Fuzzy Volatility Forecasts and Fuzzy Option Values

Transcription

1 Class of Volatility Models Fuzzy Volatility Forecasts and Fuzzy Option Values K. Thiagarajah Illinois State University, Normal, Illinois. 41st Actuarial Research Conference Montreal, Canada August 10-12, / 35

2 Class of Volatility Models Content Introduction Some Definitions Properties Fuzzy Random Variable Class of Volatility Models ARMA(l,m) Model with GARCH(p,q) Errors FC-GARCH(1,1) Model UNCV and Kurtosis Fuzzy Forecasts Call Option 2 / 35

3 Class of Volatility Models Introduction 3 / 35

4 Class of Volatility Models Volatility Important factor in Risk Management. Volatility modeling provides a simple approach to calculating Value at Risk (VaR) of financial position. Important factor in Option Trading. Black-Scholes formula for European options - Conditional variance of log return of the underlying stock (volatility) plays an important role. Modeling the volatility of a time series can improve the efficiency in parameter estimation and the accuracy in forecat intervals. 4 / 35

5 Class of Volatility Models Characteristics of Volatility Stock volatility is not directly observable. The characteristics that are commonly seen in asset returns: Volatility clusters exist. Volatility jumps are rare. Volatility varies within some fixed range. Volatility react differently to a big price increase/decrease. 5 / 35

6 Class of Volatility Models Some Definitions Properties Fuzzy Random Variable Preliminaries 6 / 35

7 Class of Volatility Models Some definitions Some Definitions Properties Fuzzy Random Variable Let A be a fuzzy subset of X. Then the support of A, S(A), is The height h(a) of A is S(A) = {x X : µ A (x) > 0} h(a) = Sup µ A (x) x X If h(a) = 1, then A is called a normal fuzzy set. The α-cut (interval of confidence at level-α) of the fuzzy set A A[α] = {x X : µ A (x) α}, α [0, 1] 7 / 35

8 Class of Volatility Models Some Definitions Properties Fuzzy Random Variable Some definitions A fuzzy number N is a fuzzy subset of R: and (i) the core of N is non-empty; (ii) α-cuts of N are all closed, bounded intervals, α (0, 1]; and (iii) the support of N is bounded. 8 / 35

9 Properties Introduction Class of Volatility Models Some Definitions Properties Fuzzy Random Variable Extension Principle: Any h : [a, b] R may be extended to H( X ) = Z as follows Z(z) = { sup X (z) h(x) = z, a x b } x for 0 α 1. z 1 (α) = min { h(x) x X (α) } z 2 (α) = max { h(x) x X (α) } 9 / 35

10 Properties Introduction Class of Volatility Models Some Definitions Properties Fuzzy Random Variable Properties: Let Ā and B be two fuzzy numbers such that Then, for all α (0, 1], (a) C[α] = Ā[α] + B[α] (b) C[α] = Ā[α] B[α] (c) C[α] = Ā[α]. B[α] (d) C[α] = Ā[α] B[α], Ā[α] = [a 1 (α), a 2 (α)] B[α] = [b 1 (α), b 2 (α)]. provided that zero does not belong to B[α] for all α (0, 1]. 10 / 35

11 Class of Volatility Models Mean, Variance and nth Moment Some Definitions Properties Fuzzy Random Variable X N( µ, σ 2 ) { } M[α] = xf (x; µ, σ 2 )dx µ µ[α], σ 2 σ 2 [α] V [α] = { } (x µ) 2 f (x; µ, σ 2 )dx µ µ[α], σ 2 σ 2 [α] Ē (X µ) n [α] = { } (x µ) n f (x; µ, σ 2 )dx µ µ[α], σ 2 σ 2 [α] 11 / 35

12 Class of Volatility Models Kurtosis and Fuzzy Paramer Some Definitions Properties Fuzzy Random Variable K[α] = Ē (X µ)4 [α] ( V [α] ) 2 { (x µ)4 f (x; µ, σ 2 )dx µ µ[α], σ 2 σ [α]} 2 = { ( ) 2 }. (x µ)2 f (x; µ, σ 2 )dx µ µ[α], σ 2 σ 2 [α] Fuzzy Parameter: Place the confidence intervals, one on top of the other, to produce a triangular shaped fuzzy number θ whose α-cuts are the confidence intervals. θ(α) = [θ 1 (α), θ 2 (α)], 0 α / 35

13 Class of Volatility Models ARMA(l,m) Model with GARCH(p,q) Errors FC-GARCH(1,1) Model Class of Volatility Models 13 / 35

14 Class of Volatility Models ARMA(l,m) with GARCH(p,q) Errors ARMA(l,m) Model with GARCH(p,q) Errors FC-GARCH(1,1) Model y t µ = j=0 ψ j a t j, where ψ j s are such that j=0 1 0 (ψ 2 j1(γ) + ψ 2 j2(γ))γ dγ <. The series {a t } is a GARCH(p,q) process given by a t = h t ɛ t, h t = p q ω + ϕ i at i 2 + β j h t j i=1 j=1 with mean zero, variance σ 2 a and kurtosis K (a). 14 / 35

15 Class of Volatility Models ARMA(l,m) with GARCH(p,q) Errors Let u t = a 2 t h t and σ 2 u = Var(u t ), then 1 p ϕ i B i i=1 where, Φ(B) = 1 r r = max(p, q). i=1 q j=1 a 2 t u t = ω + β j B j a 2 t = ω ARMA(l,m) Model with GARCH(p,q) Errors FC-GARCH(1,1) Model p ϕ i at i 2 + i=1 q β j B j u i, j=1 Φ(B)at 2 = ω + β(b)u t, r at 2 = ω + Ψ i u t i, i=1 Φ i B i, Φ i = ( ϕ i + β i ), β(b) = 1 q β j h t j, j=1 q j=1 β j B j and 15 / 35

16 Class of Volatility Models Stationary Assumptions ARMA(l,m) Model with GARCH(p,q) Errors FC-GARCH(1,1) Model (1) all the zeroes of the polynomial Φ(B) lie outside of the unit circle. (This assumption ensures that u ts are uncorrelated with zero mean and finite variance). (2) 1 (Ψ 2 i1(γ) + Ψ 2 i2(γ))γ dγ <, where Ψ(B) Φ(B) = β(b) with i=0 0 Ψ(B) = i=0 Ψ ib i. We can show that Var(y t ) = σ a 2 ψ 2 j, where σ a 2 = j=0 ω 1 r i=1 Φ i. 16 / 35

17 Class of Volatility Models Kurtosis of {y t } and Forecast Error K (y) = [ K (a) ψ 4 j j=0 ] ( ψj 2 j=0 + 6 ψ 2 i ψ 2 j i<j ARMA(l,m) Model with GARCH(p,q) Errors FC-GARCH(1,1) Model ) 2, provided j=0 ψ4 j <, where K (a) = E(ɛ 4 t ) E(ɛ 4 t ) [E(ɛ 4 t ) 1] Ψ 2 j j=0. Let y n (l) be the l-steps ahead forecast of y n+l. Then E[y n+l y n (l)] 2 ω l 1 = ψ 2 j. 1 Φ 1 Φ 2... Φ r j=0 17 / 35

18 Class of Volatility Models FC-GARCH(1,1) Model ARMA(l,m) Model with GARCH(p,q) Errors FC-GARCH(1,1) Model The classical GARCH(1,1) model takes the form y t = µ + a t, a t = h t ɛ t, h t = ω + ϕa 2 t 1 + βh t 1, where E(ɛ t ) = 0, Var(ɛ t ) = 1, E(ɛ 4 t ) = K (ɛ) + 3. K (ɛ) is the excess kurtosis of ɛ t. ω Var(a t ) = E(h t ) = 1 ϕ β ; E(a4 t ) = (K (ɛ) + 3)E(ht 2 ) E(ht 2 ω 2 (1 + ϕ + β) ) = (1 ϕ β)[1 ϕ 2 (K (ɛ) + 2) (ϕ + β) 2 ] 18 / 35

19 Class of Volatility Models Kurtosis of GARCH(1,1) Model Kurtosis of {a t }: ARMA(l,m) Model with GARCH(p,q) Errors FC-GARCH(1,1) Model K (a) = E(a4 t ) [E(a 2 t )] 2 = (K (ɛ) + 3)[1 (ϕ + β) 2 ] 1 2ϕ 2 (ϕ + β) 2 K (ɛ) ϕ 2. When ɛ t is normally distributed K (ɛ) =0. K (a) = 3[1 (ϕ + β) 2 ] 1 2ϕ 2 (ϕ + β) 2. With fuzzy Coefficients, K (a) = 3[1 ( ϕ + β) 2 ] 1 2 ϕ 2 ( ϕ + β) / 35

20 Class of Volatility Models UNCV and Forecast ARMA(l,m) Model with GARCH(p,q) Errors FC-GARCH(1,1) Model With fuzzy Coefficients, the UNC variance of {a t } is E(h t ) = E(σ t 2 ω ) = 1 ϕ β. At the forecast origin n, the one-step ahead volatility forecast is given by The l-step ahead volatility forecast is σ 2 n(1)[α] = ω + ϕā 2 n + β σ 2 n σ 2 n(l)[α] = ω + ( ϕ + β) σ 2 n(l 1), l > 1 The starting value of σ 2 0 is set to zero or E(σ2 t ). 20 / 35

21 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option 21 / 35

22 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Figure: 1 22 / 35

23 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Monthly Log Returns of IBM Stock - Fitted GARCH(1,1) model. Assume ε t are i.i.d standard normal. Fuzzy Parameters. Fuzzy UC variance. Fuzzy kurtosis Fuzzy Forecast. Fuzzy Call Option Value. 23 / 35

24 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Parameter Estimators Monthly Log Returns of IBM Stock - Fitted GARCH(1,1) model. Assume ε t are i.i.d standard normal. y t = µ + a t, a t = σ t ε t h t = σt 2 = ω + ϕat βσt 1 2 µ = (0.0023) ω = ( ) ϕ = (0.0214) β = (0.0422) 24 / 35

25 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option UNC Variance and Kurtosis E(h t ) = E(σ 2 t ) = ˆω 1 ˆϕ ˆβ = = K (a) = = 3[1 ( ˆϕ + ˆβ) ˆϕ 2 ( ˆϕ + ˆβ) 2 3[1 ( ) 2 1 2(0.0999) 2 ( ) 2 = / 35

26 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Fuzzy Estimation s with their α-cuts: [ ω = [ω 1 (α), ω 2 (α)] = Z α [ 2 ϕ = [ϕ 1 (α), ϕ 2 (α)] = Z α [ 2 β = [β 1 (α), β 2 (α)] = Z α 2 ( ), Z α 2 ] ( ) ] (0.0214), Z α (0.0214) 2 ] (0.0422), Z α (0.0422) 2 26 / 35

27 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Fuzzy UNC Variance and Kurtosis [ E[ h t ] = [E(h t,1 ), E(h t,2 )] = K (a) = [K (a) (a) 1 (α), K 2 (α)] [ = ω 1 (α) 1 ϕ 1 (α) β 1 (α), ω 2 (α) 1 ϕ 2 (α) β 2 (α) 3[1 (ϕ 1 (α) + β 1 (α)) 2 ] 1 2ϕ 2 1 (α) (ϕ 1(α) + β 1 (α)) 2, 3[1 (ϕ 2 (α) + β 2 (α)) 2 ] 1 2ϕ 2 2 (α) (ϕ 2(α) + β 2 (α)) 2 ]. ]. 27 / 35

28 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Fuzzy UNC Variance Figure: 2 28 / 35

29 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Fuzzy Kurtosis Figure: 1 29 / 35

30 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Fuzzy Forecast At the forecast origin n, the one-step ahead volatility forecast is given by σn(1) 2 = ω + ϕan 2 + βσn 2 = ( ) ( ) = With the fuzzy coefficients, σ 2 n(1)[α] = ω + ϕā 2 n + β σ 2 n = [ ω 1 (α) + ϕ 1 (α)ā 2 n + β 1 σ 2 n, ω 2 (α) + ϕ 2 (α)ā 2 n + β 2 σ 2 n]. 30 / 35

31 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Figure: 3 31 / 35

32 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Black-Scholes Formula European Call Option Value: C = S 0 N where ( log( S 0 σ2 X ) + (r + 2 )T ) σ Xe rt N T ( log( S 0 σ2 X ) + (r 2 )T ) σ T N(t) = 1 t 2π e Z2 2 dz. 32 / 35

33 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Fuzzy Call Option Values The α - cuts of call option value: and where C(α) = [C 1 (α), C 2 (α)] { } C 1 (α) = Min S 0 N (d1)) Xe rt N (d2) σ σ[α] C 2 (α) = Max d1 = d2 = { } S 0 N (d1) Xe rt N (d2) σ σ[α] ( log( S 0 σ2 X ) + (r + 2 )T ) σ, σ σ[α] T ( log( S 0 σ2 X ) + (r 2 )T ) σ, σ σ[α]. T 33 / 35

34 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Figure: 4 (S = 80, X = 90, r = 0.08, andt = 3months) 34 / 35

35 Class of Volatility Models UNCV and Kurtosis Fuzzy Forecasts Call Option Thank you for your attention! 35 / 35