A finite dimensional approximation for pricing American options on moving average Peter Tankov CMAP, Ecole Polytechnique Joint with M. Bernhart and X. Warin (EDF R&D) New advances in Backward SDEs for financial engineering applications Tamerza, Tunisia, October 25–28, 2010 Peter Tankov American options on movoing average
Introduction Surge options: American-style options whose strike is adjusted daily to the moving average of the underlying price: � t X t = 1 H t = ( S t − X t ) + , S u du . δ t − δ The strike of indexed swing options (gas market) is linked to moving averages of different oil prices. Non-Markovian dynamics of the moving average leads to an infinite-dimensional optimal stopping problem: dX t = 1 δ ( S t − S t − δ ) dt . We propose a finite-dimensional approximation allowing to price moving average options in PDE or LS Monte Carlo framework. Peter Tankov American options on movoing average
The approximation problem We consider general moving average processes of the form � ∞ M t = S t − u µ ( du ) 0 where µ is a finite possibly signed measure on [0 , ∞ ) and we set S t = S 0 for t ≤ 0. We would like to find n processes Y 1 , . . . , Y n such that ( S , Y 1 , . . . , Y n ) are jointly Markov, and M t is approximated by M n t which depends deterministically on S t , Y 1 t , . . . , Y n t . Peter Tankov American options on movoing average
Assumptions on S That the stock price S is a continuous Itˆ o process: � t � t S t = S 0 + b s ds + σ s dW s 0 0 with � � � � | σ s | 1+ γ E sup | b s | < ∞ and E sup < ∞ , γ > 0 0 ≤ t ≤ T 0 ≤ t ≤ T It can then be shown (Fischer and Nappo ’10) that the modulus of continuity of S is integrable: � � � � 2 T � E sup | S t − S s | ≤ C ε ( h ) , ε ( h ) := h ln . h t , s ∈ [0 , T ]: | t − s |≤ h Peter Tankov American options on movoing average
Comparing moving averages Lemma Let µ and ν be finite signed measures on [0 , ∞ ) such that µ + ( R + ) > 0 , and let M and N be corresponding moving average processes. Then � � sup | M t − N t | ≤ C | µ ( R + ) − ν ( R + ) | E 0 ≤ t ≤ T � T � 1 � + C ε | F µ ( t ) − F ν ( t ) | dt µ + ([0 , T ]) 0 where F ν ( t ) := ν ([0 , t ]) and F µ ( t ) := µ ([0 , t ]) . Peter Tankov American options on movoing average
Sketch of the proof We first assume that µ and ν are probability measures. Let F − 1 µ and F − 1 be generalized inverses of µ and ν respectively. Then, ν � 1 � � � � sup | M t − N t | = E sup | S t − F − 1 µ ( u ) − S t − F − 1 ( u ) | du E ν 0 ≤ t ≤ T 0 ≤ t ≤ T 0 �� 1 � | F − 1 µ ( u ) ∧ T − F − 1 ≤ C ε ν ( u ) ∧ T | du . 0 The expression inside the brackets is the Wasserstein distance between µ and ν truncated at T . Therefore, �� T � � � sup | M t − N t | ≤ C ε | F µ ( t ) − F ν ( t ) | dt . E 0 ≤ t ≤ T 0 Peter Tankov American options on movoing average
Laguerre approximation: idea We assume M n = B ⊥ Y dY t = − AYdt + 1 ( α S t dt + β dS t ) , The solution can be written as � t B ⊥ e − A ( t − s ) 1 ( α S s ds + β dS s ) M n t = −∞ � t = K n S t + h n ( t − u ) S u du , −∞ where h n is of the form (Hankel approximation) K n k � e − p k t � c k i t i , h n ( t ) = n 1 + . . . + n K + K = n k =1 i =0 In this work we focus on a subclass for which h n is of the form n − 1 � h n ( t ) = e − pt c i t i (Laguerre approximation) i =0 Peter Tankov American options on movoing average
Laguerre polynomials and functions Fix a scale parameter p > 0. The scaled Laguerre functions ( L p k ) k ≥ 0 are defined on [0 , + ∞ ) by L p � 2 p P k (2 pt ) e − pt , k ( t ) = ∀ k ≥ 0 where ( P k ) k ≥ 0 are Laguerre polynomials: � k k � ( − t ) i � P k ( t ) = , ∀ k ≥ 0 k − i i ! i =0 The Laguerre functions ( L p k ) k ≥ 0 form an orthonormal basis of L 2 ([0 , ∞ )). Peter Tankov American options on movoing average
Laguerre approximations for moving averages Let H ( x ) = µ ([ x , + ∞ )). Compute the Laguerre coefficients of the function H : A p k = � H , L p k � . n ( t ) = � n − 1 Set H p k =0 A p k L p k ( t ) and h p dt H p n ( t ) = − d n ( t ). Approximate the moving average M with � + ∞ M n , p = ( H (0) − H p h p n (0)) S t + n ( u ) S t − u du . t 0 Peter Tankov American options on movoing average
Laguerre approximations for moving averages Introduce n random processes � + ∞ X p , k L p = k ( v ) S t − v dv , k = 0 , . . . , n − 1 . t 0 They are related to the moving average approximation by n − 1 M n , p � a p k X p , k = ( H (0) − H p n (0)) S t + , ∀ t ≥ 0 . t t k =0 and have Markovian dynamics � √ 2 pS t − pX p , 0 � dX p , 0 = dt t t . . . � √ 2 pS t − 2 p � k − 1 � dX p , k i =0 X p , i − pX p , k = dt t t t with initial values √ 2 p X p , k = S 0 ( − 1) k , ∀ k ≥ 0 . 0 p Peter Tankov American options on movoing average
Convergence rate Theorem Suppose that the moving average process M is of the form � ∞ M t = K 0 S t + S t − u h ( u ) du 0 where K 0 is a constant and the function h has compact support, finite variation on R and is constant in the neighborhood of zero. Then the approximation error admits the bound � � ≤ C ε ( n − 3 | M t − M n , p 4 ) . sup | E t 0 ≤ t ≤ T Peter Tankov American options on movoing average
Approximating option prices One can approximate E [ φ ( S τ , M n , p sup E [ φ ( S τ , M τ )] by sup )] . τ τ ∈T τ ∈T Corollary Let the payoff function φ be Lipschitz in the second variable, then the pricing error admits the bound � � � ≤ C ε ( n − 3 E [ φ ( S τ , M n , p � � 4 ) . � sup E [ φ ( S τ , M τ )] − sup )] � τ � τ ∈T τ ∈T where C > 0 is a constant independent of n. Peter Tankov American options on movoing average
Uniformly weighted moving averages Let µ ( dx ) = h ( x ) dx = 1 H ( x ) = 1 δ ( δ − x ) + δ 1 [0 ,δ ] dx ⇒ The coefficients A δ, p = � H , L p k � can be computed explicitly. k We determine the optimal scale parameter p opt ( δ, n ) as � n − 1 � δ 2 � � � A δ, p � H − H p � p opt ( δ, n ) = arg min n � 2 = arg min 3 − . � � k � p > 0 p > 0 k =0 It satisfies the scaling relation p opt ( δ, n ) = p opt (1 , n ) , δ and the values p opt (1 , n ) can be tabulated. Peter Tankov American options on movoing average
Uniformly weighted moving average: illustration 115 110 105 100 95 90 Spot price S Moving average Moving average approx. with n = 1 85 Moving average approx. with n = 3 Moving average approx. with n = 7 80 0 5 0 5 0 5 0 5 0 5 0 1 1 2 2 3 3 4 4 5 Time step Simulated trajectory of the moving average process and its Laguerre approximations. Peter Tankov American options on movoing average
Least squares Monte Carlo Replace the American option by a Bermudan one with a discrete grid π = { 0 = t 0 , t 1 , . . . , t N = T } of exercise dates. Simulate M paths of stock price and Laguerre processes. Compute optimal exercise times by backward induction: Initialization: τ π, m = T , m = 1 , . . . , M 1 N Backward induction for i = N − 1 , . . . , N δ , m = 1 , . . . , M : 2 τ π, m i + τ π, m � = t i 1 A m i +1 1 ∁ A m i i � � � ��� S π, m , M n ,π, m i +1 , M n ,π A m � � ≥ E M S π i = φ φ τ π τ π t i t i t i i +1 Estimation of the option price at time 0: 3 M � � 0 = 1 � S π, m N δ , M n ,π, m V π φ τ π, m τ π, m M N δ m =1 The conditional expectations are estimated by regression on the basis functions of state variables Peter Tankov American options on movoing average
Least squares Monte Carlo In numerical examples, we find that best results are obtained if the approximate moving average M n ,π is replaced by the true moving average M π in the pay-off function, while estimating the conditional expectations by regressions on S π , X 0 ,π , . . . , X n ,π . The suboptimal approach often used by practitioners consists in estimating the conditional expectations by regression on ( S π , M π ) only. Peter Tankov American options on movoing average
Numerical examples: convergence 4.35 Lag- LS Lag- LS* 4.280 4.30 4.278 4.276 4.25 4.274 Option value 4.272 4.20 4.270 4.15 4.268 (Lag-LS*) 4.266 Benchmark by (NM-LS) 4.10 4.264 0 1 2 3 4 5 6 7 8 n 4.05 1 2 3 4 5 6 7 Left: Laguerre approximation vs. the improved method. Right: zoom for the improved method and the practitioner’ method. Peter Tankov American options on movoing average
Numerical examples: delayed options 7.00 6.50 6.00 5.50 5.00 Option value 4.50 4.00 3.50 3.00 (NM-LS) (Lag-LS*) 2.50 2.00 0 5 10 15 20 25 30 35 40 45 time lag number of time steps For moving average options with time delay whose pay-off depends � τ − l on X τ = 1 τ − l − δ S u du , the Laguerre approximation leads to a δ substantial improvement compared to the practitioner’s method. Peter Tankov American options on movoing average
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