(k 1)! e (m 1)/N v(k). (k 1)!

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Adrian Burns
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5 m N 1/N k v : N R m k 1 k B(k 1, m 1; 1/N) = ( ) ( m 1 1 k 1 N ) k 1 ( ) m k N 1. N m N (m 1)/N P ((m 1)/N; k 1) = 1 (k 1)! m EV (m) = k=1 1 (k 1)! ( ) k 1 m 1 e (m 1)/N. N ( ) k 1 m 1 e (m 1)/N v(k). m N m k > m N v(k)

6 v( ) λ (0, 1) v(k) = λ k 1 EV (m) = e (m 1)(1 λ)/n. λ m m m N λ λ = 0 λ = 0 λ = 1 λ α αn αn V (N, α) = αn e (m 1)(1 λ)/n m=1 = e1/n (1 e α(1 λ) ) e 1/N e λ/n.

7 C α N B γ α+γ α P 1 P N α

8 γ αn γn R(C) = V (N, α). N N 1 exp{ α(1 λ)} 1 λ γn αn R(B) = P α V (N, α + γ). α + γ α + γ α/(α + γ) P = 1 P P M P M = α + γ α 1 e α(1 λ). 1 e (α+γ)(1 λ) P > P M P < P M N P M (α+γ)/α λ λ P = 10 λ

9 α= γ= α= γ= λ P M λ α γ λ λ < 1 λ > 1 α γ λ α γ P M λ P M α P M γ λ α α γ γ λ

10 α γ N P λ α γ M 1 = α 1 N 1 M 2 = α 2 N 2 β i [0, 1] i j i α 1 + α 2 α i x i {C, B}

11 i R(x i, x j ) i R(C, C) = V ((1 β i )N i, α i ) + α i α i + α j V (β i N i, α i + α j ). (1 β i ) i i β α i +α j i R(B, B) = P [ ] αi α i + γ V ((1 β α i i)n i, α i + γ) + α i + α j + γ V (β in i, α i + α j + γ). P i α i γ α i /(α i + γ) 1 β i i β i i j 1 β i α i + γ β i α i + α j + γ i R(B, C) = P [ ] αi α i + γ V ((1 β α i i)n i, α i + γ) + α i + α j + γ V (β in i, α i + α j + γ). R(B, B) = R(B, C) i i j i R(C, B) = V ((1 β i )N i, α i ) + α i α i + α j + γ V ((β in i, α i + α j + γ).

P i α i γ α i /(α i + γ) 1 β i i β i i j 1 β i α i + γ β i α i + α j + γ i R(B, C) = P [ ] αi α i + γ V ((1 β α i i)n i, α

12 i j C R(C, x j ) > R(B, x j ) BR i (x j ) = B PC i P B i P i β > 0 P i C > P i B P i C = P i B P i C = P i B = V ((1 β i )N i, α i ) + αi α i+α j V (β i N i, α i + α j ) α i α V ((1 β α i+γ i)n i, α i + γ) + i α V (β i+α j+γ in i, α i + α j + γ) α V ((1 β i )N i, α i ) + i α i +α j +γ V ((β in i, α i + α j + γ) α i α i +γ V ((1 β α i)n i, α i + γ) + i α i +α j +γ V (β in i, α i + α j + γ) P i C P i B P i C > P i B β > 0 α i α i V (β i N i, α i + α j ) > α i + α j α i + α j + γ V (β in i, α i + α j + γ). V (N, x) x x = e1/n (1 e x(1 λ) (x(1 λ) + 1)) x ( ) < 0. 2 e 1/N eλ/n β = 0 β = 0

i, α i + α j + γ) α i α i +γ V ((1 β α i)n i, α i + γ) + i α i +α j +γ V (β in i, α i + α j + γ) P i C P i B P i C > P i B β > 0 α i α i V (β i N

13 β i i j i P C P B P > P C (B, B) P < P B (C, C) P B < P < P C (B, B) (C, C) P B < P < P C R(C, C) > R(B, B) P P C P B λ λ = 1 β β = 1

14 () 12 () P 8 ()() P ()() 4 () 4 () λ β 8 () 8 () 6 6 P 4 ()() P 4 ()() 2 () 2 () α γ P λ β α γ α = 10 γ = 100 λ = 0.5 β = 0.5 N = 1000

15 P B = 1 β = 0 P C = P B α α λ α α γ γ N N P = () 6 P 4 ()() 2 () P N α = 10 γ = 100 λ = 0.5 β = 0.5 N

16 i α i β i γ λ N i P α i, γ, N, P λ β i P < P B α P B < P < P C α R B β R < β B PC R > P C B P B R > P B B PB R > P > P C B

17 PC i > P j C P B i > P j B P > PB i, P j C, P j B (B, B) P < P j C, P C i, P B i (C, C) PB i > P > P j C (C, B) P j C > P > P B i (C, C) (B, B) α α A = 35, β A = 0.10 α B = 20, β B = 0.95 λ = 0.5, γ = 100 P = 3 (C, C) α A = 40 (C, B) P α (B, B)

18 A α A α A A α A

20 P M λ P M α P M γ = α + γ α α(e (α+γ)(1 λ) e α(1 λ) ) + γ(e (2α+γ)(1 λ) e (α+γ)(1 λ) ) α(e (α+γ)(1 γ) ) 2 < 0 ( = e α(2 λ) γ γe λ(α+γ) (e α+γλ e α+γ )(α(1 λ)(α + γ) + γ) ) γ a ( 2 e (1 λ)(α+γ) 1 ) 2 < 0 ( ) ( 1 e a(λ 1) e (1 λ)(α+γ) ( (α + γ)(1 λ) 1) + 1 ) = α ( e (1 λ)(α+γ) 1 ) 2 > 0 V R(C) R(B) λ V α V (N, α) α α+γ V (N, α+γ) γ P R(C) R(B) ( α α γ α α+γ V (N, α + γ) γ V N ( ( e1/n e α(1 λ) (αn 1)e λ/n αe 1/N N ) + e ) λ/n V (N, α; λ) = λ N ( e 1/N e ) λ/n 2 > 0 V (N, α) = (1 λ)e α(1 λ)+1/n α e 1/N e > 0 λ/n V (N, α + γ) α + γ ( α V (N, α + γ) α + γ N ) ) = e1/n (γ e (α+γ)(1 λ) (γ α(α + γ)(1 γ))) > 0 (e 1/N e λ/n )(α + γ) 2 ( = αe1/n e (1 λ)(α+γ) (1 λ(α + γ) + α + γ + 1) ) (α + γ) ( 2 e 1/N e ) < 0 λ/n ( ) 1 e α(1 λ) e (1+λ)/N N ( 2 e 1/N e ) λ/n 2 > 0 V (N, α) = (1 λ)

α(1 λ) (αn 1)e λ/n αe 1/N N ) + e ) λ/n V (N, α; λ) = λ N ( e 1/N e ) λ/n 2 > 0 V (N, α) = (1 λ)e α(1 λ)+1/n α e 1/N e > 0 λ/n V (N, α + γ) α + γ ( α V (N, α + γ) α + γ N ) ) = e1/n (γ e (α+γ)(1

2. Illustration of the Nikkei 225 option data

1. Introduction 2. Illustration of the Nikkei 225 option data 2.1 A brief outline of the Nikkei 225 options market τ 2.2 Estimation of the theoretical price τ = + ε ε = = + ε + = + + + = + ε + ε + ε =