ｽﾗｲﾄﾞﾀｲﾄﾙなし

Lattice Calculation
for Forward LIBOR Model
Tadashi Uratani
Hosei University
[email protected]
and
Makoto Utsunomiya
Bank of Tokyo-Mitsubishi
99/08/29
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Outline
• Interest rate sensitive security (e.g. Cap)
• Definitions: Bond Price, LIBOR
• Forward LIBOR Model
– Martingale, No Arbitrage, and
– Backward, Forward Induction Method
• Lattice Calculation
• Transitional Probability among Measures
• Valuation of Knockout Cap
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Interest rate derivatives
E.g. Cap
rate
Short term interest（LIBOR）
difference
difference
premium
Exercise rate
Company
Bank
date
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LIBOR
London Interbank Offer Rate
Typical short-term interest rate in international capital market
3 month LIBOR
6 month LIBOR
３ month
Borrowing date
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Returning date
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Forward LIBOR and Discount
Bond Price
n
t
Tn
 n1
Tn1
Tn2
３month LIBOR :  = 3/12 = 0.25
Ln (t ) : Libor starting fromTn to Tn1
Forward LIBOR
Bn (t ) : Bond of Maturity Tn at tim et
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Forward LIBOR and Bond Price
t
Tn
Tn 1
1
1
(1   n Ln (t )) 
Bn (t )
Bn 1 (t )
n
1
Bn 1 (t )  Bn ( t ) (t ) 
Tn ( t )1  t  Tn ( t )
i n ( t ) 1   i Li (t )
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Forward LIBOR Model
Bond
process
dBn (t )
  (t , Tn )dt   n (t )dW (t )
Bn (t )
Ito’s Lemma
dLn (t )   n (t )dt   n (t )dW (t )
Valuation of derivatives by Forward LIBOR
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Change of Probability Measure
dX (t )
 μX (t )dt  σX (t )dW (t )
X (t )
W (t ) : SBM on (Ω, t ,Ρ)
Girsanov Theorem
t
dP X
 1 t

2
 exp    X ( s)   ( s)  ds    x ( s)   ( s) dW ( s) 
0
dP
 2 0

t
WX (t )  W (t )   ( ( s)   X ( s))ds
0
: SBM on (, t ,  X )
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Martingale
 dX (t )
 X (t )   X (t )dt   X (t )dW (t )

 dC(t )
  C (t )dt   C (t )dW (t )

 C (t )

0t T


No Arbitrage Condition
C (t )   X (t )
  (t )
 C (t )   X (t )
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Market value of risk
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Pricing by Martingale
C (t )
X (t )
Martingale under
C (t )
X  C (T ) 
 t 

X (t )
 X (T ) 
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
X
t T
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Risk Neutral Measure
Bn ( t ) (t ) n ( t ) 1
B( t ) 
 (1  n Ln (Tn ))
B1 (t ) n 1
let X (t )  B(t )
C( t )
B  C(T ) 
 t 
tT

B( t )
 B(T ) 
P
B
 i  i (t )  n (t )
dLn (t )  
dt   n (t )dWB (t ) （
Forward.Induction）
i  n ( t ) 1   i Li (t )
n
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Risk Adjusted Measure
let X (t )  Bn1 (t )
C( t )
n 1  C(T) 
 t 
tT

Bn1 ( t )
 Bn1 (T) 
t
Tm
Tn
Martigale underP
n1
t
Ln (t )  
Ln (T )
Tn1
n 1
dLn (t )  n (t )dWn1 (t )
 i  i (t )  n (t )
dLm (t )   
dt   m (t )dWn1 (t ) （
Backward.Induction）
i  m 1 1   i Li (t )
n
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( Tm  Tn )
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Valuation of cap
Generation :F. LIBOR
X ( t )  B( t )
Choice :Meas.
Backward
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dLn (t )  n (t )dWn1(t )
dLn1(t )  n1(t )dWn2 (t )

dLN (t )  N (t )dWN 1(t )
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Lattice martingale
X (t )  Bn1 (t )
Forward
Lattice Calculation
dLn (t )  n (t )dWn1(t )
Martingale Measures
dLn1(t )  n1(t )dWn2 (t )

dLN (t )  N (t )dWN 1(t )
LN1(t )
LN (t)
L
(
t
)
n
1
L (t )
n
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Transition Probability
 g ( L (T )  
t
k
i


qk ( Li (t ), Li (T ))g ( L)dL
k
q :transition probability of measureP
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k
Relation of Transition
Probabilities
 g ( Li (T )  
k
t
k 1
t

1   k Lk (T ) 
 g ( Li (T ) 1   L (t ) 
k k


k 1
P and P
q
k 1
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k
1   k Lk (t )
( Li (t ), Li (T )) 
 qk ( Li (t ), Li (T ))
1   k Lk (T )
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One measure
TN 1
TN
Tn1
Tn
P N 1 and P n1
q
N 1
1   k Lk (T )
( Li (t ), Li (T ))  
 qn1 ( Li (t ), Li (T ))
k n 1 1   L (t )
k k
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N
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Knockout Cap
rate
LIBOR
Cap contract is
knockout
K
Ko
Year
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0
T1
Tn2
Tn1
Tn
Tn1
C (Tn )   n max Ln (Tn )  K ,0  I
def
I  I  min L1 ( T1 ), L2 ( T2 ),, Ln1 ( Tn1 )  Ko 
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Binominal tree under
each measure
Ln
Ko
Calculate the knocknout
Ln1
Ko
Unify one measure
Valuation of derivative
LN
Ko
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Knockout Cap
Underlying security:3 month LIBOR
maturity 3 years knockout cap
 i  i (t )  n (t )
dLn (t )  
dt   n (t )dWB (t ) （
forward.Induction）
i  n ( t ) 1   i Li (t )
n
 i  i (t )  n (t )
dLm (t )   
dt   m (t )dWn1 (t ) ( Backward.Induction)
i  m 1 1   i Li (t )
n
dLn (t )  n (t )dZn1 (t )
( Tm  Tn )
n (t )  Ln (t )
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Knockout Cap
14
誤差（％×１０）
12
Error
fd
λ=0.2 前進法
La
λ=0.2 格子法
λ=0.3 前進法
fd
λ=0.3 格子法
La
λ=0.4 前進法
fd
λ=0.4 格子法
fd
λ=0.5 前進法
La
λ=0.5 格子法
10
8
6
La
4
2
0
K=0.02
K=0.03
ノックアウト･レート
Knockout
rate
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Why lambda affect?
Lattice generation
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Ko
LN
Large λ
Ko
Ln1
n (t )  Ln (t )
Ln
dLn (t )  n (t )dZn1 (t )
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Ko
End
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 i  i (t )  n (t )
dLn (t )  
dt  n (t )dWB (t ) （前進法）
i n ( t ) 1   i Li (t )
n
Tn
Tn2
Tn3
t
Tn(t )
Tn
Tn(t )
L1 L2 L3
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Ln( t )
Ln3 Ln2 Ln1
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T0
T1
 i  i (t )  n (t )
dLm (t )   
dt   m (t )dWn1 (t ) （後退法）
i m1 1   i Li (t )
n
Tn
Tn2
Tn3
t
Tm
Tn
t
T2
L1 L2 L3
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Lm
Ln3 Ln2 Ln1
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T0
T1
Knockout Cap
前進法、後退法
• 単純モンテカルロ法
• 試行回数１００００回
• 時点分割１０
多重格子法
• 時点分割１０
３ヶ月LIBORを対象とした３年満期の０時点における価格
K o  0.005,0.01,0.015
λ 0.2,0.3,0.4,0.5
K  0.02
初期イールド  0.03一定( Ln (0)  0.03)
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Conclusion
• 異なった測度による推移確率関数を関係付け
ることによりいくつかの金利派生証券を評価
• 推移確率関数を格子法に利用
• Knockout Capletにおいてλが大きいとき他の方
法に比べ誤差が大きい
– 各格子の作成方法を検討
– 推移確率関数の検討
• 他の期間同士のForward LIBORの推移確率
の導入し、他の派生証券を評価
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