CH Sean Han QF, NTHU, Taiwan BFS2010. (Joint work with T.-Y. Chen and W.-H. Liu)

Transcription

1 CH Sean Han QF, NTHU, Taiwan BFS2010 (Join work wih T.-Y. Chen and W.-H. Liu)

2 Risk Managemen in Pracice: Value a Risk (VaR) / Condiional Value a Risk (CVaR) Volailiy Esimaion: Correced Fourier Transform Mehod Esimae Exreme Probabiliy by Efficien Imporance Sampling Backesing for VaR Esimaion

3 Le r( ) be an asse reurn a ime. Is α 100% VaR, denoed by VaR α, is defined by he 1 α 100% percenile of r( ). Tha is, ( ) ( ( ) ) 1 P r VaR α = α Tha is a risk conroller has a α 100% confidence ha he asse price will no drop below VaR α a ime.

4 Mahemaically, i is no a coheren risk measure* because i doesn' saisfy he risk diversificaion principal. Insead, CVaR does!. Pracically, VaR is commonly required by financial regulaions (Basel Ⅱ Accord). * Arzner P., F. Delbaen, J.-M. Eber, and D. Heah, Coheren Measures of Risk, Mahemaical Finance, 9 (1999):

5 Riskmerics: normal assumpion under EWMA model. Hisorical Simulaion: generae scenarios Model Dependen Approach: Discree-Time vs. Coninuous-Time Models

6 Assume a diffusion process du d dw ( ) = µ ( ) + σ ( ), Task: Given reurn ime series he volailiy σ ( ). u ( ), esimae * Malliavin and Mancino(2002, 2009)

7 Compue he Fourier coefficiens of du by Then, 1 2π a ( ) ( ) 0 du = du, 2π 0 1 2π a ( ) cos ( ) ( ) k du = k du, π 0 1 2π b ( ) sin ( ) ( ) k du = k du. π 0 ( ) ( ) bk du ak du u ( ) = a0 + cos( k ) + sin ( k ). k = 1 k k

8 Fourier coefficiens of variance σ N k ( 2 ) π *( ) * ( ) *( ) * a lim ( ) k σ = as du as+ k du + bs du bs k du +, N 2N + 1 N k ( 2 ) π *( ) * ( ) *( ) * b lim ( ) k σ = as du bs+ k du bs du as k du +, N 2N + 1 where n 0 s= N s= N ( ) is any posiive ineger so ha N N k k k = 0 ( ) = a ( ) ( k ) + a ( ) ( k ) σ σ cos σ sin. 2,

9 2 Reconsruc he ime series variance σ ( ). Finally, σ 2 ( ) 2 is an approximaion of σ ( ) as N N approaches infiniy, which can be given by classical Fourier-Fejer inversion formula. σ = lim σ in prob. ( ) ( ) 2 2 N N

10 We add a funcion ino he final compuaion of ime series variance in order o smooh i. N k k N k 0 2 ( ) = ( k ) a ( ) ( k ) + b ( ) ( k ) σ lim ϕ δ σ cos σ sin, = ϕ x = ( ) sin x ( ) where 2 is a funcion in order o x smooh he rajecory and δ is a smoohing parameer. Reno (2008) alers he boundary effec in he Fourier ransform mehod.

11 Idea: (Nonlinear) Leas Squares Mehod for firs-order correcion r σ δε (( a by ˆ ) ) ex p + 2 δ ε. Then by MLE o regress ou a and 2 r = a + byˆ + ε 2 ln ln. δ b

12

13 Assuming ha he driving volailiy process is governed by he Ornsein-Uhlenbeck process, dy = α m Y d + βdw ( ). (1) We use he correced esimaor a + byˆ o furher esimae model parameers αβ of Yˆ by means of maximum likelihood mehod.,,m ( )

14 For a given se of observaions he likelihood funcion is Y1, Y2,..., YN ( ) N 1 1 ( ) ( ( ) ) 2 L αβ,, m = exp Y αm + 1 α Y, = 1 2πβ 2β where denoes he lengh of discreized ime inerval.

15 By maximizing he righ hand side over he parameers ( αβ,,m), we obain he following maximum likelihood esimaors N N 1 N 1 Y Y ( N 1) YY = 2 = 1 = 1 ˆ α = 1, N 1 2 N 1 2 Y ( N 1) Y = 1 = 1 N 1 1 ˆ= β Y+ 1 ( αm + ( 1 α ) Y), N = 1 N N 1 N 1 N 1 2 Y Y Y YY = 2 = 1 = 1 = 1 ˆm=, 1 2 ˆ α N N 1 2 Y ( N 1) Y = 1 = 1 2 (3) (4) (5)

16 Le he sochasic volailiy model ds ( ) = µ Sd + exp Y 2 SdW1, dy ( - ) = α m Y d + βdw2. To empirically es our price correcion scheme, we se model parameers as follows: µ = 0.01, S = 50, Y = 2, m= 2, α=5, β = 1, 0 0 and wih he discreizaion lengh so as o generae volailiy series and asse price series. S =1/5000 σ = exp( Y 2)

17 Two crieria are used for performance comparison: Mean squared errors (MSE) and Maximum absolue errors (MAE). Comparison resuls are shown below: Fourier mehod Correced Fourier mehod Mean squared error Maximum absolue error

18 Given a Markovian dynamic model of an asse price, is reurn process is r = ln S S. S ( ) Given a loss hreshold D, he exreme probabiliy is defined by P( 0, S ) ( ) 0; D = E I rt D S0. Noe: solve VaR α from P ( 0, S ) 0; VaR α = 1 α. =. (Expeced Shorfall) CVaR E rt rt VaR α T T 0

19 Given he Black-Scholes Model under measure P, choose dp Q ( h) exp hw saisfing Then = T = T dp E [ S ] exp 0 ( ). T = S D µ D h =, σ σt he exreme probabiliy P( 0, S ) ( ) ( ) 0 = E I rt D QT h S0. 2 h T becomes (6) The unbiased imporance sampling esimaor N 1 ( i) i of P( 0, S ) is I r D Q h. (7) 0 N i = 1 ( ) ( ) T T ( ) 2

20

21 Theorem: Under he Black-Scholes model, he proposed imporance sampling esimaor is asympoically opimal or efficien under some scaling scenarios in ime and space. Proof: The variance rae of he proposed imporance sampling scheme approaches zero.

22 Sochasic volailiy model: ds = µ Sd + σsdw1 σ exp( 2) = Y dy = α( m -Y ) d + βdw 2 Ergodic propery of he averaged variance process 1 T T as 2 ε f ( Y ) d σ,for ε 0 where ε denoes a small ime scale and Y ε denoes a fas mean-revering process.

23 E.g. 1 2ν α =, β =. ε ε So ha he imporance sampling as foremenioned can be applied. CVaR esimaion can be easily solved.

24 c ρ = VaR 99% Given model parameers of sochasic volailiy: m 5, α 5, β 1, S0 50, = = = = = Y 0 3, µ =0 ρ c = VaR 99% CVaR N. Approx. IS (7.5073E-05) (7.3833E-05) (7.2498E-05) (7.2739E-05) (6.9643E-05)

25 Daa sample period:

26 Daa sample period:

27 Daa sample period: RiskMerics Significance 1% Significance 5% LRuc Rejec VaR Model LRuc Rejec VaR Model LRind Rejec VaR Model LRind Don' Rejec VaR Model LRcc Rejec VaR Model LRcc Rejec VaR Model Hisorical Simulaion Significance 1% Significance 5% LRuc Rejec VaR Model LRuc Rejec VaR Model LRind Don' Rejec VaR Model LRind Don' Rejec VaR Model LRcc Rejec VaR Model LRcc Rejec VaR Model SV Significance 1% Significance 5% LRuc Don' Rejec VaR Model LRuc Rejec VaR Model LRind Don' Rejec VaR Model LRind Don' Rejec VaR Model LRcc Don' Rejec VaR Model LRcc Rejec VaR Model GARCH(1,1) Significance 1% Significance 5% LRuc Rejec VaR Model LRuc Rejec VaR Model LRind Don' Rejec VaR Model LRind Rejec VaR Model LRcc Rejec VaR Model LRcc Rejec VaR Model

28 Remove boundary effec of Fourier ransform mehod for volailiy esimaion. (efficien) imporance sampling mehods are invesigaed. VaR backesing resuls for FX and equiy daa.

29 Thank You