m , k , can be approximated by 2 J − 1 c m , k ≈ 2 m / 2 � (2 j − 1) π 2 m � � � i k π (2 j − 1) � ˆ ℜ f . e 2 J 2 J − 1 2 J j =1 Putting everything together gives the following approximation of f k 2 � f ( x ) ≈ c m , k ϕ m , k ( x ) . k = k 1 ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 8 / 24
SWIFT option valuation formulas Truncating the integration range on [ a , b ] and replacing density f by the SWIFT approximation, k 2 v ( x , t 0 ) ≈ e − rT � c m , k V m , k , k = k 1 where, � b V m , k := v ( y , T ) ϕ m , k ( y | x ) d y . a By employing the Vieta’s formula again and interchanging summation and integration operations, we obtain that 2 J − 1 � b V m , k ≈ 2 m / 2 � 2 j − 1 � � π (2 m y − k ) v ( y , T ) cos d y . 2 J − 1 2 J a j =1 ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 9 / 24
SWIFT for Asian options under exponential L´ evy models Exponential L´ evy models: log S ( t ), follows a L´ evy process. The L´ evy dynamics have a stationary and i.i.d. increments, fully described from its characteristic function. But, for arithmetic Asian options, the derivation of the corresponding characteristic function is rather involved. Lets start by defining the return or increment process R i , � S ( t i ) � i = 1 , . . . , N . R i := log S ( t i − 1 ) Based on R i , we define a new process Y i := R N +1 − i + Z i − 1 , i = 2 , . . . , N , � 1 + e Y i � where Y 1 = R N and Z i := log , ∀ i . ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 10 / 24
SWIFT for Asian options under exponential L´ evy models Applying the Carverhill-Clewlow-Hodges factorization to Y i , N � 1 + e Y N � 1 S 0 � S ( t i ) = . N + 1 N + 1 i =0 Thus, the option price for arithmetic Asian contracts can be now expressed in terms of the transitional density of the Y N as � v ( x , t 0 ) = e − rT v ( y , T ) f Y N ( y ) d y , R where x = log S 0 and the call payoff function is given by � + � S 0 (1 + e y ) v ( y , T ) = − K N + 1 Again, the probability density function f Y N is generally not known, even for L´ evy processes. However, as the process Y N is defined in a recursive manner, the characteristic function of Y N can be computed iteratively as well. ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 11 / 24
Characteristic function of Y N By the definition of Y i , the initial and recursive characteristic functions are f Y 1 ( ξ ) = ˆ ˆ f R N ( ξ ) = ˆ f R ( ξ ) , f Y i ( ξ ) = ˆ ˆ f R N +1 − i + Z i − 1 ( ξ ) = ˆ f R N +1 − i ( ξ ) · ˆ f Z i − 1 ( ξ ) = ˆ f R ( ξ ) · ˆ f Z i − 1 ( ξ ) . By definition, the characteristic function of Z i − 1 reads � (1 + e x ) − i ξ f Y i − 1 ( x ) d x . � e − i ξ log ( 1+ e Yi − 1 ) � ˆ f Z i − 1 ( ξ ) := E = R We can again apply the wavelet approximation to f Y i − 1 as k 2 � ˆ (1 + e x ) − i ξ � f Z i − 1 ( ξ ) ≈ c m , k ϕ m , k ( x ) d x R k = k 1 k 2 � ( e x + 1) − i ξ sinc (2 m x − k ) d x . m � = 2 c m , k 2 R k = k 1 ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 12 / 24
Characteristic function of Y N The integral on the right hand side needs to be computed efficiently to make the method easily implementable, robust and very fast. State-of-the-art methods from the literature rely on solving the integral by means of quadratures. Theorem (Theorem 1.3.2 of [3]) Let f be defined on R and let its Fourier transform ˆ f be such that for some positive constant d, | ˆ � e − d | ω | � f ( ω ) | = O for ω → ±∞ , then as h → 0 1 � � e − π d � f ( x ) S j , h ( x ) d x − f ( jh ) = O , h h R � x � where S j , h ( x ) = sinc h − j for j ∈ Z . Theorem 1 allows us to approximate the integral above provided that g ( x ) := ( e x + 1) − i ξ satisfies the hypothesis. ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method July 10, 2018 13 / 24
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