Vol. 3 No (Mar. 2010) An Option Valuation Model Based on an Asset Pricing Model Incorporating Investors Beliefs Kentaro Tanaka, 1 Koich

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1 Vol. 3 No (Mar An Option Valuation Model Based on an Asset Pricing Model Incorporating Investors Beliefs Kentaro Tanaka, 1 Koichi Miyazaki 1 and Koji Nishiki 1 Preceding researches for an option valuation model based on an asset pricing model incorporating investors beliefs on a drift of dividend process adopt an approach to derive the partial differential equation that the option price should satisfy. Though the approach is mathematically elegant, it is quite difficult for us to intuitively capture how parameters related to the beliefs affect to the option pricing and also to evaluate complex options. In this research, we propose the option valuation model based on a simulation approach and discuss the mechanism that the parameters generate volatility smile based on numerical experiments. 1 Department of Systems ngineering, The University of lectro-communications 1. Black-Scholes 1 BS BS BS IV IV BS BS Dupire Rubinstein Derman Heston Britten-Jones Merton Naik Wang Zapatero Veronesi Veronesi 2000 David Li c 2010 Information Processing Society of Japan

2 52 IV IV IV Veronesi (1 ] u (c, s ds (1 t u (c, t c t (1 D (t c (2 P (t = ( u t c (D (s,s D (s ds F (t (2 u c (D (t,t F (t t u c (D (t,t u (c, t c c = D (t (2 t s D (s u c (D (s,s/u c (D (t,t (3 u (c, t =e φt c 1 γ (3 1 γ γ φ (3 (2 P (t (4 ( 1 γ P (t D (s D (t = exp ( φ (s t ds F (t] (4 D (t t (4 D (5 dd = θ(tddt + σ DDdW D (5 θ (t σ D W D θ (t Veronesi θ (t Θ={θ 1,...,θ n} t θ (t (6 θ (t e de = θ (t dt + σ edw e (6 σ e W e W D dw DdW e =0 σ e θ (t

3 π i (t (7 t θ (t =θ i i n Θ={θ 1,...,θ n} θ (t t Θ n πi (t =1 Θ={θ 1,...,θ n} π i (t 1 π i (t θ i π i (t (8 Veronesi 2000 Liptser π i(t =prob (θ (t =θ i F (t (7 ( ] 1 dπ i(t = p (f i π i(t π σ 2 σ 2 D e i(t(θ i m θ (t (θ(t m θ (t dt +π i(t(θ i m θ (t ( (8 1 σ D dw D + 1 σ e dw e p f i π i m θ (9 m θ (t =(θ (t F (t = n π i (t θ i (9 (10 (5 (11 d W D = 1 ( dd σ D D m θ(tdt = (θ (t m θ(t dt + dw D σ D (10 dd = m θ (tddt + σ DDd W D (11 (10 e d W e = 1 σ e (de m θ (tdt (8 (12 ( 1 dπ i (t =p (f i π i (t dt + π i (t(θ i m θ (t d σ W D + 1 d D σ W e (12 e d W D d W e (9 0 dw D dw e 2.3 Veronesi 2000 P (t (13 D (t n πi(tci ( n P (t =D (t π i(tc i (13 C i (4 θ (t =θ i (14 (14 (15 ( ] 1 γ D (s C i = exp ( φ (s t ds θ (t =θ i (14 D (t C i = t 1 (φ + p +(γ 1 θ i +0.5γ (γ 1 σ 2 D (1 ph (15 H (16 H = n j=1 f j (φ + p +(γ 1 θ j +0.5γ (γ 1 σ 2 D (16 (15 (16 C i φ p θ i γ σ D π i f i (13 (17 ( n ( n n dp (t = π i(tc i dd (t+ dπ i(tc i D (t+ C i (dπ i(tdd (t = Pμ pdt + Pσ Dd W n ( 1 D + D C iπ i (θ i m θ d σ W D + 1 d D σ W e (17 e μ p σ p (18 (19 ( μ p = m θ + D n P p C i (f i π D n i+ C iπ i (θ i m θ (18 P

4 54 σ P = σ 2 D + V θ ( 1 2+V θ + 1 σd 2 σe 2 ] (19 P (t call = e φ(t t ( D (T D (t γ Max D (T n ] ] π i (T C i K, 0 F (t (24 (19 V θ (20 n V θ = πici(θi m θ n (20 j=1 πjcj Veronesi (4 D (t (21 ( γ D (s P (t = exp ( φ (s t D (s ds F (t] (21 D (t t ( γ (21 s D (s t exp ( φ (s t D(s D(t P (t T b (D(T P (t b t T s T D (s =0 (22 P (t b = exp ( φ (T t ( γ D (T b (D (T F (t] D (t T P (t call (22 (23 (24 n ] b (D(T = MaxP (T K, 0] = Max D (T π i (T C i K, 0 (23 (22 (24 D(T D(t πi (T D(T D(t (25 (25 T m t θ (s ds π i (T ( T D (T D (t =exp π i (T t m θ (s ds 1 2 σ2 D (T t+σ D ( WD (T W D (t (25 (1 m m =40 Δt Δt = T t m (10 d W D d W e (2 (12 t Δt π i (t dπ i (t (3 (2 dπ i (t π i (t +Δt = π i (t+dπ i (t (4 (1 (2 (3 t T π i T π i (T 1 Longstaff T m t θ (s ds (1 s π i (s m θ (s (2 (1 t T t T m θ (s (3 (2 m θ (s T m t θ (s ds π 1,,π n 1

5 55 2 (24 BS IV T 1 P (t disc (22 b (D(T = 1 (26 ( γ D (T P (t disc = exp ( φ (T t F (t] D (t (26 r(t (27 (26 r (t =φ + γm θ (t 1 2 γ(γ +1σ2 D ( Black-Scholes BS BS BS (28 dp =(r d Pdt+ σpdw Q (28 d W Q BS 1 BS γ 0 π i (0 π 1 (0 = 1 π 2 (0 = 0,π n (0 = 0 1 f i f 1 =1,f 2 =0, f n =0 (12 p (f i π i (t dt i = 1, 2,,n f i = π i (t 0 (12 π i (t(θ i m θ ( 1 σ D d W D + 1 σ e d W e i =1 π1 =1 θ 1 = m θ 0 i = 2,,n π i =0 0 ( π 1 (t =1,π 2 (t =0,,π n (t =0 p θ 1 = m θ (11 (29 dd = θ 1Ddt + σ DDd W D (29 m θ = θ 1 f i = π i (t π i (t =0 i =2,,n (18 μ P μ P = θ 1 (19 σ P m θ = θ 1 π i (t =0 i =2,,n V θ 0 σ P = σ D 1 (30 dp (t =θ 1Pdt+ σ DPd W D (30 (30 (30 (31 dp (t =θ Q 1 Pdt+ σdpd W Q D (31 (31 θ Q 1 R 0 (32 ] dpt + Ddt dr] = rdt = ( θ Q 1 + d r =0 (32 P t θ Q 1 θq 1 = r d e 1 BS BS BS

6 56 Fig. 1 1 Distribution of equity return up to option maturity. 3 π i (t Fig. 3 Dynamics of investors beliefs. 2 Fig. 2 Illustration of IV. IV 2 IV 2 putotm ATM callotm 2 IV IV IV P (T =D(T π i (T C i (13 ln P (T P (0 =lnd(t +ln π i (T C i D(0 π i (0C i ln P (T P (0 IV ln D(T D(0 ln π i (T C i π i (0C i IV π i (T C i IV D (0 150 φ 1.50% T 0.3 γ 0 θ i π i (0 π i f i p σ D σ e IV

7 % , γ 0 1 IV 1 P (0 10,000 IV ( π i 1 IV 3.2 IV π i (0 1-a 8% 1 +8% 2 1-a 1 f i 2 +8% 1-a f i +8% 8% 4.1 f i IV 2 IV 1 Table 1 Parameter sets. i f i θ i π i (0 p σ D σ e % 8.00% 90.00% % 10% a skew(π i % 8.00% 10.00% % 8.00% 10.00% % 10% % 8.00% 90.00% % 12.00% 12.50% % 0.00% 75.00% % 10% % 10.00% 12.50% % 18.00% 7.50% b smile(θ i % 0.00% 85.00% % 10% % 13.00% 7.50% % 24.00% 5.00% % 0.00% 90.00% % 10% % 16.00% 5.00% % 24.00% 5.00% % 0.00% 90.00% % 10% % 16.00% 5.00% % 24.00% 5.00% % 0.00% 90.00% % 10% c smile(p % 16.00% 5.00% % 24.00% 5.00% % 0.00% 90.00% % 10% % 16.00% 5.00% % 24.00% 5.00% % 0.00% 90.00% % 10% % 16.00% 5.00% % 24.00% 5.00% d smile(σ e % 0.00% 90.00% % % 16.00% 5.00% % 24.00% 5.00% e smile(σ D % 0.00% 90.00% % % 16.00% 5.00% π i (

8 a IV Fig. 4 IV under the set of Table 1-a. 5 1-a Fig. 5 quity return distribution under the set of Table 1-a. 6 1-a Fig. 6 Distribution of dividend growth rate under the set of Table 1-a. 7 1-a Fig. 7 Distribution of price-dividend-ratio growth rate under the set of Table 1-a. ln 2 1-a Table 2 ach moment value under the set of Table 1-a. ( P (T ln P (0 ( D(T ln D(0 ( π i (T C i π i (0C i 1 2 x] 0.74% 0.56% (x μ 3] % % (x μ 2] 1.45% 1.19% (x μ 4] 3σ % % x] 2.01% 1.65% (x μ 3] % % (x μ 2] 0.36% 0.36% (x μ 4] 3σ % % x] 1.28% 2.21% (x μ 3] % % (x μ 2] 0.54% 0.38% (x μ 4] 3σ % % ln P (T D(T ln P (0 D(0 ln π i (T C i π i (0C i (x μ 4] 3σ % % D (T (11 m θ θ i π i (0 m θ

9 D (T n =2 θ i 8% 8% θ i 8% 8% C i (15 ( πi (0 Ci C1 = % C2 = % 2 πi (T Ci π i (T π i (0 f i C 1 =62.42 π 1 (T 90% C 2 = π 2 (T 10% 2 πi (T Ci 2 πi (0 Ci π 2 (T θ i IV 3.3 θ i 3 1-b Smile(θ i 3 1 θ i 12% 0% 10% 2 θ i 18% 0% 13% 8 1-b IV Fig. 8 IV under the set of Table 1-b. 3 θ i 24% 0% 16% 1 3 θ i π i ( % 75% 12.5% 2 7.5% 85% 7.5% 3 5% 90% 5% 8 1 IV IV b

10 60 ln ( P (T ln P (0 ( D(T ln D(0 ( π i (T C i π i (0C i 3 1-b Table 3 ach moment value under the set of Table 1-b x] 0.65% 0.81% 1.00% (x μ 3] % % % (x μ 2] 1.36% 1.68% 2.06% (x μ 4] 3σ % % % x] 0.23% 0.27% 0.28% (x μ 3] % % % (x μ 2] 0.36% 0.37% 0.38% (x μ 4] 3σ % % % x] 0.42% 0.54% 0.72% (x μ 3] % % % (x μ 2] 0.43% 0.63% 0.91% (x μ 4] 3σ % % % 1 3 n =3 θ i 3 0 θ 1 θ 3 C 1 C 3 (15 ( θ 1 θ 3 C 1 C 3 θ 2 =0 C i θ 1 θ 3 C 1 C 3 π i (0 π i (T π i (T π i (0 f i 3 θ 2 =0 C i C 2 f 2 90% π 2 (0 9 1-c IV Fig. 9 IV under the set of Table 1-c. 3 90% 1 75% 3 π 2 (T θ 1 θ 3 C 1 C θ 1 θ 3 π 1 (T π 3 (T p p θ i π i (0 p 1-c p c Smile(p p p p

11 61 ln ( P (T ln P (0 ( D(T ln D(0 ( π i (T C i π i (0C i 4 1-c Table 4 ach moment value under the set of Table 1-c. p =0.3 p =0.6 p =0.9 p =1.2 x] 0.97% 0.43% 0.30% 0.26% (x μ 3] % % % % (x μ 2] 2.02% 0.92% 0.65% 0.55% (x μ 4] 3σ % % % % x] 0.27% 0.28% 0.26% 0.26% (x μ 3] % % % % (x μ 2] 0.37% 0.37% 0.37% 0.37% (x μ 4] 3σ % % % % x] 0.70% 0.16% 0.04% 0.00% (x μ 3] % % % % (x μ 2] 0.88% 0.20% 0.07% 0.04% (x μ 4] 3σ % % % % p p n =3 θ i 3 θ i p θ i C i 0 θ 1 θ 3 C 1 C 3 (15 ( θ 1 θ 3 C 1 C 3 θ 2 =0 C i θ 1 θ 3 C 1 C d IV Fig. 10 IV under the set of Table 1-d. π i (T π i (0 f i C 1 C 3 1 C 1 C C 1 C σ e σ e σ e 1 σ e 7.5% 10% 20% 50% 4 10 σ e d Smile(σ e

12 62 ln ( P (T ln P (0 ( D(T ln D(0 ( π i (T C i π i (0C i 5 1-d Table 5 ach moment value under the set of Table 1-d. σ e =7.5% σ e = 10% σ e = 20% σ e = 50% x] 1.14% 0.97% 0.82% 0.77% (x μ 3] % % % % (x μ 2] 2.35% 2.02% 1.67% 1.59% (x μ 4] 3σ % % % % x] 0.27% 0.27% 0.28% 0.27% (x μ 3] % % % % (x μ 2] 0.37% 0.37% 0.37% 0.37% (x μ 4] 3σ % % % % x] 0.87% 0.70% 0.55% 0.50% (x μ 3] % % % % (x μ 2] 1.20% 0.88% 0.54% 0.45% (x μ 4] 3σ % % % % n =3 θ i 3 θ i C i π i (0 π i (T π i (t (12 σ e π i (T θ i C 2 C 1 C 3 θ 1 θ 3 C e IV Fig. 11 IV under the set of Table 1-e. C σ D σ D σ D 1-e Smile(σ D σ D 10% 20% 30% 3 γ =0 σ D P (0 σ D 11 σ D e Smile(σ D σ D σ D σ D σ D

13 63 ln ( P (T ln P (0 ( D(T ln D(0 ( π i (T C i π i (0C i 6 1-e Table 6 ach moment value under the set of Table 1-e. σ D = 10% σ D = 20% σ D = 30% x] 0.97% 1.27% 2.01% (x μ 3] % % % (x μ 2] 2.02% 2.70% 4.32% (x μ 4] 3σ % % % x] 0.27% 0.74% 1.50% (x μ 3] % % % (x μ 2] 0.37% 1.39% 3.07% (x μ 4] 3σ % % % x] 0.70% 0.53% 0.51% (x μ 3] % % % (x μ 2] 0.88% 0.55% 0.49% (x μ 4] 3σ % % % σ D n =3 θ i θ i C i π i (0 π i (T π i (t (12 σ D σ D π i (T θ i C 2 C 1 C 3 θ 1 θ 3 C 1 C 3 5. BS 1 BS IV IV IV IV 2 1 Black, F. and Scholes, M.: The Pricing of Options and Corporate Liabilities, Journal of Political conomy, Vol.81, pp ( Dupire, B.: Pricing with a smile, Risk, July pp ( Rubinstein, M.: Implied binomial trees, Journal of Finance, Vol.49, pp ( Derman,. and Kani, I.: Riding on a smile, Risk, February, pp ( Heston, S.L.: A Closed-form solution for options, Review of Financial and Studies, Vol.13, pp ( Britten-Jones, M. and Neuberger, A.: Option prices, implied price processes, and stochastic volatility, Journal of Finance, Vol.55, pp ( Merton, R.C.: Option pricing when under lying stock returns are discontinuous,

14 64 Journal of Financial conomics, Vol.3, pp ( Naik, V. and Lee, M.: General equilibrium pricing of options on the market portfolio with discontinuous returns, Review of Financial Studies, Vol.3, pp ( Wang, J.: A model of intertemporal asset prices under asymmetric information, Review of conomic Studies, Vol.60, ( Zapatero, F.: ffects of financial innovations on market volatility when beliefs are heterogeneous, Journal of conomic Dynamics and Control, Vol.22, pp ( Veronesi, P.: How does information quality affect stock returns, Journal of Finance, Vol.55, pp ( David, A. and Veronesi, P.: Option prices with uncertain fundamentals, Workingpaper, University of Chicago ( Li, T.: Heterogeneous beliefs, option prices, and volatility smiles, Working Paper, The Chinese University of HongKong ( Liptser, R. and Shiryayev, A.N.: Statistics of Random Processes, I, II, Springer- Verlag, NewYork ( F.A. Longstaff and.s. Schwartz: Valuing American options by simulation: A simple least-squares approach, Review of Financial Studies, Vol.14, pp (2001. ( ( ( JAF 59 20

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