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master thesis presentation for pricing theory under negative interest rate environment

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Stephan Chang
Stephan Chang

This document contains a mix of text, numbers, formulas and code snippets related to quantitative finance modeling. It discusses topics like derivative pricing under negative interest rates, the SABR model, implied volatility, and mean reversion. Various formulas are presented for pricing derivatives, calculating implied volatility, and estimating parameters for models like SABR.

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This picture was .aken by Aaron Davis. # Black Swan
# British Exit This picture was .aken by Paul Lloyd.
The Nominal Short Rate Cannot Be Negative. /* Fischer Black, 1995 */
/* 2008 ~ 2015 */
- %Hello world;)
2016/06/30
Derivative Pricing Under Negative Interest Rate environment- %Hello world;)
- %Hello world;)
- %Hello world;) - %Hello world;)
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
1 2 3 4 5 6 7 8 0.20 9 10 0.12 30 0.79 0.55 0.08 0.43 0.72 0.98 1.51 0.05 0.35 0.50 0.63 0.83 0.94 0.95 1.58 0.01 0.18 0.28 0.41 0.58 0.73 0.88 1.56 0.02 0.13 0.25 0.31 0.42 0.68 0.78 0.92 1.56 0.15 0.34 0.48 0.68 0.88 1.04 1.20 2.03 0.19 0.34 0.54 0.75 0.90 1.06 1.21 2.00 0.06 0.22 0.45 0.62 0.81 0.93 1.07 1.66 0.19 0.31 0.51 0.74 0.88 0.88 1.02 1.71
master thesis presentation for pricing theory under negative interest rate environment
log(F0/K) Implied
master thesis presentation for pricing theory under negative interest rate environment
- %Hello world;) - %Hello world;)
dFt = FtdWt Ft = F0exp( Wt 2 t 2 ) Vc = P(0, T)[F0 (d1) K (d2)]
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
Vp (0) = P(0, T)[K ( d2) F0 ( d1)] log( F0 K )K 0 d1 = log(F0/K) + ( 2 2 )T T d2 = d1 T d1 d2 ( d1) 0 ( d2) 0 Vp = 0
Vp (0) = P(0, T)[K ( d2) F0 ( d1)] log( F0 K )K 0 d1 = log(F0/K) + ( 2 2 )T T d2 = d1 T d1 d2 ( d1) 0 ( d2) 0 Vp = 0
f( ) = P(0, T)[K ( d2) F0 ( d1)] Vp black Vp market = 0
master thesis presentation for pricing theory under negative interest rate environment
Implied(K, T)
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
dFt = LV(t, Ft)FtdWt LV(T, K) = 2 C(T,K) T K2 2C(T,K) 2K
dFt = LV(t, Ft)FtdWt LV(T, K) = 2 C(T,K) T K2 2C(T,K) 2K c LV Vc LV F = B F + B B B F = B c + B B F
dFt = LV(t, Ft)FtdWt LV(T, K) = 2 C(T,K) T K2 2C(T,K) 2K c LV Vc LV F = B F + B B B F = B c + B B F
dFt = F ( ) t dWt 0 1 p(t, f, F0) (p)t 1 2 (F(2 ) )FF = 0
master thesis presentation for pricing theory under negative interest rate environment
pA(t, f) = 1 1 f1 2 t ( f f0 ) 1 2 e q2+q2 0 2t I| |( qq0 t ) pR(t, f) = 1 1 f1 2 t ( f f0 ) 1 2 e q2+q2 0 2t I ( qq0 t )
dFt = tFt dW (1) t d t = tdW (2) t dW (1) t dW (2) t = dt F0 = F 0 = 0 1
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
H a(K)b(T, K)( c(K) g(c(K)) ) a(K) = [(FK) (1 ) 2 (1 + (1 )2 24 log2 F K + (1 )4 1920 log4 F k )] 1 b(T, K) = [1 + ( (1 )2 24 2 (FK)1 + 4(FK)(1 )/2 + 2 3 2 24 2 )T] c(K) = (FK)(1 )/2 log F K g(x) = log( 1 2 x + x2 + x 1 )
ATM H F1 [1 + ( (1 )2 24 2 (F)2 2 + 4(F)1 + 2 3 2 24 2 )T]
BSABR c = VC SABR(K, F, , T) = H(K, F; , , , ) ( , , ) = arg min , , i [ M i H(Fi, Ki; , , )]2
= 2 T 1
master thesis presentation for pricing theory under negative interest rate environment
Vc (t) Vp (t) = P(t, T)(Ft K) fT P(t, T) = 2 Vc SABR 2K = 2 Vp SABR 2K
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
q(t, f) = 1 2 (pR(t, f) + pA(t, f)) pR(t, f) pA(t, f)
dFt = tFt dW (1) t d t = tdW (2) t dW (1) t dW (2) t = 0 F0 = F 0 =
dFt = tFt dW (1) t d t = tdW (2) t dW (1) t dW (2) t = 0 F0 = F 0 = dFt = tFt dW (1) t d t = tdW (2) t dW (1) t dW (2) t = dt d ?Ft = ?t ?Ft ? d ?Wt (1) d?t = ??td ?Wt (2) d ?Wt (1) d ?Wt (2) = 0
master thesis presentation for pricing theory under negative interest rate environment
- %Hello world;) - %Hello world;)
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
dFt = tdW (1) t F0 = F d t = tdW (2) t 0 = dW (1) t dW (2) t = dt N(K) = ( ) (1 + 2 3 2 24 2 T) = (F0 K) ( ) = log( 1 2 + 2 + 1 )
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
d ?Ft = t ?Ft dW (1) t d t = tdW (2) t dW (1) 1 dW (2) t = dt F0 = ?F 0 = ?F = F + s atm shift ?F1 [1 + ( (1 )2 24 2 (?F)2 2 + 4(?F)1 + 2 3 2 24 2 )T]
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
CN = P(0, T)[(F0 K) (d) + T (d)] CD = P(0, T)[ ?F0 ( ?d1) ?K ( ?d2)]
F > 0 < F < F > s
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
q(t, f) = 1 2 (pR(t, f) + pA(t, f)) q(t, f) = 1 2 (pR(t, f) pA(t, f)) f > 0 f < 0
master thesis presentation for pricing theory under negative interest rate environment
dFt = t|Ft| dW (1) t d t = tdW (2) t dW (1) t dW (2) t = dt F0 = F 0 = 0 < 1 2 FBSABR = (F0 K)(1 ) F0/|F0| K/|K| ， x ， (1 + T( (2 ) 2 24|Fav|2 2 + sign(Fav) 4|Fav|1 ))
master thesis presentation for pricing theory under negative interest rate environment
- %Hello world;) - %Hello world;)
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
? log ATM H log (1 ) log F ? ( , , ) = arg min , , i [ M i H(Fi, Ki; , , )]2
? log ATM H log (1 ) log F ? ( , , ) = arg min , , i [ M i H(Fi, Ki; , , )]2
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master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
- %Hello world;) - %Hello world;)
- %Hello world;)
master thesis presentation for pricing theory under negative interest rate environment
master thesis presentation for pricing theory under negative interest rate environment
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master thesis presentation for pricing theory under negative interest rate environment
# Mean Reversion ?
master thesis presentation for pricing theory under negative interest rate environment

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master thesis presentation for pricing theory under negative interest rate environment

  • 1. This picture was .aken by Aaron Davis. # Black Swan
  • 2. # British Exit This picture was .aken by Paul Lloyd.
  • 3. The Nominal Short Rate Cannot Be Negative. /* Fischer Black, 1995 */
  • 4. /* 2008 ~ 2015 */
  • 7. Derivative Pricing Under Negative Interest Rate environment- %Hello world;)
  • 9. - %Hello world;) - %Hello world;)
  • 13. 1 2 3 4 5 6 7 8 0.20 9 10 0.12 30 0.79 0.55 0.08 0.43 0.72 0.98 1.51 0.05 0.35 0.50 0.63 0.83 0.94 0.95 1.58 0.01 0.18 0.28 0.41 0.58 0.73 0.88 1.56 0.02 0.13 0.25 0.31 0.42 0.68 0.78 0.92 1.56 0.15 0.34 0.48 0.68 0.88 1.04 1.20 2.03 0.19 0.34 0.54 0.75 0.90 1.06 1.21 2.00 0.06 0.22 0.45 0.62 0.81 0.93 1.07 1.66 0.19 0.31 0.51 0.74 0.88 0.88 1.02 1.71
  • 17. - %Hello world;) - %Hello world;)
  • 18. dFt = FtdWt Ft = F0exp( Wt 2 t 2 ) Vc = P(0, T)[F0 (d1) K (d2)]
  • 21. Vp (0) = P(0, T)[K ( d2) F0 ( d1)] log( F0 K )K 0 d1 = log(F0/K) + ( 2 2 )T T d2 = d1 T d1 d2 ( d1) 0 ( d2) 0 Vp = 0
  • 22. Vp (0) = P(0, T)[K ( d2) F0 ( d1)] log( F0 K )K 0 d1 = log(F0/K) + ( 2 2 )T T d2 = d1 T d1 d2 ( d1) 0 ( d2) 0 Vp = 0
  • 23. f( ) = P(0, T)[K ( d2) F0 ( d1)] Vp black Vp market = 0
  • 28. dFt = LV(t, Ft)FtdWt LV(T, K) = 2 C(T,K) T K2 2C(T,K) 2K
  • 29. dFt = LV(t, Ft)FtdWt LV(T, K) = 2 C(T,K) T K2 2C(T,K) 2K c LV Vc LV F = B F + B B B F = B c + B B F
  • 30. dFt = LV(t, Ft)FtdWt LV(T, K) = 2 C(T,K) T K2 2C(T,K) 2K c LV Vc LV F = B F + B B B F = B c + B B F
  • 31. dFt = F ( ) t dWt 0 1 p(t, f, F0) (p)t 1 2 (F(2 ) )FF = 0
  • 33. pA(t, f) = 1 1 f1 2 t ( f f0 ) 1 2 e q2+q2 0 2t I| |( qq0 t ) pR(t, f) = 1 1 f1 2 t ( f f0 ) 1 2 e q2+q2 0 2t I ( qq0 t )
  • 34. dFt = tFt dW (1) t d t = tdW (2) t dW (1) t dW (2) t = dt F0 = F 0 = 0 1
  • 38. H a(K)b(T, K)( c(K) g(c(K)) ) a(K) = [(FK) (1 ) 2 (1 + (1 )2 24 log2 F K + (1 )4 1920 log4 F k )] 1 b(T, K) = [1 + ( (1 )2 24 2 (FK)1 + 4(FK)(1 )/2 + 2 3 2 24 2 )T] c(K) = (FK)(1 )/2 log F K g(x) = log( 1 2 x + x2 + x 1 )
  • 39. ATM H F1 [1 + ( (1 )2 24 2 (F)2 2 + 4(F)1 + 2 3 2 24 2 )T]
  • 40. BSABR c = VC SABR(K, F, , T) = H(K, F; , , , ) ( , , ) = arg min , , i [ M i H(Fi, Ki; , , )]2
  • 43. Vc (t) Vp (t) = P(t, T)(Ft K) fT P(t, T) = 2 Vc SABR 2K = 2 Vp SABR 2K
  • 46. q(t, f) = 1 2 (pR(t, f) + pA(t, f)) pR(t, f) pA(t, f)
  • 47. dFt = tFt dW (1) t d t = tdW (2) t dW (1) t dW (2) t = 0 F0 = F 0 =
  • 48. dFt = tFt dW (1) t d t = tdW (2) t dW (1) t dW (2) t = 0 F0 = F 0 = dFt = tFt dW (1) t d t = tdW (2) t dW (1) t dW (2) t = dt d ?Ft = ?t ?Ft ? d ?Wt (1) d?t = ??td ?Wt (2) d ?Wt (1) d ?Wt (2) = 0
  • 50. - %Hello world;) - %Hello world;)
  • 56. dFt = tdW (1) t F0 = F d t = tdW (2) t 0 = dW (1) t dW (2) t = dt N(K) = ( ) (1 + 2 3 2 24 2 T) = (F0 K) ( ) = log( 1 2 + 2 + 1 )
  • 59. d ?Ft = t ?Ft dW (1) t d t = tdW (2) t dW (1) 1 dW (2) t = dt F0 = ?F 0 = ?F = F + s atm shift ?F1 [1 + ( (1 )2 24 2 (?F)2 2 + 4(?F)1 + 2 3 2 24 2 )T]
  • 63. CN = P(0, T)[(F0 K) (d) + T (d)] CD = P(0, T)[ ?F0 ( ?d1) ?K ( ?d2)]
  • 64. F > 0 < F < F > s
  • 67. q(t, f) = 1 2 (pR(t, f) + pA(t, f)) q(t, f) = 1 2 (pR(t, f) pA(t, f)) f > 0 f < 0
  • 69. dFt = t|Ft| dW (1) t d t = tdW (2) t dW (1) t dW (2) t = dt F0 = F 0 = 0 < 1 2 FBSABR = (F0 K)(1 ) F0/|F0| K/|K| ， x ， (1 + T( (2 ) 2 24|Fav|2 2 + sign(Fav) 4|Fav|1 ))
  • 71. - %Hello world;) - %Hello world;)
  • 76. ? log ATM H log (1 ) log F ? ( , , ) = arg min , , i [ M i H(Fi, Ki; , , )]2
  • 77. ? log ATM H log (1 ) log F ? ( , , ) = arg min , , i [ M i H(Fi, Ki; , , )]2
  • 88. - %Hello world;) - %Hello world;)
  • 92. ?