Technical analyis and optimal allocation strategies in the presence - PowerPoint PPT Presentation

Technical analyis and optimal allocation strategies in the presence of changes of instantaneous return rates and transaction costs C. Blanchet-Scalliet, R. Gibson Brandon, B. de Saporta, D. Talay, E. Tanré INRIA Sophia Antipolis, France TOSCA Project-team H. PHAM’09 – 18/09/09

Outline Introduction Our Model Main Result Numerical simulations

Outline Introduction Our Model Main Result Numerical simulations

Technical Analysis Why is technical analysis used? ◮ Technical analysis provides decision rules based on past prices behavior. It avoids model specification and thus model risk. ◮ With access to intra-daily financial data, short term traders in pursuit of “quick trades” use chartist methods of price regime changes detection. Few mathematical studies ◮ Pastukhov (2004): mathematical properties of volatility indicators. ◮ Shiryaev and Novikov (2008): exhibit an optimal one-time rebalancing strategy in the Black-Scholes model when the drift term of the stock may change its value spontaneously at some random non-observable time, ◮ Blanchet et al. (2007): a framework to compare the performances obtained by various strategies derived from erroneously calibrated mathematical models and from technical analysis; comparisons when the exact model is a diffusion model with one and only one change of stock returns at a random time.

Outline Introduction Our Model Main Result Numerical simulations

Our model The market: a deterministic short term rate r , a non risky asset with price process S 0 , and a stock with price process S whose instantaneous trend may only take two values µ 1 and µ 2 with µ 1 < r < µ 2 . The changes of trend may occur at random times τ 0 = 0 , τ n := ν 1 + · · · + ν n , where the ν j are independent, the ν 2 n + 1 (resp., ν 2 n ) are i.i.d., exponential with parameter λ 1 (resp., λ 2 ). Thus the trend process is � µ 1 if τ 2 n ≤ θ < τ 2 n + 1 , µ ( θ ) := µ 2 if τ 2 n + 1 ≤ θ < τ 2 n + 2 , and the stock price process is dS θ = µ ( θ ) d θ + σ dB θ . S θ

Admissible strategies Set: ◮ π θ : proportion of wealth invested at time θ in S , ◮ U : a utility function, ◮ W π : wealth process resulting from the strategy π . An investment strategy ( π θ ) over [ t , T ] is said admissible if it is a piecewise constant càdlàg process taking values in { 0 ; 1 } which is progressively measurable w.r.t the filtration F S := ( F S θ , 0 ≤ θ ≤ T ) and satisfies E | U ( W π T ) | < + ∞ . The set of such admissible strategies is denoted by A t .

The Optional Projection process is F θ := P ( µ ( θ ) = µ 1 | F S θ ) . Notice that � θ � � µ 1 F s + µ 2 ( 1 − F s ) − σ 2 � � B θ := 1 log S θ − ds σ S 0 2 0 is a F S Brownian motion, and that dF θ = ( − λ 1 F θ + λ 2 ( 1 − F θ )) d θ + µ 1 − µ 2 F θ ( 1 − F θ ) dB θ . σ We have: dS θ = ( µ 1 F θ + µ 2 ( 1 − F θ )) d θ + σ dB θ , S θ from which F S = F B .

We add proportional transaction costs . ◮ Given an amount W to transfer from the bank account to the stock, the cost is g 01 W . ◮ If W is transfered from the stock to the bank account, then the cost is g 10 W . Thus dW π θ = ( π θ ( µ 1 F θ + µ 2 ( 1 − F θ ) − r ) + r ) d θ W π θ − + π θ σ dB θ − g 01 I ∆π θ =  − g 10 I ∆π θ = −  . The continuous part Z of V π satisfies π θ ( µ 1 F θ + µ 2 ( 1 − F θ ) − σ 2 � � dZ π θ = 2 − r ) + r d θ + π θ σ dB θ .

Initial conditions, value functions Given t ∈ [ 0 , T ] , i ∈ { 0 , 1 } and π in A t , let ( F t , f , Z t , z , f ,π , W t , x , f , i ,π ) be issued at time t from f ∈ [ 0 , 1 ] , z ∈ R , and from x > 0 if π t = i , and from x ( 1 − g ij ) if π t = j = 1 − i . Denote by ξ t , i ,π the purely discontinuous part of W t , x , f , i ,π : ξ t , i ,π = − log ( 1 − g 01 ) I π t − i =  − log ( 1 − g 10 ) I π t − i = −  θ � [ log ( 1 − g 01 ) I ∆π s =  + log ( 1 − g 10 ) I ∆π s = −  ] − t < s ≤ θ we have W t , x , f , i ,π = x exp ( Z t , 0 , f ,π − ξ t , i ,π ) . θ θ θ Now set ∀ π ∈ A t , J i ( t , x , f , π ) := E [ U ( W t , x , f , i ,π )] T and define the value functions as V i ( t , x , f ) := sup J i ( t , x , f , π ) . π ∈A t

An elementary inequality Consider a utility function U which is, either the logarithmic utility function, or an element of the set U of the increasing and concave functions of class C 1 (( 0 , + ∞ ); R ) which satisfy: U ( 0 ) = 0 , and there exist real numbers C > 0 and 0 ≤ α ≤ 1 such that 0 < U ′ ( x ) ≤ C ( 1 + x − α ) for all x > 0 . Then there exists C > 0 such that, for all real numbers z, ˜ z and all positive real numbers x, ˜ x, and ζ , � � z − ζ �� xe ˜ � xe z − ζ � − U ˜ � U � � � � 1 + x − α e − α z + ˜ z � � e z + e ˜ z � x − α e − α ˜ ≤ C ( | x − ˜ x | + ( x + ˜ x ) | z − ˜ z | ) , where α = 1 if U ( x ) = log ( x ) .

Outline Introduction Our Model Main Result Numerical simulations

A system of variational inequalities − ∂ V 0  � � − L 0 V 0 ; V 0 ( t , x , f ) − V 1 ( t , x ( 1 − g 01 ) , f ) min = 0 ,   ∂ t  − ∂ V 1 � � − L 1 V 1 ; V 1 ( t , x , f ) − V 0 ( t , x ( 1 − g 10 ) , f ) min = 0 ,    ∂ t with the boundary condition V 0 ( T , x , f ) = V 1 ( T , x , f ) = U ( x ) , where L 0 ϕ ( t , x , f ) := xr ∂ϕ ∂ x ( t , x , f ) − ( λ 1 f − λ 2 ( 1 − f )) ∂ϕ ∂ f ( t , x , f ) � 2 f 2 ( 1 − f ) 2 ∂ 2 ϕ + 1 � µ 1 − µ 2 ∂ f 2 ( t , x , f ) , 2 σ and L 1 ϕ ( t , x , f ) := x ( µ 1 f + µ 2 ( 1 − f ) − r ) ∂ϕ ∂ x ( t , x , f ) − ( λ 1 f − λ 2 ( 1 − f )) ∂ϕ ∂ f ( t , x , f � 2 2 x 2 σ 2 ∂ 2 ϕ f 2 ( 1 − f ) 2 ∂ 2 ϕ + 1 ∂ x 2 ( t , x , f ) + 1 � µ 1 − µ 2 ∂ f 2 ( t , x , f ) 2 σ + x ( µ 1 − µ 2 ) f ( 1 − f ) ∂ 2 ϕ ∂ x ∂ f ( t , x , f ) .

Definition of viscosity solutions A pair of continuous functions ( V 0 , V 1 ) from [ 0 , T ] × ( 0 , + ∞ ) × [ 0 , 1 ] to R is a viscosity upper solution to the above system if V 0 ( T , x , f ) = V 1 ( T , x , f ) = U ( x ) and if, for all i � = j in { 0 , 1 } , all bounded function φ of class C 1 , 2 ([ 0 , T ] × R + × [ 0 , 1 ]) with bounded f ) of V i − φ , one has x , ˆ derivatives, and all local minimum (ˆ t , ˆ � − ∂φ � ∂ t (ˆ x , ˆ f ) − L i φ (ˆ x , ˆ f ); V i (ˆ x , ˆ f ) − V j (ˆ x ( 1 − g ij ) , ˆ t , ˆ t , ˆ t , ˆ t , ˆ min f ) ≥ 0 . A viscosity lower solution is defined analogously. Finally, a viscosity solution is both a upper and lower viscosity solution.

Theorem Let V α be the class of functions Υ which are continuous on [ 0 , T ] × [ 0 , + ∞ ) × [ 0 , 1 ] and satisfy: for all ( t , f ) ∈ [ 0 , T ] × [ 0 , 1 ] , Υ( t , 0 , f ) = 0 and there exists C > 0 such that | Υ( t , x , f ) | ≤ C ( 1 + x − α + x ) for all ( t , x , f ) ∈ [ 0 , T ] × ( 0 , + ∞ ) × [ 0 , 1 ] . Then the pair of value functions ( V 0 , V 1 ) is the unique viscosity solution of the above system in V α satisfying V 0 ( T , x , f ) = V 1 ( T , x , f ) = U ( x ) . If U is logarithmic, ( V 0 , V 1 ) is the unique viscosity solution in the set of function { log ( x ) + ¯ V ( t , f ) } where ¯ V is continuous on [ 0 , T ] × [ 0 , 1 ] .

A digression on the numerical resolution The preceding theorem allows one to use numerical solutions to π such that J i ( t , x , f , ¯ construct Markov allocation strategies ¯ π ) is close to V i ( t , x , f ) . To implement such a strategy, the investor needs to estimate F t at each time t from the observation of the prices ( S θ ; θ ≤ t ) . For some smooth functions α 1 and α 2 , dF θ = α 1 ( F θ ) d θ + α 2 ( F θ ) dS θ . S θ ◮ One can discretize this equation by using, e.g., the Euler scheme. ◮ Martinez, Rubenthaler and Tanré (2009) approximate F by a method based on filtering theory which is more accurate than the Euler approximation.

Some references We therefore had to solve a stochastic control problem which, to the best of our knowledge, had not been solved in the literature so far. Related works actually concern other dynamics: ◮ Tang and Yong (1993) study optimal switching and impulse controls. ◮ Brekke and Øksendal (1994) consider optimal switching in an economic activity. ◮ Pham (2007), Ly Vath, Pham (2007), Ly Vath, Pham, Villeneuve (2008) obtained results on families of models which do not include our model.

Continuity of the Value Functions Theorem: There exists C > 0 such that, for all i ∈ { 0 ; 1 } , 0 ≤ t ≤ ˆ t ≤ T, x and x in ( 0 , + ∞ ) , f and ˆ ˆ f in [ 0 , 1 ] , one has � � � V i (ˆ x , ˆ f ) − V i ( t , x , f ) t , ˆ � � � ≤ C ( 1 + x − α + ˆ � � x )( | ˆ x − α ) f − f | + | ˆ t − t | 1 / 2 ) | ˆ x − x | + ( x + ˆ . Corollary: For all β ≥ α , 0 ≤ s ≤ t ≤ T, and i, x, f, for all admissible control π ∈ A t , β � � ) − V i ( s , x e − ξ s , i ,π � V i ( t , W s , x , f , i ,π , F s , f < C ( t − s ) β/ 2 . , f ) E � t � t t �