Asian SWIFT method Efficient wavelet-based valuation of arithmetic Asian options ´ Alvaro Leitao, Luis Ortiz-Gracia and Emma I. Wagner Seminar - UB December 19, 2018 ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method December 19, 2018 1 / 45
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 December 19, 2018 2 / 45
Outline Definitions 1 Problem formulation 2 The SWIFT method 3 SWIFT for Asian options 4 Numerical results 5 Conclusions 6 ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method December 19, 2018 3 / 45
Definitions Option A contract that offers the buyer the right, but not the obligation, to buy (call) or sell (put) a financial asset at an agreed-upon price (the strike price) during a certain period of time or on a specific date (exercise date). Investopedia. Option price The fair value to enter in the option contract. In other (mathematical) words, the (discounted) expected value of the contract. v ( S , t ) = D ( t ) E [ P ( S ( t ))] where P is the payoff function, S the underlying asset, t the exercise time and D ( t ) the discount factor. ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method December 19, 2018 4 / 45
Definitions (II) Pricing techniques Stochastic process, S ( t ), governed by a SDE. Underlying models: Black-Scholes, L´ evy-based, Heston, etc. Simulation: Monte Carlo method. PDEs: Feynman-Kac theorem. Fourier inversion techniques: characteristic function. Types of options - payoff function Vanilla: involves only the value of S at exercise. Standardized. Exotic: involves more complicated features. Over the counter. Path-dependent: Asian, Barrier, Lookback, . . . ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method December 19, 2018 5 / 45
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 December 19, 2018 6 / 45
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 December 19, 2018 7 / 45
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 . 1 2 0.8 1.5 0.6 1 0.4 0.5 0.2 0 0 -0.2 -0.5 -10 -5 0 5 10 -10 -5 0 5 10 ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method December 19, 2018 8 / 45
The SWIFT method 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. 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 , m → + ∞ . The infinite series is well-approximated (see Lemma 1 of [3]) by 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 December 19, 2018 9 / 45
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 By using the classical Vieta’s formula, + ∞ � π x � � ϕ ( x ) = sinc ( x ) = cos . 2 j j =1 We truncate the infinite product into a finite product with J terms, then, thanks to the cosine product-to-sum identity, 2 J − 1 J � π x 1 � 2 j − 1 � � � � cos = cos π x . 2 J − 1 2 j 2 J j =1 j =1 Then, 2 J − 1 ϕ m , k ( x ) d x ≈ 2 m / 2 � � � 2 j − 1 � � π (2 m x − k ) f ( x ) ¯ f ( x ) cos d x . 2 J − 1 2 J R R j =1 ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method December 19, 2018 10 / 45
The SWIFT method � � ˆ � Noting that ℜ f ( ξ ) = R f ( x ) cos( ξ x ) d x and � � � ξ x − k π 2 j − 1 f ( ξ ) e i k π 2 j − 1 e − i ˆ = f ( x ) d x . 2 J 2 J R Thus, we have, 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 December 19, 2018 11 / 45
SWIFT option valuation formulas Truncating the integration range on [ a , b ] in the risk-neutral valuation formula, 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 December 19, 2018 12 / 45
SWIFT option sensitivities Under the SWIFT framework, the estimation of the option price sensitivities, the so-called Greeks . The Greeks are defined as the partial derivatives of the option price with respect to some market/model parameter. They can be efficiently calculated by constructing similar series expansions. Generally, two possible situations can appear: the option price depends only on the parameter of interest either through the density function or payoff function. The partial derivative of the characteristic function and, hence, the density coefficients and the payoff function can be analytically computed for many financial models and option contracts. ´ A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method December 19, 2018 13 / 45
SWIFT option sensitivities We firstly assume that the option price depends on the parameter of interest only through the density function, 2 J − 1 c m , k ( ξ, ς ) = 2 m / 2 i k ξ � � � ˆ ℜ f ( ξ ; ς ) e , 2 m 2 J − 1 j =1 where ξ = (2 j − 1) π 2 m and ς the parameter of interest. 2 J By differentiating ( n times) the characteristic function, the “Greek” density coefficients 2 J − 1 � ∂ n ˆ � m , k ( ξ ) := ∂ n c m , k ( ξ, ς ) = 2 m / 2 f ( ξ ; ς ) i k ξ c ( n ) � ℜ . e 2 m ∂ς n 2 J − 1 ∂ς n j =1 For example, the so-called Delta , ∆, and Gamma , Γ, the first and second derivatives w.r.t. S 0 , are computed by plugging the c ( n ) m , k , k 2 k 2 c (1) c (2) � � ∆ := e − rT Γ := e − rT m , k V m , k , m , k V m , k . ´ k = k 1 k = k 1 A. Leitao & L. Ortiz-Gracia & E. Wagner Asian SWIFT method December 19, 2018 14 / 45
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