Vol. 3 No. 2 51 64 (Mar. 2010 1 1 1 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 1994 2 Rubinstein 1994 3 Derman 1994 4 Heston 1993 5 Britten-Jones 2000 6 Merton 1976 7 Naik 1990 8 Wang 1993 9 Zapatero 1998 10 Veronesi 2000 11 Veronesi 2000 David 2002 12 Li 2007 13 51 c 2010 Information Processing Society of Japan
52 IV IV IV 3 4 2. Veronesi 2000 11 2.1 (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 2000 11 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
53 2.2 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 1977 14 π i(t =prob (θ (t =θ i F (t (7 ( ] 1 dπ i(t = p (f i π i(t + + 1 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
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 3. 3.1 Veronesi 2000 11 (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 2001 15 1 50 0.3 40 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
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 (27 3.2 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 (12 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 3.3 1 BS BS
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 3 4. 4.1 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
57 225 1.5% 225 2009 10,000 150 0.5 0.1 0.3 γ 0 1 IV 1 P (0 10,000 IV (24 4.2 4.2.1 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 4 4 1 IV 2 IV 1 Table 1 Parameter sets. i f i θ i π i (0 p σ D σ e 1 1 61.33% 8.00% 90.00% 0.3 10% 10% a skew(π i 2 38.67% 8.00% 10.00% 2 1 65.33% 8.00% 10.00% 0.3 10% 10% 2 34.67% 8.00% 90.00% 1 6.44% 12.00% 12.50% 1 2 90.00% 0.00% 75.00% 0.3 10% 10% 3 3.56% 10.00% 12.50% 1 6.71% 18.00% 7.50% b smile(θ i 2 2 90.00% 0.00% 85.00% 0.3 10% 10% 3 3.29% 13.00% 7.50% 1 7.15% 24.00% 5.00% 3 2 90.00% 0.00% 90.00% 0.3 10% 10% 3 2.85% 16.00% 5.00% 1 2.73% 24.00% 5.00% 1 2 96.27% 0.00% 90.00% 0.3 10% 10% 3 1.00% 16.00% 5.00% 1 1.36% 24.00% 5.00% 2 2 97.58% 0.00% 90.00% 0.6 10% 10% c smile(p 3 1.06% 16.00% 5.00% 1 1.02% 24.00% 5.00% 3 2 97.98% 0.00% 90.00% 0.9 10% 10% 3 1.00% 16.00% 5.00% 1 0.97% 24.00% 5.00% 4 2 97.98% 0.00% 90.00% 1.2 10% 10% 3 1.06% 16.00% 5.00% 1 2.73% 24.00% 5.00% d smile(σ e 2 96.27% 0.00% 90.00% 0.3 10% 3 1.00% 16.00% 5.00% 1 2.73% 24.00% 5.00% e smile(σ D 2 96.27% 0.00% 90.00% 0.3 10% 3 1.00% 16.00% 5.00% π i (0 5 6 7 1
58 4 1-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] 0.6524% 0.5271% (x μ 2] 1.45% 1.19% (x μ 4] 3σ 4 0.0312% 0.0268% x] 2.01% 1.65% (x μ 3] 0.0032% 0.0049% (x μ 2] 0.36% 0.36% (x μ 4] 3σ 4 0.0000% 0.0000% x] 1.28% 2.21% (x μ 3] 0.0736% 0.0508% (x μ 2] 0.54% 0.38% (x μ 4] 3σ 4 0.0113% 0.0086% ln P (T D(T ln P (0 D(0 ln π i (T C i π i (0C i 2 3 1 3 4 (x μ 4] 3σ 4 4 5 1 2 2 1 3 2 6 2 1 2.01% 2 1.65% D (T (11 m θ θ i π i (0 m θ 1 0.64 2 0.64 1 2
59 2 3 4 0 1 2 D (T n =2 θ i 8% 8% 2 7 2 1 2 2 1 3 2 1 2 θ i 8% 8% C i (15 (16 1 62.42 104.91 2 41.33 69.48 1 2 πi (0 Ci C1 =62.42 90% C2 =104.91 10% 2 πi (T Ci π i (T π i (0 f i C 1 =62.42 π 1 (T 90% C 2 =104.91 π 2 (T 10% 2 πi (T Ci 2 πi (0 Ci π 2 (T 2 1 4.2.2 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 (0 1 12.5% 75% 12.5% 2 7.5% 85% 7.5% 3 5% 90% 5% 8 1 IV 2 10000 IV 3 2 4.2.1 3 1-b 1 4 4 3 4 1 3 1 3 1 3 3 3 4 0 4 1 3
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. 1 2 3 x] 0.65% 0.81% 1.00% (x μ 3] 0.0010% 0.0321% 0.0835% (x μ 2] 1.36% 1.68% 2.06% (x μ 4] 3σ 4 0.0044% 0.0403% 0.1541% x] 0.23% 0.27% 0.28% (x μ 3] 0.0002% 0.0006% 0.0012% (x μ 2] 0.36% 0.37% 0.38% (x μ 4] 3σ 4 0.0000% 0.0001% 0.0001% x] 0.42% 0.54% 0.72% (x μ 3] 0.0003% 0.0134% 0.0373% (x μ 2] 0.43% 0.63% 0.91% (x μ 4] 3σ 4 0.0010% 0.0124% 0.0583% 1 3 n =3 θ i 3 0 θ 1 θ 3 C 1 C 3 (15 (16 1 47.16 95.42 2 41.37 110.70 3 36.74 131.54 1 3 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 1 3 1 1 3 θ 1 θ 3 C 1 C 3 3 1 3 θ 1 θ 3 π 1 (T π 3 (T 1 4.2.3 p p θ i π i (0 p 1-c 0.3 1.2 9 p 4.2.2 4 1-c Smile(p 1 4 4 4 p p p
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] 0.1920% 0.2109% 0.1050% 0.0666% (x μ 2] 2.02% 0.92% 0.65% 0.55% (x μ 4] 3σ 4 0.1481% 0.0217% 0.0070% 0.0035% x] 0.27% 0.28% 0.26% 0.26% (x μ 3] 0.0057% 0.0065% 0.0050% 0.0047% (x μ 2] 0.37% 0.37% 0.37% 0.37% (x μ 4] 3σ 4 0.0001% 0.0001% 0.0001% 0.0001% x] 0.70% 0.16% 0.04% 0.00% (x μ 3] 0.0280% 0.0161% 0.0047% 0.0019% (x μ 2] 0.88% 0.20% 0.07% 0.04% (x μ 4] 3σ 4 0.0564% 0.0042% 0.0007% 0.0002% 4 3 4 0 4 p p n =3 θ i 3 θ i p θ i C i 0 θ 1 θ 3 C 1 C 3 (15 (16 1 36.74 131.54 2 47.78 87.79 3 52.80 80.78 4 55.71 76.83 4 1 θ 1 θ 3 C 1 C 3 θ 2 =0 C i θ 1 θ 3 C 1 C 3 10 1-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 3 36.74 131.54 C 1 C 3 4.2.4 σ e σ e σ e 1 σ e 7.5% 10% 20% 50% 4 10 σ e 4.2.3 5 1-d Smile(σ e 1 4 5 4
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] 0.0055% 0.0057% 0.0084% 0.0061% (x μ 2] 2.35% 2.02% 1.67% 1.59% (x μ 4] 3σ 4 0.2571% 0.1481% 0.0627% 0.0451% x] 0.27% 0.27% 0.28% 0.27% (x μ 3] 0.0055% 0.0057% 0.0084% 0.0061% (x μ 2] 0.37% 0.37% 0.37% 0.37% (x μ 4] 3σ 4 0.0001% 0.0001% 0.0001% 0.0001% x] 0.87% 0.70% 0.55% 0.50% (x μ 3] 0.0491% 0.0280% 0.0185% 0.0080% (x μ 2] 1.20% 0.88% 0.54% 0.45% (x μ 4] 3σ 4 0.1181% 0.0564% 0.0160% 0.0089% 5 3 4 0 4 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 1 11 1-e IV Fig. 11 IV under the set of Table 1-e. C 3 4.2.5 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 4.2.4 6 1-e Smile(σ D 1 4 6 4 σ D σ D σ D 6 3 4 0 4 σ D
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] 0.1920% 0.0821% 0.0550% (x μ 2] 2.02% 2.70% 4.32% (x μ 4] 3σ 4 0.1481% 0.0685% 0.0607% x] 0.27% 0.74% 1.50% (x μ 3] 0.0057% 0.0032% 0.0020% (x μ 2] 0.37% 1.39% 3.07% (x μ 4] 3σ 4 0.0001% 0.0005% 0.0023% x] 0.70% 0.53% 0.51% (x μ 3] 0.0280% 0.0126% 0.0109% (x μ 2] 0.88% 0.55% 0.49% (x μ 4] 3σ 4 0.0564% 0.0178% 0.0132% σ D n =3 θ i 3 4.2.4 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 2 1 1. IV 2 1 Black, F. and Scholes, M.: The Pricing of Options and Corporate Liabilities, Journal of Political conomy, Vol.81, pp.637 654 (1973. 2 Dupire, B.: Pricing with a smile, Risk, July pp.18 20 (1994. 3 Rubinstein, M.: Implied binomial trees, Journal of Finance, Vol.49, pp.771 818 (1994. 4 Derman,. and Kani, I.: Riding on a smile, Risk, February, pp.32 39 (1994. 5 Heston, S.L.: A Closed-form solution for options, Review of Financial and Studies, Vol.13, pp.585 625 (2000. 6 Britten-Jones, M. and Neuberger, A.: Option prices, implied price processes, and stochastic volatility, Journal of Finance, Vol.55, pp.839 866 (2000. 7 Merton, R.C.: Option pricing when under lying stock returns are discontinuous,
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