The influence of risk-neutral measures

Recall Model (6) and let the intensity rate be a constant θ. By Proposition 3.2, if the jump-size is N (µy, σ_y²) distributed under the P measure, then Yt,n

i.i.d

∼ N (˜µy, σ²_y) under the Q_cmeasure, where Q_c denotes the risk-neutral measure corresponding to a specific constant c satisfying (23) and (24), ˜µy = µy + δ^L_tσ²_y, and δ_t^L is obtained from solving (11) with a fixed c. In addition, the intensity rate of the Poisson process Nt under Qc becomes ˜θ = M_Yt,n|Ft−1(δ^L_t)θ. Since different choices of c correspond to different risk-neutral measures Q_c, we investigate the influence of the choices of c on the CatEPut prices. Note that the parameters δ_t^G and δ_t^L defined in (10) and (11), respectively, when proceeding of the the Esscher transform in the GARCH and Jump parts, are determined by c.

1. If c = θ[exp(−µ_y+ 0.5σ²_y) − 1], which corresponds to the Merton measure, then δ_t^L= 0 and δ_t^G = −{θ[exp(−µy+ 0.5σ_y²) − 1] + λσt}/σ_t².

2. If c = 0, which corresponds to the risk-neutral measure in Amin (1993) when T = 1, then δ^L_t = 0.5−µy/σ_t² and δ^G_t = −λ/σt, which is the same as the results of Siu, et al. (2004) and Huang, et al. (2012) in the GARCH framework.

3. In particular, if δ_t^L= δ_t^G= δ^∗_t, then by (10) and (11) δ_t^∗satisfies exp[µy(δ_t^∗−1)+

0.5σ²_y(δ_t^∗−1)²]−exp[µ_yδ^∗_t+0.5σ²_y(δ_t^∗)²] = −(λσ_t+δ_t^∗σ_t²)/θ and c = −(λσ_t+δ_t^∗σ²_t).

Table 3 presents the CatEPut prices under various k = c {θ[exp(−µy + 0.5σ_y²) − 1]}⁻¹ with θ = 0.4, µ_y = 0.02, 0.04, 0.08, σ²_y = 0.4 and T = 1, 3, 5. For the case of δ_t^L = δ^G_t = δ_t^∗, we report the values of k with c = −(λσ1 + δ₁^∗σ₁²), that is, k = −0.3322, −0.3401 and −0.3608 if µ_y = 0.02, 0.04 and 0.08, respectively. The CatEPut prices presented in Table 3 versus k are shown in Figure 8, which indicates that the CatEPut prices decrease as k (or c, since c is proportional to k) increasing.

That is, different Qcmeasure computes different CatEPut prices. From Table 3, we find that the CatEPut prices increase in T for all the different Q_c measures, which is the same as the phenomenon observed in Section 4.3, where only Merton measure is considered. Interestingly, as µ_y increasing, the CatEPut prices increase for all the different Qcmeasures if T = 1 while the CatEPut prices decrease for all Qcif T = 5.

5 Conclusion

This study establishes a general approach to derive a risk-neutral GARCH-Jump model with stochastic intensity rate by the Esscher transform. The CatEPut option prices are computed under the GARCH-Jump model. Sensitivity analysis of the influence of the model parameters on the CatEPut prices are conducted by sim-ulation. Numerical results indicate that there is no significant difference between the GARCH-Jump model and jump-diffusion model for CatEPut prices. Table 4 summarizes the influence of the intensity rate of the occurrence of catastrophes, the distribution assumption of the jump size, the time to maturity, and the selection of risk-neutral measure on pricing CatEPut options in GARCH-Jump model. In general, the CatEPut prices increase as the intensity rate, the expected value of the jump size, the variance of the jump size or the time to maturity increasing.

Appendix

Proof of Lemma 3.1. Let η_t = (log θ_t)/σ_θ and α = (µ_θ− 0.5σ²_θ)/σ_θ. By (7), we have

dη_t= αdt + dW_t. (19)

Further let

M_t= max{η_s : s ∈ [t − 1, t]}. (20) By (19), (20) and Corollary 7.2.2 in Shreve (2004), the pdf of Mt is

f_Mt(m) = 2

√2πe^{−0.5(m−α)2} − 2α e^{2α m}Φ(−m − α), (21)

where m ≥ 0 and Φ(·) denotes the cdf of N (0, 1). By Taylor’s expansion and (20), we have

M_θ^∗

t|Ft−1(h) =

∞

X

j=0

h^j

j !E_t−1[(θ^∗_t)^j] ≤

∞

X

j=0

h^j

j !E_t−1[e^jσθMt]. (22) By (21) and noting that

Z ∞ 0

e^{jσθm−0.5(m−α)2}dm = O(e^{jασθ+0.5j2σ2θ}) and

Z ∞ 0

e^jσθm+2αmΦ(−m − α)dm = O(j⁻¹e^{jασθ+0.5j2σ2θ}), the right-hand-side of (22) is finite. Consequently, M_θ^∗_t|Ft−1(h) exists.

However, it is difficult to obtain closed-form representation of M_θ^∗

t|Ft−1(h). In this lemma, we approximate it by the following second order Taylor’s expansion:

M_θ^∗

t|Ft−1(h) ≈ e^hµθ∗t(1 + h² 2σ_θ^2∗

t), where µ_θt^∗ = E_t−1(θ_t^∗) and σ_θ^2∗

t = Var_t−1(θ^∗_t) are the conditional mean and variance of θ^∗_t given F_t−1, respectively. In the following, we derive the closed-form represen-tations for µ_θt^∗ and σ²_θ^∗

t.

Let Xn= n⁻¹Pn−1

where the inequality holds by Cauchy-Schwarz inequality. By Theorem 4.5.4 in Chung (2001), E_t−1(θ^∗_t)^r = lim

n→∞E_t−1(|X_n|^r), for 0 < r ≤ 2. Therefore, the condi-tional mean and condicondi-tional variance of θ_t^∗ given F_t−1 can be computed as µ_θ^∗

t = Proof of Proposition 3.1. Let Λ_T be a Radon-Nikod´ym derivative and the corre-sponding risk-neutral measure Q be defined by dQ = ΛTdP . Recall the scheme of the Esscher transform introduced in Section 2.1 and the equation, R_t= R^G_t − L_t, in Model (6). By using similar arguments as in Shreve (2004), ΛT is decomposed by

Λ_T = Λ^R_T^G× Λ^L_T,

where Λ^R_T^G and Λ^L_T are two independent Radon-Nikod´ym derivatives such that E^Q_t−1(e^RtG) = E_t−1Λ^R_t^G

Λ^R_t−1^G e^RGt 

= e^r−c (23)

and

Consequently, the Radon-Nikod´ym derivative Λ^R_T^G derived by the Esscher transform is In Model (6), the conditional density of Lt|Ft−1 under P is

f_Lt|F_t−1(x_t) = Define a new conditional pdf of Lt|Ft−1 with a parameter δt

f_Lt|F_t−1(x_t; δ_t) = e^xtδt

M_Lt|Ft−1(δ_t)f_Lt|F_t−1(x_t), (31)

Then, choose a δt = δ_t^L in (31) for 0 ≤ t ≤ T such that (24) holds where δ^L_t is the solution of (11). Consequently, the Radon-Nikod´ym derivative Λ^L_T derived by the Esscher transform is

Λ^L_T =

T

Y

t=1

e^{δLt Lt}

E_t−1[e^{h(δLt) θt∗}], (32) where h(δ^L_t) = M_Yt,n|Ft−1(δ_t^L) − 1 and E_t−1[e^{h(δLt) θ∗t}] can be approximated either by Lemma 3.1 or Monte Carlo simulation. In addition, if Λ^L_T in (32) is further decomposed by

Λ^L_T = Λ^L|θ_T ^∗× Λ^θ_T^∗, (33) where Λ^L|θ_T ^∗ is the Radon-Nikod´ym derivative of Q with respect to P for LT condi-tional on θ^∗_T, and Λ^θ_T^∗ is the Radon-Nikod´ym derivative for θ^∗_T. Note that Λ^L|θ_t ^∗ can be obtained by similar arguments used in (29)-(32), that is,

Λ^L|θ_T ^∗ =

T

Y

t=1

e^{δLt Lt}

e^{h(δLt) θ∗t}. (34)

By (32)-(34), we have

Λ^θ_T^∗ =

T

Y

t=1

e^{h(δLt) θ∗t}

E_t−1[e^{h(δLt) θt∗}]. (35)

Finally, the desire result holds by (33)-(35). 

Proof of Proposition 3.2. By (28) in the proof of Proposition 3.1, the change of measure processes Λ^R_t^G be defined by Λ^R_t^G = E_t(Λ^R_T^G) for t = 1, . . . , T . Choosing δ^G_t in (10) the conditional mgf of R^G_t |F_t−1 under measure Q is

M^Q_RG

t|Ft−1(z; δ_t^G) = E^Q_t−1(e^{z RGt} ) = E_t−1(Λ^R_t^G Λ^R_t−1^Ge^{z RGt} )

= e^{(r−c−0.5σt2)z+0.5σ2tz2}, (36) which has the same form as the conditional mgf of R^G_t|F_t−1 under P given in (25).

That is, R^G_t|F_t−1 is normally distributed with mean r − c + σ²_t/2 and variance σ²_t/2 under Q.

By (34) and (35) in the proof of Proposition 3.1, the change of measure processes Λ^L|θ_t ^∗ and Λ^θ_t^∗ are defined by

Λ^L|θ_t ^∗ = Et(Λ^L|θ_T ^∗) and Λ^θ_t^∗ = Et(Λ^θ_T^∗), (37) for t = 1, . . . , T . By (33), the conditional mgf of L_t|F_t−1 under measure Q is M^Q_L

t|Ft−1(z; δ_t^L) = E^Q_t−1(e^{z Lt})

= E_t−1( Λ^L_t

Λ^L_t−1e^{z Lt}) = E_t−1[ Λ^θ_t^∗

Λ^θ_t−1^∗ E_t−1(e^{z Lt}Λ^L|θ_t ^∗ Λ^L|θ_t−1^∗|θ_t^∗)]

= E_t−1n e^{h(δLt) θ∗t}

E_t−1(e^{h(δLt) θt∗})exp{θ^∗_tM_Yt,n|F_t−1(δ_t^L)[M_Yt,n|F_t−1(z + δ_t^L)

M_Yt,n|Ft−1(δ^L_t) − 1]}o (38) where the last equality holds by (34), (35) and (37).

In addition, let ψ_θ^∗

t|F_t−1(x) denote the conditional pdf of θ^∗_t|F_t−1 under P . By (35) and (37), the conditional density of θ_t^∗|Ft−1 under Q can be represented as

ψ˜_θ^∗

t|Ft−1(x) = e^{h(δtL) x}

E_t−1(e^{h(δtL) θt∗})ψ_θ^∗

t|Ft−1(x). (39)

By (39) and let ˜θ_t^∗ = θ_t^∗M_Yt,n|Ft−1(δ^L_t), the conditional mgf obtained in (38) can be rewritten as

M^Q_L

t|Ft−1(z; δ^L) = E^Q_t−1{exp[˜θ^∗_t(M_Yt,n|Ft−1(z + δ_t^L)

M_Yt,n|F_t−1(δ_t^L) − 1)]}. (40)



Table 1: The CatEPut prices and P (L₁ > L) (in parentheses) versus intensity rates under the GARCH-Jump model with constant, gamma and normally distributed jump sizes in the Merton measure with T = 1.

Case 1 Case 2 Case 3

θ A = 0.02 Ga(10, 0.002) N (0.02, 4 × 10⁻⁵) Ga(0.1, 0.2) N (0.02, 4 × 10⁻³) Ga(10⁻³, 20) N (0.02, 4 × 10⁻¹)

0.2 9.1009 9.1013 9.0808 3.1014 5.9534 0.0663 5.6801

(0.1813) (0.1813) (0.1803) (0.0566) (0.1098) (0.0016) (0.0926)

0.4 17.4452 17.3270 17.0452 4.9773 10.2147 0.1987 10.0838

(0.3297) (0.3275) (0.3219) (0.0924) (0.194) (0.0029) (0.1678)

0.6 23.1722 21.8047 21.6306 6.4104 13.9292 0.3020 14.3979

(0.4512) (0.4236) (0.4172) (0.1202) (0.2584) (0.0041) (0.2289)

0.8 28.3179 23.1340 23.4905 7.6496 16.1245 0.3949 17.6613

(0.5507) (0.4488) (0.4554) (0.1428) (0.3074) (0.0052) (0.2786)

1 13.5460 22.1135 22.6213 8.7856 17.9866 0.4018 20.8239

(0.2642) (0.4321) (0.4458) (0.162) (0.3447) (0.0063) (0.319)

1.2 17.7409 21.9585 21.9910 9.5113 19.7644 0.4333 22.5657

(0.3374) (0.4204) (0.4246) (0.1784) (0.3729) (0.0073) (0.3518)

Table 2: The CatEPut prices under GARCH-Jump model with stochastic intensity rate and jump sizes is N (0.02, 0.4) distributed in the Merton measure.

θ0

T (µθ, σθ) µθ− 0.5σ²_θ 0.2 0.4 0.6 0.8 1 (0.1, 0) 0.1 6.2771 11.3545 15.4044 18.9942

(0.1, 0.3) 0.055 6.2392 11.1265 15.1625 18.5459 (0.1, 0.4) 0.02 6.0868 10.9545 15.0098 18.3812 (0.2, 0.6) 0.02 6.5768 11.5850 15.4185 18.2758 3 (0.1, 0) 0.1 14.7899 22.6858 26.8348 29.3594 (0.1, 0.3) 0.055 14.3457 21.6676 25.6937 28.5816 (0.1, 0.4) 0.02 13.9540 21.0800 25.2308 27.6481 (0.2, 0.6) 0.02 14.5843 21.1960 24.8104 27.2773 5 (0.1, 0) 0.1 19.1988 25.8191 28.1968 29.1458 (0.1, 0.3) 0.055 18.1266 24.2215 26.9488 28.4127 (0.1, 0.4) 0.02 17.4332 23.4287 26.2077 27.5207 (0.2, 0.6) 0.02 17.7179 23.0994 25.7644 27.1952

Table 3: The CatEPut prices versus k under normally distributed jump sizes, where k = c{θ[exp(−µ_y+ 0.5σ_y²) − 1]}⁻¹ with θ = 0.4 and σ_y² = 0.4.

µy= 0.02 k -0.3322 0 0.25 0.5 0.75 1

T = 1 14.4340 13.3650 12.5233 11.6947 11.0095 10.3423 T = 3 28.0865 25.8917 24.3212 22.7853 21.3443 20.1021 T = 5 31.8047 29.4976 28.0237 26.1909 24.6224 23.2531

µy= 0.04 k -0.3401 0 0.25 0.5 0.75 1

T = 1 14.3354 13.2432 12.4245 11.7216 11.1213 10.5368 T = 3 27.6199 25.5841 24.1402 22.7616 21.6054 20.4345 T = 5 31.2082 29.1659 27.3588 26.1471 24.6395 23.2393

µy= 0.08 k -0.3608 0 0.25 0.5 0.75 1

T = 1 13.9434 13.1623 12.4645 11.9243 11.5145 10.9894 T = 3 26.2002 24.6243 23.6016 22.6098 21.6747 20.7056 T = 5 29.5389 27.5814 26.5712 25.3519 24.2673 23.0902

Table 4: Summary of sensitivity analysis on the CatEPut prices.

Yt,n

Constant Ga(α, β) N (µy, σ²_y)

Merton measure θ ↑ ↓ ↑ ↑

and constant intensity µy( if T = 1) ↑ ↑ ↑

σ_y² – ↓ ↑

T ↑ ↓ ↑ ↑

Merton measure θ0 – – ↑

and stochastic intensity µθ− 0.5σ_θ² – – ↑

Qcand constant intensity c – – ↓

Figure 1: The CatEPut prices under various initial stock price S₀ and intensity rate θ in the GARCH-Jump model, where T = 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

θ P( L 1 > L)

Figure 2: The value of P (L₁ > L) versus the intensity rate θ in the case of constant jump size.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 5 10 15 20 25 30

P( L 1 > L )

CatEPut price

Figure 3: The CatEPut prices given in Table 1versus P (L₁ > L).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure 4: The probability of P (L1 > L) versus the intensity rate θ in the cases of normally (blue circle) and gamma (red dot) distributed jump size. The upper panel is the case of Ga(10,0.0002) and N(0.02,4 × 10⁻⁵). The middle panel is the case of Ga(0.1,0.2) and N(0.02,4 × 10⁻³). The lower panel is the case of Ga(10⁻³,20) and GARCH-Jump model with gamma and normally distributed jump sizes in the Merton mea-sure, where µ_y = 0.01, 0.02, 0.05, 0.5, θ = 0.4, 0.8 and T = 1.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 6: The CatEPut prices versus the intensity rate θ = 0.2, 0.4, 0.6, 0.8 under GARCH-jump model and jump-diffusion (JD) model with constant jump size in the Merton measure, where T = 1, 3, 5 .

0.2 0.4 0.6 0.8

0 10 20

σy2 = 4*10−3 CatEPut price

Gamma distribution σy2 = 4*10−1 CatEPut price

0.2 0.4 0.6 0.8

Figure 7: The CatEPut prices versus the intensity rate θ under GARCH-Jump model and jump-diffusion (JD) model in the Merton measure, where the notations for T = 1, 3, 5 are the same as those in Figure 6.

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2

Figure 8: The CatEPut prices given in Table 4 versus k.

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在文檔中 考慮有隨機跳躍頻率強度的GARCH模型之巨災選擇權定價 (頁 19-32)