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An Asymptotic Expansion Approach in Finance
Akihiko TakahashiGraduate School of Economics, University of Tokyo
February 12, 2009
Abstract
This presentation reviews an asymptotic expansion approach to numerical prob-lems for pricing and hedging derivatives.
Table of Contents
1. Motivation
2. Related Literatures
3. Outline of an Asymptotic Expansion Approach
4. Examples
5. Computation of Higher-Orders
1
1 Motivation
• Recently, there are many liquid products in OTC(over-the-counter) deriva-tive markets:
For European type contracts,
“Plain vanilla” call/put options(equity, foreign exchange), bond options(treasury),cap/floor, swaptions(interest rates), and average options(commodities).
• Simple and liquid contracts, but models are sometimes complicating dueto smiles or/and long-term contracts.
Examples:
Average options(commodities):
Pricing under a model with stochastic volatility
Long-term foreign exchange options:
Pricing under a model with stochastic volatility, and term structures ofdomestic/foreign interest rates.
• It is difficult to obtain explicit (“closed-form”) formulas in many cases.
• On the other hand, trading liquid products requires fast computation ofvalues and risk-indicators (“Greeks”), as well as calibration to the marketprices.
– Greeks(Delta, Gamma, Vega, Theta...)
– Calibration: search parameters in a model so that the model repro-duces liquid market prices.
• Fast and accurate approximation scheme is desired.
• An asymptotic expansion approach
This method is applicable in the unified manner to pricing of above prod-ucts in the economy evolved by broad class of Ito processes;almost the same procedure can be applied.
2
2 Related Literatures
• Pricing Average options: Kunitomo and Takahashi(1992), Yoshida(1992b)
• Mathematical Validity based on Watanabe theory(Watanabe(1987)) inMalliavin calculus: Yoshida(1992a,b)
In applications to finance; Takahashi(1995,1999), Kunitomo-Takahashi(2003),Takahashi-Yoshida(2004,2005)
• Pricing in diffusion models: Takahashi(1995,1999) Takahashi-Uchida(2006),Takahashi-Saito(2003)
• Pricing in Heath-Jarrow Morton model: Takahashi(1995), Kunitomo-Takahashi(2001),Takahashi-Matsushima(2004)
• Dynamic Optimal Portfolio based on Clark-Ocone formula:
Takahashi-Yoshida(2004), Kobayashi-Takahashi-Tokioka(2003)
• Variance Reduction in Monte Carlo: Takahashi-Yoshida(2005)
• Greeks: Matsuoka-Takahashi-Uchida(2006), Takahashi(2005)
• Pricing in Jump-diffusion and Levy processes: Kunitomo-Takahashi(2004),Takahashi(2007)
• Pricing in Market Model: Kawai(2003), Takahashi-Takehara(2007a,b,c)
• Pricing in Credit model: Muroi(2005)
• Pricing in Cross-currency models: Takahashi(1995), Osajima(2006),Takahashi-Takehara(2007a,b,c)
• Random limit(expansion around a log-normal distribution):
e.g. Fournie-Lasry-Touzi(1997), Lutkebohmert(2004),Kunitomo-Kim(2005), Takahashi-Yoshida(2004), Takahashi-Toda-Yamada(2008)
3
3 Outline of an Asymptotic Expansion Approach
• First, I consider a d-dimensional diffusion process X(ε), which is the strongsolution to the following stochastic differential equation:
dX(ε)t = V0(X
(ε)t , ε)dt+ εV (X(ε)
t )dWt; X(ε)0 = x0, t ∈ [0, T ],
where W denotes a m-dimensional standard Wiener process and ε ∈ [0, 1]is a known parameter.
Suppose that coefficients V0 : Rd × [0, 1] → Rd, V : Rd → Rd × Rm aresmooth and satisfy regularity conditions.
• Next, suppose that a function g : Rd → R to be smooth and all derivativeshave polynomial growth orders. Then, for ε ↓ 0, g(X(ε)
T ) has its asymptoticexpansion;
g(X(ε)T ) = g0T + εg1T + ε2g2T + ε3g3T + o(ε3).
– The justification of this and the following asymptotic expansions wasprovided in Watanabe(1987) and Yoshida(1992a,b) based on Malli-avin Calculus.(See Kunitomo-Takahashi(2003), Takahashi(1995), Takahashi-Yoshida(2004,2005)in the context of finance.)
– This talk concentrates on applications of the method.
4
• The coefficients in the expansion, g0T , g1T , g2T · · · can be obtained by Tay-lor’s formula and represented based on multiple Wiener-Ito integrals.
• In particular, let Dt = ∂X(ε)t
∂ε |ε=0, Et = ∂2X(ε)t
∂ε2 |ε=0 and Ft = ∂3X(ε)t
∂ε3 |ε=0
then g0T , g1T , g2T and g3T can be written as
g0T = g(X(0)T ), g1T =
d∑i=1
∂ig(X(0)T )Di
T ,
g2T =12
d∑i,j=1
∂i∂jg(X(0)T )Di
TDjT +
12
d∑i=1
∂ig(X(0)T )Ei
T ,
g3T =16
d∑i,j,k=1
∂i∂j∂kg(X(0)T )Di
TDjTD
kT +
12
d∑i,j=1
∂i∂jg(X(0)T )Ei
TDjT
+16
d∑i=1
∂ig(X(0)T )F i
T ,
where ∂k = ∂∂xk
.
• Here, Dit, E
it and F i
t , i = 1, · · · , d denote the i-th elements of Dt, Et andFt respectively. Dt, Et and Ft are represented by
Dt =∫ t
0
YtY−1u [∂εV0(X(0)
u , 0)du+ V (X(0)u )dWu],
Et =∫ t
0
YtY−1u
(d∑
j,k=1
∂j∂kV0(X(0)u , 0)Dj
uDkudu+ ∂2
εV0(X(0)u , 0)du+ 2
d∑j=1
∂ε∂jV0(X(0)u , 0)Dj
udu
+ 2d∑
j=1
∂jV (X(0)u )Dj
udWu
),
Ft =∫ t
0
YtY−1u
(d∑
j,k,l=1
∂j∂k∂lV0(X(0)u , 0)Dj
uDkuD
ludu+ 3
d∑j,k=1
∂j∂kV0(X(0)u , 0)Ej
uDkudu
+ 3d∑
j,k=1
∂j∂k∂εV0(X(0)u , 0)Dj
uDkudu+ 3
d∑j=1
∂j∂εV0(X(0)u , 0)Ej
udu
+ 3d∑
j=1
∂j∂2εV0(X(0)
u , 0)Djudu+ ∂3
εV0(X(0)u , 0)du
+3d∑
j,k=1
∂j∂kV (X(0)u )Dj
uDkudWu + 3
d∑j=1
∂jV (X(0)u )Ej
udWu
).
where Y denotes the solution to the differential equation;
dYt = ∂V0(X(0)t , 0)Ytdt; Y0 = Id,
where ∂V0 denotes the d × d matrix whose (j, k)-element is ∂kVj0 , V j
0 isthe j-th element of V0, and Id denotes the d× d identity matrix.
5
• Next, normalize g(X(ε)T ) as
G(ε) =g(X(ε)
T ) − g0T
ε
for ε ∈ (0, 1].
• Moreover, letat = (∂g(X(0)
T ))′[YTY
−1t V (X(0)
t )]
and make the following assumption:
(Assumption 1) ΣT =∫ T
0
ata′tdt > 0.
• Since ΣT is the variance of the random variable g1T , which follows a normaldistribution, Assumption 1 means the condition that the distribution ofg1T does not degenerate. In application, it is easy to check this conditionin most cases.
6
• Next, the characteristic function of G(ε), ψG(ε)(ξ) is approximated by
ψG(ε)(ξ) = E[exp(iξG(ε))]= E[exp(iξg1T )] + ε(iξ)E[exp(iξg1T )g2T ]
+ ε2(iξ)E[exp(iξg1T )g3T ] +ε2
2(iξ)2E[exp(iξg1T )g2
2T ] + o(ε2).
• Moreover,
ψG(ε)(ξ) = exp(
(iξ)2ΣT
2
)+ ε(iξ)E [exp(iξg1T )E[g2T |g1T ]]
+ ε2(iξ)E [exp(iξg1T )E[g3T |g1T ]] +ε2
2(iξ)2E
[exp(iξg1T )E[g2
2T |g1T ]]+ o(ε2).
Note that E[g2T |g1T ], E[g22T |g1T ] and E[g3T |g1T ] are polynomial functions
of g1T .
• Then, the inversion of the approximated characteristic function providesan approximated probability density function of Gε:
fGε(x) = n[x; 0,ΣT ] + ε
[− ∂
∂xp2(x)n[x; 0,ΣT ]
]
+ ε2[− ∂
∂xp3(x)n[x; 0,ΣT ]
]+
12ε2[∂2
∂x2p22(x)n[x; 0,ΣT ]
]+ o(ε2),
where p2(x) = E[g2T |g1T = x], p22(x) = E[g22T |g1T = x]
and p3(x) = E[g3T |g1T = x]. Also, n[x; 0,ΣT ] denotes the density functionof the normal distribution with mean 0 and variance ΣT :
n[x; 0,ΣT ] =1√
2πΣT
exp−x2
2ΣT
.
7
• Now, given a smooth function φ : R → R of which all derivatives havepolynomial growth orders.
• Then, the expectation E[φ(G(ε))IB(G(ε))] has its asymptotic expansion as
E[φ(G(ε))IB(G(ε))] = Φ0 + εΦ1 + ε2Φ2 + o(ε2).
(B denotes a Borel set in R, and IB(G(ε)) = 1 ifG(ε) ∈ B and IB(G(ε)) = 0otherwise.)
Here, each term of the expansion can be expressed based on the expecta-tions of polynomials of coefficients in the asymptotic expansion of g(X(ε)
T )conditional on a normal random variable.
For example, Φ0, Φ1 and Φ2 are written as
Φ0 =∫
B
φ(x)n[x; 0,ΣT ]dx,
Φ1 = −∫
B
φ(x)∂xE[g2T |g1T = x]n[x; 0,ΣT ]dx,
Φ2 =∫
B
(12φ(x)∂2
xE[g22 |g1T = x]n[x|0,ΣT ] − φ(x)∂xE[g3|g1T = x]n[x|0,ΣT ]
)dx,
– Note that E[g2T |g1T = x], E[g22T |g1T = x] and E[g3T |g1T = x] are
polynomial functions of x and hence the computations of the expec-tations become easier; the formula of the conditional expectationsgiven in the following can be used.
• (Example) For a call option on a security with maturity T , its maturityprice g(Xε
T ) and a strike price K = g(X0T ) − εy for arbitrary y ∈ R, the
payoff is expressed as
maxg(XεT ) −K, 0 = εφ(Gε)IB(Gε),
where φ(x) = (x+ y) and B = Gε ≥ −y.
8
• I list up some formulas of conditional expectations used in asymptoticexpansions. W = (W 1
t , · · · ,Wmt ) : 0 ≤ t ≤ T denotes a m-dimensional
Wiener process. Let qi : [0, T ] → Rm, i = 1, 2, 3, 4, 5 are non-randomfunctions and we define Σ as
Σ =∫ T
0
q′1vq1vdv,
where z′
is the transpose of z. We assume that 0 < Σ < ∞ and integra-bility in the following formulas.
Moreover, Hn(x; Σ) denotes the Hermite polynomial of degree n:
Hk(x; Σ) := (−Σ)kex2/2Σ dk
dxke−x2/2Σ.
For example，H0(x; Σ) = 1, H1(x; Σ) = x, H2(x; Σ) = x2−Σ, H3(x; Σ) =x3 − 3Σx, H4(x; Σ) = x4 − 6Σx2 + 3Σ2.
1.
E
[∫ T
0
q′2tdWt|
∫ T
0
q′1vdWv = x
]=
(∫ T
0
q′2tq1tdt
)H1(x; Σ)
Σ
2.
E
[∫ T
0
∫ t
0
q′2udWuq
′3tdWt|
∫ T
0
q′1vdWv = x
]=
(∫ T
0
∫ t
0
q′2uq1uduq
′3tq1tdt
)H2(x; Σ)
Σ2
3.
E
[(∫ T
0
q′2udWu
)(∫ T
0
q′3sdWs
)|∫ T
0
q′1vdWv = x
]=
(∫ T
0
q′2uq1udu
)(∫ T
0
q′3sq1sds
)H2(x; Σ)
Σ2
+∫ T
0
q′2tq3tdt
4.
E
[∫ T
0
∫ t
0
∫ s
0
q′2udWuq
′3sdWsq
′4tdWt|
∫ T
0
q′1vdWv = x
]=
(∫ T
0
q′4tq1t
∫ t
0
q′3sq1s
∫ s
0
q′2uq1ududsdt
)H3(x; Σ)
Σ3
5.
E
[∫ T
0
(∫ t
0
q′2udWu
)(∫ t
0
q′3sdWs
)q′4tdWt|
∫ T
0
q′1vdWv = x
]=
∫ T
0
(∫ t
0
q′2uq1udu
)(∫ t
0
q′3sq1sds
)q′4tq1tdt
H3(x; Σ)
Σ3
+
(∫ T
0
∫ t
0
q′2uq3uduq
′4tq1tdt
)H1(x; Σ)
Σ
9
6.
E
[(∫ T
0
∫ t
0
q′2sdWsq
′3tdWt
)(∫ T
0
q′4udWu
)|∫ T
0
q′1vdWv = x
]=
(∫ T
0
q′3tq1t
∫ t
0
q′2sq1sdsdt
)(∫ T
0
q′4uq1udu
)H3(x; Σ)
Σ3
+
(∫ T
0
q′3tq1t
∫ t
0
q′2sq4sdsdt+
∫ T
0
q′3tq4t
∫ t
0
q′2sq1sdsdt
)H1(x; Σ)
Σ
7.
E
[(∫ T
0
∫ t
0
q′2sdWsq
′3tdWt
)(∫ T
0
∫ r
0
q′4udWuq
′5rdWr
)|∫ T
0
q′1vdWv = x
]=
(∫ T
0
q′3tq1t
∫ t
0
q′2sq1sdsdt
)(∫ T
0
q′5rq1r
∫ r
0
q′4uq1ududr
)H4(x; Σ)
Σ4
+
∫ T
0
q′3tq1t
∫ t
0
q′5rq1r
∫ r
0
q′2uq4ududrdt+
∫ T
0
q′5tq1t
∫ t
0
q′3rq1r
∫ r
0
q′2uq4ududrdt
+∫ T
0
q′3tq1t
∫ t
0
q′2rq5r
∫ r
0
q′4uq1ududrdt+
∫ T
0
q′3tq5t
(∫ t
0
q′2sq1sds
)(∫ t
0
q′4uq1udu
)dt
+∫ T
0
q′5rq1r
∫ r
0
q′3uq4u
∫ u
0
q′2sq1sdsdudr
H2(x; Σ)
Σ2
+∫ T
0
∫ t
0
q′2uq4uduq
′3tq5tdt
8.
E
[(∫ T
0
q′2tdWt
)(∫ T
0
q′3sdWs
)(∫ T
0
∫ r
0
q′4udWuq
′5rdWr
)|∫ T
0
q′1vdWv = x
]=
(∫ T
0
q′2tq1tdt
)(∫ T
0
q′3sq1sds
)(∫ T
0
q′5rq1r
∫ r
0
q′4uq1ududr
)H4(x; Σ)
Σ4
+
(∫ T
0
q′3sq1sds
)(∫ T
0
q′2rq5r
∫ r
0
q′4uq1ududr
)+
(∫ T
0
q′2tq3tdt
)(∫ T
0
q′5rq1r
∫ r
0
q′4uq1ududr
)
+
(∫ T
0
q′2tq1tdt
)(∫ T
0
q′5rq1r
∫ r
0
q′3sq4sdsdr
)+
(∫ T
0
q′3tq1tdt
)(∫ T
0
q′5rq1r
∫ r
0
q′2uq4ududr
)
+
(∫ T
0
q′2tq1tdt
)(∫ T
0
q′3rq5r
∫ r
0
q′4uq1ududr
)H2(x; Σ)
Σ2
+∫ T
0
q′2tq5t
∫ t
0
q′3sq4sdsdt+
∫ T
0
q′3rq5r
∫ r
0
q′2uq4ududr
10
4 Average Option
• Shiraya-Takahashi[2008]
4.1 λ-SABR model
• Examples of Stochastic Volatility Models
Heston[1993], SABR (Hagan-Kumar-Lesniewski-Woodward[2002]), λ- SABR(Labordere[2005]):
Pricing European Plain Vanilla Options.
• I show an approximation of an average option under λ- SABR modelfollowing Shiraya-Takahashi[2008].
• λ-SABR model
The dynamics of the underlying asset’s price S(t), t ∈ [0, T ] is describedby a two-dimensional diffusion process (S, σ) : (S(t), σ(t)), 0 ≤ t ≤ Tthat is a solution of the following stochastic differential equation underthe equivalent martingale measure(EMM):
S(t) = S(0) + α
∫ t
0
S(u)du+∫ t
0
σ(u)S(u)βdW1(u)
σ(t) = σ(0) +∫ t
0
λ(θ − σ(u))du+∫ t
0
ν1σ(u)dW1(u) +∫ t
0
ν2σ(u)dW2(u)
Here, S(0) and σ(0) are positive constants; β ∈ [0, 1] is a constant; λ andθ are nonnegative constants; α is a constant; ν1 = ρν，ν2 = (
√1 − ρ2)ν,
where ν ≥ 0, ρ ∈ [−1, 1] are constants; W = (W1,W2) is a two-dimensionalWiener Process.
• An average call option’s payoff with maturity T and strike price K:
C(T ) = max X(T ) −K, 0 ,
where
X(T ) =1T
∫ T
0
S(t)dt.
• Then, C(0), the price at t = 0 of the average call option is expressed as:
C(0) = e−rT E[C(T )],
where r denotes the risk-free rate that is a nonnegative constant.
11
4.2 Asymptotic Expansion of an Average Call Option
• λ-SABR model is rewritten in the asymptotic expansion’s framework:
For ε ∈ [0, 1],
S(ε)(t) = S(0) + α
∫ t
0
S(ε)(u)du+ ε
∫ t
0
σ(ε)(u)S(ε)(u)βdW1(u) (1)
σ(ε)(t) = σ(0) +∫ t
0
λ(θ − σ(ε)(u))du
+ε(∫ t
0
ν1σ(ε)(u)dW1(u) +
∫ t
0
ν2σ(ε)(u)dW2(u)
).
• An average call option’s payoff with maturity T and strike price K whereK = X
(0)T − εy for arbitrary y ∈ R:
C(T ) = maxX(ε)(T ) −K, 0
,
where
X(ε)(T ) :=1T
∫ T
0
S(ε)(t)dt.
12
• Then, the asymptotic expansion of X(ε)(T ) when α = 0 is given by:
X(ε)(T ) = X(0)(T ) + εX(1)(T ) + ε2X(2)(T ) + ε3X(3)(T ) + o(ε3),
where X(0)(T ) = X(ε)(T )|ε=0, X(k)(T ) = 1k!
∂kX(ε)(T )∂εk |ε=0, k = 1, 2, 3 and
they are given as follows:
X(0)(T ) =eαT − 1αT
S(0)
X(1)(T ) =∫ T
0
f11(s)′dW (s)
X(2)(T ) =2∑
i=1
∫ T
0
∫ s
0
f2i(u)′dW (u)g2i(s)′dW (s)
X(3)(T ) =
(3∑
i=1
∫ T
0
∫ s
0
∫ u
0
f3i(v)′dW (v)g3i(u)′dW (u)h3i(s)′dW (s)
+2∑
i=1
∫ T
0
(∫ s
0
g4i(u)′dW (u))(∫ s
0
f4i(u)′dW (u))h4i(s)′dW (s)
),
where x′ denotes the transpose of x.
f11(t), f2i(t)(i = 1, 2), f3i(t)(i = 1, 2, 3), f4i(i = 1, 2); g2i(t)(i = 1, 2),g3i(t)(i = 1, 2, 3), g4i(t)(i = 1, 2), h3i(t)(i = 1, 2, 3), and h4i(t)(i = 1, 2)are given as follows:
f21(t) = f31(t) = f41(t) = g41(t) = g42(t) =(e−αt (S(0)eαt)β (
θ + (σ(0) − θ)e−λt)
0
),
f11(t) =eα(T−t) − 1
αTeαtf21(t),
f22(t) = f32(t) = f33(t) = f42(t) =(ν1(θeλt + σ(0) − θ
)ν2(θeλt + σ(0) − θ
) ) ,g31(t) =
(β (S(0)eαt)β−1 (
θ + (σ(0) − θ)e−λt)
0
), h32(t) =
eα(T−t) − 1αT
g31(t),
g21(t) = h31(t) = h32(t), g32(t) =(
(S(0)eαt)βe−(α+λ)t
0
),
g22(t) = h33(t) =eα(T−t) − 1
αTg32(t), g33(t) =
(ν1ν2
),
h41(t) =(
eα(T−t)−12αT β(β − 1) (S(0)eαt)β−2 (
θ + (σ(0) − θ)e−λt)eαt
0
),
h42(t) =(
eα(T−t)−1αT β (S(0)eαt)β−1
e−λt
0
). (2)
Remark 1 When α = 0, the asymptotic expansion of X(ε)(T ) is obtainedas α→ 0 above.
13
• Then, C(0), the price at t = 0 of the average call option is expressed asfollows:
C(0) = e−rT
(ε
(y
∫ ∞
−y
n[x; 0,Σ]dx+∫ ∞
−y
x n[x; 0,Σ]dx)
+ε2∫ ∞
−y
E[X(2)(T )
∣∣X(1)(T ) = x]n[x; 0,Σ]dx
+ε3(∫ ∞
−y
E[X(3)(T )
∣∣X(1)(T ) = x]n[x; 0,Σ]dx
+12E[(X(2)(T )
)2 ∣∣X(1)(T ) = −y]n[y; 0,Σ]
))
+o(ε3),
where Σ =∫ T
0|f11(s)|2 ds and n[x; 0,Σ] = 1√
2πΣexp
−x2
2Σ
.
14
• More concrete approximation of the price is obtained by formulas of theconditional expectations:
Theorem 1 Suppose that the underlying asset price follows (1) for ε ∈(0, 1] under the equivalent martingale measure. Then, the asymptotic ex-pansion up to the ε3-order of C(0), the price of a average call option atthe contract date with the maturity date T and the strike price K whereK = X
(0)T − εy for arbitrary y ∈ R is given by:
C(0) = e−rT
[ε
yN
(y√Σ
)+ Σn[y; 0,Σ]
+ε2∫ ∞
−y
C1H2(x; Σ)
Σ2n[x; 0,Σ]dx
+ε3∫ ∞
−y
C2H3(x; Σ)
Σ3n[x; 0,Σ]dx+ C3n[y; 0,Σ]
+(C4H4(y; Σ)
Σ4+ C5
H2(y; Σ)Σ2
+ C6
)n[y; 0,Σ]
]+o(ε3), (3)
where N(x) denotes the distribution function of a standard normal distri-bution. Moreover,
C1 =2∑
i=1
∫ T
0
f11(s)′g2i(s)∫ s
0
f11(u)′f2i(u)duds
C2 =3∑
i=1
∫ T
0
f11(s)′h3i(s)∫ s
0
f11(u)′g3i(u)∫ u
0
f11(v)′f3i(v)dvduds
+12
2∑i=1
∫ T
0
f11(s)′h4i(s)∫ s
0
f11(u)′g4i(u)du∫ s
0
f11(u)′f4i(u)duds
C3 =3∑
i=1
∫ T
0
f11(s)′h4i(s)∫ s
0
g4i(u)′f4i(u)duds
C4 =3∑
i=1
∫ T
0
f11(s)′k5i(s)∫ s
0
f11(s)′h5i(u)duds
C5 =3∑
i=1
∫ T
0
f11(s)′k5i(s)∫ s
0
f11(u)′g5i(u)∫ u
0
f5i(v)′h5i(v)dvduds
+∫ T
0
f11(s)′g5i(s)∫ s
0
f11(u)′k5i(u)∫ u
0
f5i(v)′h5i(v)dvduds
+∫ T
0
f11(s)′g5i(s)∫ s
0
f5i(u)′k5i(u)∫ u
0
f11(v)′h5i(v)dvduds
+∫ T
0
g5i(s)′k5i(s)∫ s
0
f11(u)′h5i(u)du∫ s
0
f11(u)′f5i(u)duds
+∫ T
0
f11(s)′k5i(s)∫ s
0
g5i(u)′h5i(u)∫ u
0
f11(v)′f5i(v)dvduds
C6 =3∑
i=1
∫ T
0
g5i(s)′k5i(s)∫ s
0
f11(u)′h5i(u)duds.
Here, f11(t), f2i(t)(i = 1, 2), f3i(t)(i = 1, 2, 3), f4i(i = 1, 2); g2i(t)(i =
15
1, 2), g3i(t)(i = 1, 2, 3), g4i(t)(i = 1, 2); h3i(t)(i = 1, 2, 3), h4i(t)(i = 1, 2)are given in (2) above.
Moreover, f5i(t)(i = 1, 2, 3), g5i(i = 1, 2, 3), h5i(t)(i = 1, 2, 3), k5i(t)(i =1, 2, 3) are given by f51(t) = f53(t) = h51(t) = f21(t), f52(t) = h52(t) =h53(t) = f22(t), g51(t) = g53(t) = k51(t) = g21(t), g52(t) = k52(t) =12k53(t) = g22(t).
Remark 2 The following relation may be useful to compute the integralson the right hand side of the equation (3).∫ ∞
−y
1Σk
Hk(x; Σ)n[x; 0,Σ]dx =1
Σk−1Hk−1(−y; Σ)n[y; 0,Σ] (k ≥ 1)
4.3 Heston model
• The underlying asset price S:
dS(t) = αS(t)dt+ S(t)√V (t)dW 1(t), (4)
dV (t) = κ(θ − V (t))dt+ ν1√V (t)dW 1(t) + ν2
√V (t)dW 2(t),
where ν ≥ 0, ν1 = ρν，ν2 = (√
1 − ρ2)ν and ρ(∈ [−1, 1]).
• An asymptotic expansion of an average option price is obtained in thesimilar way as in λ-SABR model．In paritcular, changing the coefficientsof the equations in theorem 1 to the following provides an approximationformula under Heston model.
f21(t) = f31(t) = g31(t) = f41(t) =( √
θ + (V (0) − θ) e−κt
0
),
f11(t) = g21(t) = h31(t) = 2h32(t) =eα(T−t) − 1
αTS(0)f21(t),
f22(t) = f32(t) = f33(t) = f42(t) = g41(t) = g42(t) =( √
θ + (V (0) − θ) e−κtν1eκt√
θ + (V (0) − θ) e−κtν2eκt
),
g32(t) =
(e−κt√
θ+(V (0)−θ)e−κt
0
), g22(t) = 2h33(t) = h41(t) =
eα(T−t) − 12αT
S(0)g32(t),
g33(t) =
⎛⎝ ν1√
θ+(V (0)−θ)e−κt
ν2√θ+(V (0)−θ)e−κt
⎞⎠ , h42(t) =
⎛⎝ −(eα(T−t)−1)e−2κtS(0)
8αT(√
θ+(V (0)−θ)e−κt)3 ,
0
⎞⎠ .
16
4.4 Numerical Examples
• λ-SABR model
• Benchmark values:
Monte Carlo Simulation(5 million trials, 2000 time steps)
S0 = 100, α = r = 0, ε = 1, and σ(0) are determined such that thecoefficient of the diffusion term is equivalent to that of log-normal processat time 0: for example in Case i where the log-normal volatility is 30%,σ(0)S0.5
0 = 0.3S0.
Table 1:
Parameter λ σ(0) β ρ θ ν T
i 0.5 3.0 0.5 -0.3 3.0 0.3 1ii 0.5 0.3 1.0 -0.3 0.3 0.3 1iii 0.5 12.0 0.2 -0.3 12.0 0.3 1iv 0.5 3.0 0.5 -0.7 3.0 0.3 1v 0.5 3.0 0.5 0 3.0 0.3 1vi 0.5 2.0 0.5 -0.3 2.0 0.3 1vii 0.5 5.0 0.5 -0.3 3.0 0.3 1viii 0.5 3.0 0.5 -0.3 3.0 0.1 1ix 0.5 3.0 0.5 -0.3 3.0 0.7 1x 0.1 3.0 0.5 -0.3 3.0 0.3 1xi 1.0 3.0 0.5 -0.3 3.0 0.3 1xii 0.5 3.0 0.5 -0.3 3.0 0.3 2
Figure 1: Approximated Density of X(T )(Case ix)
17
Asymptotic Expansion DifferenceCase Strike MC 1st 2nd 3rd 1st 2nd 3rd
i 70 30.237 30.293 30.229 30.242 0.056 -0.007 0.00590 12.958 13.031 12.950 12.958 0.072 -0.008 -0.000
100 6.919 6.910 6.910 6.915 -0.009 -0.009 -0.004120 1.183 1.066 1.163 1.175 -0.117 -0.020 -0.008150 0.024 0.010 0.017 0.021 -0.014 -0.007 -0.003
ii 70 30.132 30.293 30.091 30.134 0.160 -0.042 0.00190 12.774 13.031 12.775 12.770 0.257 0.001 -0.003
100 6.904 6.910 6.910 6.900 0.005 0.005 -0.005120 1.400 1.066 1.376 1.394 -0.334 -0.024 -0.006150 0.069 0.010 0.033 0.056 -0.059 -0.036 -0.014
iii 70 30.335 30.300 30.320 30.339 -0.034 -0.015 0.00490 13.104 13.058 13.083 13.103 -0.045 -0.021 -0.001
100 6.963 6.943 6.943 6.959 -0.020 -0.020 -0.004120 1.082 1.082 1.052 1.076 0.001 -0.029 -0.006150 0.012 0.010 0.008 0.011 -0.002 -0.004 -0.001
iv 70 30.308 30.293 30.330 30.318 -0.015 0.021 0.01090 13.051 13.031 13.077 13.053 -0.020 0.026 0.002
100 6.888 6.910 6.910 6.885 0.022 0.022 -0.003120 0.999 1.066 1.009 0.989 0.067 0.010 -0.010150 0.006 0.010 0.006 0.004 0.004 -0.001 -0.002
v 70 30.185 30.293 30.154 30.189 0.108 -0.031 0.00490 12.879 13.031 12.855 12.877 0.151 -0.024 -0.003
100 6.933 6.910 6.910 6.929 -0.023 -0.023 -0.005120 1.315 1.066 1.278 1.309 -0.250 -0.037 -0.007150 0.047 0.010 0.026 0.039 -0.037 -0.021 -0.009
18
Asymptotic Expansion DifferenceCase Strike MC 1st 2nd 3rd 1st 2nd 3rdvi 70 30.018 30.017 30.014 30.018 -0.000 -0.003 0.001
90 11.230 11.234 11.216 11.230 0.004 -0.014 -0.000100 4.618 4.607 4.607 4.616 -0.012 -0.012 -0.003120 0.222 0.195 0.207 0.219 -0.027 -0.016 -0.004150 0.000 0.000 0.000 0.000 -0.000 -0.000 -0.000
vii 70 31.554 31.937 31.586 31.568 0.383 0.031 0.01490 16.493 16.716 16.517 16.493 0.223 0.025 -0.000
100 10.982 11.000 11.000 10.977 0.018 0.018 -0.005120 4.089 3.773 4.099 4.074 -0.316 0.010 -0.015150 0.612 0.381 0.585 0.597 -0.231 -0.026 -0.015
viii 70 30.181 30.293 30.179 30.179 0.112 -0.002 -0.00290 12.878 13.031 12.887 12.874 0.153 0.009 -0.003
100 6.904 6.910 6.910 6.899 0.006 0.006 -0.004120 1.234 1.066 1.240 1.230 -0.169 0.006 -0.004150 0.028 0.010 0.023 0.028 -0.018 -0.005 -0.000
ix 70 30.420 30.293 30.330 30.453 -0.127 -0.090 0.03490 13.185 13.031 13.077 13.205 -0.154 -0.108 0.020
100 7.009 6.910 6.910 7.011 -0.099 -0.099 0.002120 1.188 1.066 1.009 1.163 -0.123 -0.180 -0.025150 0.045 0.010 0.006 0.027 -0.035 -0.040 -0.018
x 70 30.245 30.293 30.236 30.252 0.048 -0.009 0.00790 12.969 13.031 12.958 12.969 0.061 -0.011 0.000
100 6.921 6.910 6.910 6.917 -0.011 -0.011 -0.004120 1.179 1.066 1.153 1.170 -0.113 -0.025 -0.009150 0.024 0.010 0.016 0.020 -0.015 -0.008 -0.004
xi 70 30.228 30.293 30.223 30.232 0.065 -0.006 0.00490 12.947 13.031 12.942 12.946 0.084 -0.005 -0.001
100 6.917 6.910 6.910 6.912 -0.007 -0.007 -0.004120 1.188 1.066 1.173 1.181 -0.123 -0.015 -0.007150 0.024 0.010 0.018 0.021 -0.014 -0.006 -0.003
xii 70 31.099 31.306 31.096 31.119 0.207 -0.003 0.02090 15.445 15.575 15.439 15.448 0.131 -0.006 0.004
100 9.783 9.772 9.772 9.779 -0.011 -0.011 -0.004120 3.104 2.860 3.073 3.088 -0.244 -0.031 -0.015150 0.322 0.186 0.279 0.303 -0.136 -0.043 -0.019
19
Computation with Calibrated Parameters
• F (t, T ) denotes the futures price at time t with maturity T .The process of F (t, T ) under the Equivalent Maringale Measure(EMM):
λ-SABR model;
dF (t, T ) = µ(t, T )σ(t)F (t, T )βdW 1(t),dσ(t) = λ(θ − σ(t))dt+ ν1σ(t)dW 1(t) + ν2σ(t)dW 2(t),
and Heston model;
dF (t, T ) = µ(t, T )F (t, T )√V (t)dW 1(t),
dV (t) = κ(θ − V (t))dt+ ν1√V (t)dW 1(t) + ν2
√V (t)dW 2(t),
where µ(t, T ) := 1t<T.
• The call price at time 0 with maturity T ∗(≤ T ):
C(0) = e−r∗T∗E
[max
1T ∗
∫ T∗
0
F (t, T )dt−K, 0
].
Table 2:
StrikeDate Contract Month Put-1 Put-2 Call-1 Call-2 Call-3 Futures Price Maturity Interest Rate
2007/10/01 Z7 50 70 80 90 110 79.28 0.12 5.19 ％M8 50 70 80 90 110 76.05 0.62 5.06 ％Z8 40 60 70 80 100 74.25 1.13 4.83 ％M9 40 60 70 80 100 73.10 1.62 4.67 ％
2008/07/01 Z8 110 130 140 150 170 142.47 0.38 2.98 ％M9 110 130 140 150 170 142.18 0.87 3.27 ％Z9 110 130 140 150 170 140.90 1.38 3.40 ％M0 110 130 140 150 170 139.79 1.88 3.49 ％
2008/10/01 Z8 70 90 100 110 130 97.92 0.13 4.06 ％M9 70 90 100 110 130 99.34 0.62 4.04 ％Z9 70 90 100 110 130 100.99 1.13 3.96 ％M0 70 90 100 110 130 102.03 1.62 3.55 ％
Table 3:
Heston λ-SABRDate κ V (0) ρ θ ν λ σ(0) β ρ θ ν
2007/10/01 1.180 0.082 -0.408 0.032 0.560 1.590 0.281 1.000 -0.341 0.133 1.1562008/07/01 0.463 0.218 -0.028 0.040 0.811 0.877 0.808 0.889 0.056 0.184 1.0812008/10/01 1.128 0.328 -0.050 0.011 0.700 0.837 0.575 1.000 -0.023 0.025 0.706
20
Average Option: λ-SABR model
Asymptotic Expansion DifferenceDate Maturity MC 1st 2nd 3rd 1st 2nd 3rd
2007/10/01 0.5Y 55 Put 0.04 0.01 0.02 0.04 -0.03 -0.02 0.0075 Call 3.70 3.64 3.65 3.70 -0.06 -0.05 0.0195 Call 0.06 0.02 0.01 0.05 -0.04 -0.05 -0.01
1Y 55 Put 0.24 0.11 0.15 0.27 -0.12 -0.09 0.0375 Call 3.65 3.54 3.54 3.66 -0.11 -0.11 0.0195 Call 0.20 0.08 0.05 0.16 -0.12 -0.15 -0.04
1.5Y 55 Put 0.67 0.54 0.57 0.70 -0.13 -0.09 0.0475 Call 3.10 3.00 2.98 3.10 -0.10 -0.12 0.0095 Call 0.16 0.05 0.03 0.12 -0.11 -0.13 -0.04
2008/07/01 0.5Y 120 Put 2.24 2.53 1.91 2.28 0.29 -0.33 0.04140 Call 11.09 10.92 10.83 11.10 -0.17 -0.26 0.01160 Call 4.37 3.43 4.01 4.35 -0.93 -0.36 -0.02
1Y 120 Put 4.75 5.00 4.11 4.87 0.25 -0.64 0.11140 Call 13.58 13.06 13.01 13.63 -0.53 -0.57 0.05160 Call 7.07 5.48 6.32 7.06 -1.59 -0.74 -0.00
1.5Y 120 Put 3.28 3.55 2.96 3.32 0.28 -0.32 0.04140 Call 9.82 9.55 9.56 9.82 -0.27 -0.26 0.00160 Call 3.77 2.80 3.40 3.75 -0.97 -0.38 -0.02
2008/10/01 0.5Y 80 Put 1.55 2.04 1.44 1.57 0.49 -0.11 0.02100 Call 8.08 7.97 8.00 8.08 -0.10 -0.07 -0.00120 Call 2.57 1.82 2.43 2.56 -0.75 -0.15 -0.01
1Y 80 Put 2.82 3.55 2.62 2.87 0.73 -0.21 0.05100 Call 11.35 11.24 11.19 11.37 -0.11 -0.17 0.01120 Call 5.15 4.00 4.89 5.13 -1.14 -0.25 -0.02
1.5Y 80 Put 3.66 4.50 3.37 3.74 0.84 -0.29 0.08100 Call 13.31 13.19 13.06 13.34 -0.12 -0.25 0.03120 Call 6.87 5.53 6.52 6.87 -1.35 -0.35 -0.01
21
Average Option: Heston model
Asymptotic Expansion DifferenceDate Maturity MC 1st 2nd 3rd 1st 2nd 3rd
2007/10/01 0.5Y 55 Put 0.03 0.02 0.03 0.04 -0.01 0.00 0.0175 Call 3.74 3.85 3.86 3.74 0.11 0.12 -0.0095 Call 0.05 0.04 0.02 0.03 -0.01 -0.03 -0.01
1Y 55 Put 0.22 0.20 0.26 0.27 -0.02 0.04 0.0575 Call 3.68 3.96 3.95 3.67 0.28 0.27 -0.0195 Call 0.17 0.14 0.09 0.11 -0.02 -0.07 -0.05
1.5Y 55 Put 0.66 0.63 0.70 0.71 -0.02 0.04 0.0575 Call 3.12 3.41 3.38 3.10 0.29 0.27 -0.0195 Call 0.13 0.10 0.06 0.08 -0.03 -0.07 -0.04
2008/07/01 0.5Y 120 Put 2.26 2.94 2.29 2.21 0.67 0.03 -0.05140 Call 11.14 11.51 11.42 11.12 0.37 0.28 -0.02160 Call 4.34 3.90 4.49 4.34 -0.43 0.15 0.01
1Y 120 Put 4.85 6.17 5.18 4.70 1.32 0.33 -0.15140 Call 13.66 14.45 14.40 13.61 0.79 0.74 -0.05160 Call 7.06 6.67 7.60 7.07 -0.39 0.54 0.01
1.5Y 120 Put 3.30 3.97 3.36 3.25 0.67 0.06 -0.05140 Call 9.84 10.11 10.12 9.83 0.28 0.29 -0.01160 Call 3.72 3.21 3.83 3.73 -0.50 0.11 0.01
2008/10/01 0.5Y 80 Put 1.56 2.19 1.57 1.54 0.63 0.01 -0.02100 Call 8.08 8.19 8.22 8.08 0.11 0.14 -0.00120 Call 2.55 1.96 2.58 2.56 -0.59 0.04 0.01
1Y 80 Put 2.87 3.97 2.98 2.80 1.10 0.11 -0.07100 Call 11.39 11.79 11.73 11.37 0.40 0.34 -0.02120 Call 5.15 4.44 5.38 5.17 -0.70 0.23 0.02
1.5Y 80 Put 3.75 5.20 3.97 3.63 1.45 0.22 -0.12100 Call 13.38 14.05 13.92 13.34 0.68 0.54 -0.04120 Call 6.90 6.28 7.35 6.93 -0.62 0.45 0.03
22
5 Currency Option under a Libor Market Modelof Interest Rates and a Stochastic Volatility ofa Spot Exchange Rate
Takahashi and Takehara[2007]
5.1 Cross-Currency Libor Market Models
• Let (Ω,F , Ft0≤t≤T , P ) be a filtered probability space satisfying theusual conditions, Wt0≤t≤T be a D-dimensional Ft-standard Wienerprocess.
• The following notations are firstly prepared.
– 0 = T0 < T1 < · · · < TN < TN+1 = T : Resetting time of forwardLibor rates
– fdj(t)Nj=0: Domestic forward Libor rates on the period [Tj , Tj+1]
– ffj(t)Nj=0: Foreign forward Libor rates on the period [Tj , Tj+1]
– Pd(t, s): A domestic zero coupon bond with maturity s > t
– Pf (t, s): A domestic zero coupon bond with maturity s > t
– S(t): A spot exchange rate
• Then, I consider the pricing problem
V (0) = Pd(0, T )EPNd [maxS(T ) −K, 0]
= Pd(0, T )EPNd [maxFT (T ) −K, 0] (5)
where
FT (t) :=Pf (t, T )Pd(t, T )
S(t) (6)
is the forward exchange rate(forward forex) with maturity T and PNd is
the EMM whose numeraire is given by Pd(0, T ) = Pd(0, TN+1).
• First, under PNd , fdj(t) are assumed to follow
fdj(t) = fdj(0) +N∑
i=j+1
∫ t
0
g0di(u)
′γdj(u)fdj(u)du+
∫ t
0
fdj(u)γ′dj(u)dWu, (7)
where Wt is aD-dimensional standard Wiener process under PNd , g0
dj(t) :=−τjfdj(t)1+τjfdj(t)
γdj(t), τj := Tj+1−Tj and γdj(t)Nj=0 are deterministic functions
of t.
• Similarly, under the EMM PNf whose numeraire is Pf (0, T ), ffj(t) are
assumed to follow
ffj(t) = ffj(0) +N∑
i=j+1
∫ t
0
g0fi(u)
′γfj(u)ffj(u)du+
∫ t
0
ffj(u)γ′fj(u)dW
fu , (8)
where W ft is a D-dimensional standard Wiener process under PN
f ,
g0fj(t) := −τf ffj(t)
1+τjffj(t)γfj(t) and γfj(t)N
j=0 are also deterministic functionsof t.
23
• Finally, it is assumed that under the domestic risk-neutral measure thespot forex S(t) and its volatility σ(t) follow
S(t) = S(0)
+∫ t
0
(rd(u) − rf (u))S(u)du+∫ t
0
f(σ(u))σ′S(u)dWu (9)
σ(t) = σ(0) +∫ t
0
µ(u, σ(u))du+∫ t
0
ω′(u, σ(u))dWu (10)
where Wt is a D-dimensional standard Wiener process under that mea-sure, and σ is some constant vector satisfying ‖σ‖ = 1.
• Unifying (7), (8) and (10) into the processes under the same EMM PNd ,
fdj(t), ffj(t) and σ(t) are the solutions of the following graded systemof stochastic differential equations(SDEs):
fdj(t) = fdj(0) +N∑
i=j+1
∫ t
0
g0di(u)
′γdj(u)fdj(u)du+
∫ t
0
fdj(u)γ′dj(u)dWu, (11)
σ(t) = σ(0) +∫ t
0
µ(u, σ(u), fdj(u))du+∫ t
0
ω′(u, σ(u))dWu, (12)
ffj(t) = ffj(0) −j∑
i=0
∫ t
0
g0fi(u)
′γfj(u)ffj(u)du+
N∑i=0
∫ t
0
g0di(u)
′γfj(u)ffj(u)du
−∫ t
0
f(σ(u))σ′γfj(u)ffj(u)du+
∫ t
0
ffj(u)γ′fj(u)dWu, (13)
where
µ(t, σ(t), fdj(t)) := µ(t, σ(t)) + ω′(t, σ(t))
N∑i=0
g0di(t).
• Then, the forward forex FT (t) is a martingale under PNd and follows
FT (t) = FT (0) +∫ t
0
σ′F (u)F (u)dWu (14)
where
σF (t) :=N∑
j=0
(g0
fj(t) − g0dj(t)
)+ f(σ(t)). (15)
24
5.2 An Asymptotic Expansion of the Forward Forex
According to the framework in the previous section, I expand the underlyingforward forex FT (t).
• The underlying system of (11), (12), (13) and (14) are rewritten in anasymptotic expansion’s framework with ε ∈ [0, 1]:
f(ε)dj (t) = fdj(0) + ε2
N∑i=j+1
∫ t
0
g0,(ε)di (u)
′γdj(u)f
(ε)dj (u)du+ ε
∫ t
0
f(ε)dj (u)γ
′dj(u)dWu, (16)
σ(t) = σ(0) +∫ t
0
µ(u, σ(ε)(u), f (ε)dj (u))du+ ε
∫ t
0
ω′(u, σ(ε)(u))dWu, (17)
f(ε)fj (t) = ffj(0) − ε2
j∑i=0
∫ t
0
g0,(ε)fi (u)
′γfj(u)f
(ε)fj (u)du+ ε2
N∑i=0
∫ t
0
g0,(ε)di (u)
′γfj(u)f
(ε)fj (u)du
−ε2∫ t
0
f(σ(ε)(u))σ′γfj(u)f
(ε)fj (u)du+ ε
∫ t
0
f(ε)fj (u)γ
′fj(u)dWu, (18)
F(ε)T (t) = FT (0) + ε
∫ t
0
σ(ε)F (u)
′F (ε)(u)dWu (19)
where
σ(ε)F (t) :=
N∑j=0
(g0,(ε)fj (t) − g
0,(ε)dj (t)
)+ f(σ(ε)(t)),
g0,(ε)dj (t) :=
−τjf (ε)dj (t)
1 + τjf(ε)dj (t)
γdj(t),
g0,(ε)fj (t) :=
−τjf (ε)fj (t)
1 + τjf(ε)fj (t)
γfj(t).
• Then, the explicit expansion of F (ε)T (T ) up to the third order of ε can be
obtained by the methodology already presented in the previous section:
F(ε)T (T ) = FT (0) + εA1
T + ε2A2T + ε3A3
T + o(ε3). (20)
• Assume ε ∈ (0, 1] and let Kε := FT (0) − εy for arbitrary y ∈ R. Thenfrom the expansion above, under the condition ΣT > 0 the price of thecall currency option with maturity T and strike rate Kε is given by
V (0;T,K) = P d(0, T )[ε
∫ ∞
−y
(x+ y)n[x; 0,ΣT ]dx+ ε2∫ ∞
−y
E[A2T |A1
T = x]n[x; 0,ΣT ]dx
+ε3∫ ∞
−y
E[A3T |A1
T = x]n[x; 0,ΣT ]dx +ε3
2E[(A2
T
)2 |A1T = −y]n[y; 0,ΣT ]
]+ o(ε3)
(21)
where ΣT :=∫ T
0‖σ(0)
F (u)‖2du and n[x;m, v] := 1√2πv
exp(− (x−m)2
2v
).
• Note that since the conditional expectations such as E[A2T |A1
T = x] aregiven by linear combinations of Hermite polynomials of x with varianceΣT , this pricing formula can be solved explicitly.
25
Table 4: Initial domestic/foreign forward interest rates and their volatilities
fd γ∗d ff γ∗
f
Case (i) 0.05 0.2 0.05 0.2
Case (ii) 0.02 0.5 0.05 0.2
Case (iii) 0.05 0.2 0.02 0.5
5.3 Numerical Examples
• In this subsection I confirm the accuracy of the method in this cross-currency framework. Especially, the process of the volatility of the spotforex is specified by
µ(t, x) = κ(θ − x),ω(t, x) = ω
√x,
(22)
with some constants κ, θ and a constant vector ω.
• Moreover, the parameters are set as follows. D = 4, ε = 1, σ(0) = θ = 0.1,and κ = 0.1; ω = ω∗Vσ where ω∗ = 0.1 and Vσ denotes a four dimensionalconstant vector given below.
• For f(x) it is assumed to be f(x) = minx,Kmax with some constantKmax, while the approximation with an asymptotic expansion is exercisedwith replacement of f(x) by f(x) := x to avoid complexity in calculation;Kmax is set to be 106 in numerical examples below.
• I further suppose that initial term structures of domestic and foreignforward interest rates are flat, and their volatilities have flat structuresand are constant over time: that is, for all j, fdj(0) = fd, ffj(0) = ff ,γdj(t) = γ∗dVd1t≤Tj(t) and γfj(t) = γ∗fVf1t≤Tj(t). Here, γ∗d and γ∗f areconstant scalars, and Vd and Vf denote four dimensional constant vectors.I consider three cases for fd, γ∗d , ff and γ∗f as in Table 2.
• Moreover, given a correlation matrix C among all four factors, I can de-termine the constant vectors Vd, Vf , VS and Vσ to satisfy ‖Vd‖ = ‖Vf‖ =‖VS‖ = ‖Vσ‖ = 1 and V ′V = C where V := (Vd, Vf , VS , Vσ), and VS ≡ σ;σ was defined right after the equation (9).
• Additionally, I make an assumption that γdn(t)−1(t) and γfn(t)−1(t), volatil-ities of the domestic and foreign interest rates applied to the period fromt to the next fixing date Tn(t), are set to be zero for arbitrary t ∈ [t, Tn(t)]where n(t) := minj; t ≤ Tj.
• Finally, for correlations the following four sets of parameters are consid-ered:
– “Corr.1”: All the factors are independent:– “Corr.2”: There exists only the correlation of -0.5 between the spot
exchange rate and its volatility (i.e. V′SVσ = −0.5) while there are
no correlations among the others:– “Corr.3”: The correlation between interest rates and the spot ex-
change rate are allowed while there are no correlations among theothers: the correlation between domestic ones and the spot forex is0.5(V
′dVS = 0.5) and the correlation between foreign ones and the
spot forex is -0.5(V′fVS = −0.5).
26
– “Corr.4”: Correlations among most factors are considered; V′dVf =
0.3 between the domestic and foreign interest rates; V′dVS = 0.5,
V′fVS = −0.5 between interest rates and the spot forex; and V
′SVσ =
−0.5 between the spot forex and its volatility.
• Monte Carlo simulations are exercised with the following settings.
– Using Euler-Maruyama scheme with time step of 0.05
– Also using the Antithetic Variable Method
– 1,000,000 trials
T Int. Corr. K/F0 Estimated Values Differences Relative DiffenrencesM.C. 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd
5y Case (i) Corr.1 0.75 20.70 20.94 20.41 20.79 0.24 -0.30 0.09 1.1% -1.4% 0.4%1 7.68 7.43 7.43 7.69 -0.25 -0.25 0.01 -3.3% -3.3% 0.2%
1.25 2.32 1.41 1.95 2.33 -0.91 -0.38 0.01 -39.2% -16.1% 0.3%Corr.2 0.75 21.02 20.94 20.97 21.14 -0.08 -0.06 0.12 -0.4% -0.3% 0.6%
1 7.52 7.43 7.43 7.53 -0.09 -0.09 0.01 -1.2% -1.2% 0.1%1.25 1.63 1.41 1.39 1.56 -0.22 -0.25 -0.07 -13.6% -15.1% -4.3%
Corr.3 0.7 24.56 25.03 24.29 24.64 0.47 -0.27 0.08 1.9% -1.1% 0.3%1 8.95 8.75 8.75 8.96 -0.20 -0.20 0.01 -2.2% -2.2% 0.2%
1.3 2.67 1.60 2.34 2.68 -1.08 -0.33 0.01 -40.2% -12.5% 0.3%Corr.4 0.7 24.75 24.97 24.72 24.89 0.22 -0.03 0.13 0.9% -0.1% 0.5%
1 8.66 8.63 8.63 8.68 -0.03 -0.03 0.02 -0.4% -0.4% 0.2%1.3 1.98 1.53 1.78 1.95 -0.45 -0.20 -0.04 -22.6% -10.1% -1.9%
Case (ii) Corr.1 0.75 20.71 20.95 20.39 20.81 0.24 -0.31 0.10 1.1% -1.5% 0.5%1 7.71 7.43 7.43 7.75 -0.28 -0.28 0.03 -3.6% -3.6% 0.4%
1.25 2.38 1.42 1.97 2.39 -0.97 -0.42 0.00 -40.6% -17.5% 0.1%Corr.2 0.75 21.03 20.95 20.95 21.17 -0.08 -0.08 0.14 -0.4% -0.4% 0.7%
1 7.55 7.43 7.43 7.58 -0.12 -0.12 0.03 -1.5% -1.5% 0.3%1.25 1.70 1.42 1.41 1.63 -0.29 -0.29 -0.08 -16.9% -17.3% -4.6%
Corr.3 0.7 24.54 25.04 24.22 24.61 0.50 -0.32 0.07 2.1% -1.3% 0.3%1 8.96 8.76 8.76 8.98 -0.20 -0.20 0.02 -2.2% -2.2% 0.2%
1.3 2.78 1.61 2.43 2.82 -1.17 -0.35 0.04 -42.2% -12.5% 1.6%Corr.4 0.7 24.77 24.98 24.66 24.88 0.21 -0.11 0.11 0.8% -0.4% 0.5%
1 8.67 8.64 8.64 8.69 -0.03 -0.03 0.02 -0.3% -0.3% 0.2%1.3 2.08 1.54 1.86 2.08 -0.54 -0.22 -0.00 -26.0% -10.7% -0.1%
Case (iii) Corr.1 0.75 24.04 24.27 23.67 24.15 0.23 -0.37 0.11 1.0% -1.5% 0.4%1 8.94 8.61 8.61 8.98 -0.32 -0.32 0.04 -3.6% -3.6% 0.5%
1.25 2.70 1.64 2.24 2.72 -1.06 -0.46 0.02 -39.2% -16.9% 0.9%Corr.2 0.75 24.40 24.27 24.32 24.55 -0.13 -0.09 0.15 -0.5% -0.4% 0.6%
1 8.77 8.61 8.61 8.80 -0.15 -0.15 0.03 -1.7% -1.7% 0.4%1.25 1.90 1.64 1.60 1.83 -0.26 -0.31 -0.07 -13.7% -16.0% -3.7%
Corr.3 0.7 28.55 29.02 28.25 28.68 0.47 -0.30 0.12 1.6% -1.0% 0.4%1 10.38 10.15 10.15 10.40 -0.23 -0.23 0.02 -2.2% -2.2% 0.2%
1.3 3.06 1.86 2.63 3.05 -1.19 -0.43 -0.00 -39.1% -14.0% -0.2%Corr.4 0.7 28.79 28.95 28.73 28.93 0.16 -0.06 0.14 0.5% -0.2% 0.5%
1 10.07 10.02 10.02 10.11 -0.06 -0.06 0.03 -0.6% -0.6% 0.3%1.3 2.26 1.79 2.00 2.20 -0.47 -0.26 -0.06 -20.9% -11.4% -2.5%
Table 5: A comparison of estimators by asymptotic expansions to by montecarlo simulations: 5y.
27
T Int. Corr. K/F0 Estimated Values Differences Relative DiffenrencesM.C. 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd
10y Case (i) Corr.1 0.6 25.68 26.42 25.25 25.89 0.74 -0.43 0.21 2.9% -1.7% 0.8%1 10.17 9.70 9.70 10.28 -0.47 -0.47 0.11 -4.6% -4.6% 1.1%
1.4 3.85 2.01 3.18 3.82 -1.84 -0.67 -0.03 -47.8% -17.4% -0.9%Corr.2 0.6 25.92 26.42 25.79 26.16 0.50 -0.13 0.24 1.9% -0.5% 0.9%
1 9.90 9.70 9.70 9.99 -0.20 -0.20 0.09 -2.0% -2.0% 0.9%1.4 3.06 2.01 2.64 3.01 -1.05 -0.41 -0.04 -34.2% -13.5% -1.4%
Corr.3 0.5 31.63 33.08 31.22 31.77 1.44 -0.41 0.14 4.6% -1.3% 0.4%1 12.45 12.21 12.21 12.52 -0.24 -0.24 0.07 -2.0% -2.0% 0.6%
1.5 4.92 2.56 4.42 4.97 -2.36 -0.50 0.05 -47.9% -10.2% 1.1%Corr.4 0.5 31.70 32.82 31.58 31.87 1.11 -0.13 0.16 3.5% -0.4% 0.5%
1 11.73 11.77 11.77 11.81 0.04 0.04 0.08 0.4% 0.4% 0.7%1.5 3.78 2.30 3.54 3.83 -1.48 -0.24 0.05 -39.1% -6.4% 1.2%
Case (ii) Corr.1 0.6 25.67 26.44 25.10 26.05 0.77 -0.58 0.38 3.0% -2.2% 1.5%1 10.32 9.73 9.73 10.56 -0.59 -0.59 0.24 -5.7% -5.7% 2.4%
1.4 4.30 2.03 3.37 4.32 -2.28 -0.93 0.02 -52.9% -21.7% 0.4%Corr.2 0.6 25.92 26.44 25.64 26.35 0.52 -0.29 0.43 2.0% -1.1% 1.7%
1 10.01 9.73 9.73 10.20 -0.28 -0.28 0.19 -2.8% -2.8% 1.9%1.4 3.51 2.21 2.83 3.55 -1.31 -0.68 0.04 -37.2% -19.4% 1.0%
Corr.3 0.5 31.58 33.10 30.90 31.78 1.52 -0.68 0.21 4.8% -2.2% 0.6%1 12.48 12.25 12.25 12.57 -0.24 -0.24 0.09 -1.9% -1.9% 0.7%
1.5 5.44 2.59 4.79 5.68 -2.86 -0.65 0.23 -52.5% -12.0% 4.3%Corr.4 0.5 31.68 32.84 31.33 31.97 1.16 -0.35 0.29 3.6% -1.1% 0.9%
1 11.76 11.81 11.81 11.84 0.05 0.05 0.08 0.4% 0.4% 0.7%1.5 4.24 2.32 3.83 4.46 -1.92 -0.41 0.22 -45.2% -9.8% 5.3%
Case (iii) Corr.1 0.6 34.96 35.50 34.14 35.33 0.54 -0.82 0.37 1.6% -2.3% 1.1%1 13.86 13.06 13.06 14.18 -0.80 -0.80 0.32 -5.8% -5.8% 2.3%
1.4 5.15 2.72 4.09 5.28 -2.43 -1.07 0.12 -47.2% -20.7% 2.4%Corr.2 0.6 35.29 35.50 34.86 35.66 0.22 -0.42 0.37 0.6% -1.2% 1.1%
1 13.56 13.06 13.06 13.90 -0.50 -0.50 0.34 -3.7% -3.7% 2.5%1.4 4.08 2.72 3.37 4.16 -1.36 -0.72 0.08 -33.3% -17.5% 1.9%
Corr.3 0.5 43.00 44.45 42.39 43.36 1.45 -0.61 0.36 3.4% -1.4% 0.8%1 16.77 16.45 16.45 16.88 -0.32 -0.32 0.11 -1.9% -1.9% 0.7%
1.5 6.43 3.75 5.53 6.50 -2.68 -0.89 0.08 -41.7% -13.9% 1.2%Corr.4 0.5 43.01 44.10 42.76 43.34 1.08 -0.25 0.32 2.5% -0.6% 0.8%
1 15.86 15.85 15.85 16.02 -0.00 -0.00 0.17 0.0% 0.0% 1.1%1.5 4.95 3.12 4.46 5.03 -1.83 -0.50 0.08 -37.0% -10.0% 1.6%
Table 6: A comparison of estimators by asymptotic expansions to by montecarlo simulations: 10y.
28
6 Computation of Higher-Orders
6.1 Expansion around a Normal Distribution
Takahashi-Toda(2008)
• A N -dimensional process S(ε)t = (S(ε),1
t , · · · , S(ε),Nt ):
dS(ε),it = ε
d∑j=1
V ij (S(ε)
t , t)dW jt (i = 1, · · · , N)
S(ε)0 = S0 ∈ RN
where W = (W 1, · · · ,W d) is a d-dimensional standard Wiener process,and Vj = (V 1
j , · · · , V Nj ): RN ×R+ → RN satisfies some regularity condi-
tions.
Remark 3
dS(ε),it = κi(t)(θi(t) − S
(ε),it )dt+ ε
d∑j=1
V ij (S(ε)
t , t)dW jt (i = 1, · · · , N)
where κi, θi(i = 1, · · · , N) are deterministic functions of t.
Define S(ε)t = (S(ε),1
t , · · · , S(ε),Nt ) as
S(ε),it := e
∫ t
0κi(s)ds
S(ε),it −
∫ t
0
e
∫ s
0κi(u)du
κi(s)θi(s)ds (i = 1, · · · , N).
Then,
dS(ε),it = ε
d∑j=1
V ij (S(ε)
t , t)dW jt (i = 1, · · · , N),
V ij (S(ε)
t , t) := e
∫ t
0κi(s)ds
V ij (S(ε)
t , t).
• Asymptotic expansion of S(ε)t :
S(ε)t = S
(0)t + εA1t +
ε2
2A2t +
ε3
6A3t +
ε4
24A4t + o(ε4), as ε ↓ 0,
where S(0)t = S0 and
dA1t =d∑
j=1
Vj(S(0)t , t)dW j
t
dA2t = 2d∑
j=1
N∑i=1
Ai1t
∂
∂xiVj(S
(0)t , t)dW j
t
dA3t = 3
N∑
i=1
Ai2t
∂
∂xiVj(S
(0)t , t) +
N∑i=1
N∑k=1
Ai1tA
k1t
∂2
∂xi∂xkVj(S
(0)t , t)
dW j
t
dA4t = 4N∑
i=1
Ai3t
∂
∂xiVj(S
(0)t , t) + 3
N∑i=1
N∑k=1
Ai1tA
k2t
∂2
∂xi∂xkVj(S
(0)t , t)
+N∑
i=1
N∑k=1
N∑l=1
Ai1tA
k1tA
l1t
∂3
∂xi∂xk∂xlVj(S
(0)t , t)dW j
t .
29
• For ξ ∈ R, define a complex-valued stochastic process Zt := Z〈ξ〉t as
Z〈ξ〉t := exp(iξ)A1
1t −12(iξ)2〈A1
1·〉t.
Then, Z〈ξ〉t is a martingale and
dZ〈ξ〉t = (iξ)
d∑j=1
V 1j (S(0)
t , t)Z〈ξ〉t dW j
t
and note that
exp(iξ)A11t = Z
〈ξ〉t exp−ξ
2
2Σt,
where
Σt :=∫ t
0
d∑j=1
V 1j (S(0)
s , t)2ds.
• Normalization of S(ε),1:
X(ε)t :=
S(ε),1t − S
(0),1t
ε= A1
1T +ε
2A1
2T +ε2
6A1
3T +ε3
24A1
4T + o(ε3)
30
• Asymptotic expansion of the characteristic function of X(ε)T up to the ε3-
order:
ψX
(ε)T
(ξ) := E[exp(iξ)X(ε)T ]
= E[exp(iξ)A11T exp ε
2(iξ)A1
2T +ε2
6(iξ)A1
3T +ε3
24(iξ)A1
4T + o(ε3)]
= E[exp(iξ)A11T ] + ε
(iξ)2E[A1
2T exp(iξ)A11T ]
+ε2
(iξ)6E[A1
3T exp(iξ)A11T ] +
(iξ)2
8E[(A1
2T )2 exp(iξ)A11T ]
+ε3
(iξ)24
E[A14T exp(iξ)A1
1T ] +(iξ)2
12E[A1
2TA13T exp(iξ)A1
1T ]
+(iξ)3
48E[(A1
2T )3 exp(iξ)A11T ]
+ o(ε3)
=
1 + ε
(iξ)2E[A1
2TZT ] + ε2
(iξ)6E[A1
3TZT ] +(iξ)2
8E[(A1
2T )2ZT ]
+ε3
(iξ)24
E[A14TZT ] +
(iξ)2
12E[A1
2TA13TZT ] +
(iξ)3
48E[(A1
2T )3ZT ]
× exp−ξ2
2ΣT
+o(ε3)
• Define ηi1,1, η
i2,1, η
i,k2,2, η
i3,1, η
i,k3,2, η
i,k3,3, η
i4,1, η
i,k4,2,1, η
i,k4,2,2, η
i,k5,2, η
i,k5,3, and ηi,k,l
6,3
as
ηi1,1(t) := E[Ai
1tZt]
ηi2,1(t) := E[Ai
2tZt]
ηi,k2,2(t) := E[Ai
1tAk1tZt]
ηi3,1(t) := E[Ai
3tZt]
ηi,k3,2(t) := E[Ai
1tAk2tZt]
ηi,k,l3,3 (t) := E[Ai
1tAk1tA
l1tZt]
ηi4,1(t) := E[Ai
4tZt]
ηi,k4,2,1(t) := E[Ai
1tAk3tZt]
ηi,k4,2,2(t) := E[Ai
2tAk2tZt]
ηi,k,l4,3 (t) := E[Ai
1tAk1tA
l2tZt]
ηi,k5,2(t) := E[Ai
2tAk3tZt]
ηi,k,l5,3 (t) := E[Ai
1tAk2tA
l2tZt]
ηi,k,l6,3 (t) := E[Ai
2tAk2tA
l2tZt].
31
• Consider the evaluation of η12,1(T ) = E[A1
2TZT ] which appears in the ε-order.
d(A12tZt) = A1
2tdZt + ZtdA12t + dA1
2tdZt
=
⎧⎨⎩2(iξ)
d∑j=1
V 1j (S(0)
t , t)Zt
N∑i=1
Ai1t
∂
∂xiV 1
j (S(0)t , t)
⎫⎬⎭ dt
+d∑
j=1
(iξ)V 1
j (S(0)t , t)A1
2tZt + 2N∑
i=1
∂
∂xiV 1
j (S(0)t , t)Ai
1tZt
dW j
t
• Since the second and third terms are martingales, taking the expectationon both sides, we have the following ordinary differential equation of η1
2,1:
d
dtη12,1(t) = 2(iξ)
d∑j=1
N∑i=1
V 1j (S(0)
t , t)∂
∂xiV 1
j (S(0)t , t)ηi
1,1(t)
• Here, ηi1,1(i = 1, · · · , N) appearing in the right hand side of above ODE is
evaluated in the similar manner:
d(Ai1tZt) = Ai
1tdZt + ZtdAi1t + dAi
1tdZt
= (iξ)d∑
j=1
V 1j (S(0)
t , t)V ij (St(0), t)Ztdt
+d∑
j=1
(iξ)V 1
j (S(0)t , t)Ai
1tZt + V ij (S(0)
t , t)Zt
dW j
t
Hence,d
dtηi1,1(t) = (iξ)
d∑j=1
V 1j (S(0)
t , t)V ij (S(0)
t , t)E[Zt]
Since E[Zt] = 1, we have
ηi1,1(t) = (iξ)
∫ t
0
d∑j=1
V 1j (S(0)
s , s)V ij (S(0)
s , s)ds.
• Higher order terms can be evaluated in the similar way.
The key observation is that each ODE does not involve any higher orderterms, and only lower terms appears in the r.h.s. of the ODE. So, one caneasily solve the system of ODEs and evaluate expectations.
32
• Proposition 1 The asymptotic expansion of the characteristic functionof X(ε)
T up to the ε3-order is expressed as
ψX
(ε)T
(ξ) =
1 + ε
(iξ)2η12,1(T ) + ε2
(iξ)6η13,1(T ) +
(iξ)2
8η1,14,2,2(T )
+ε3
(iξ)24
η14,1(T ) +
(iξ)2
12η1,15,2(T ) +
(iξ)3
48η1,1,16,3 (T )
× exp−ξ2
2ΣT + o(ε3),
where η12,1(T ), η1
3,1(T ) ,η1,14,2,2(T ), η1,1
4,1(T ), η1,15,2(T ) and η1,1
6,3(T ) are obtainedby the following system of ordinary differential equations:
ηi1,1(t) = (iξ)
∫ t
0
d∑j=1
V 1j (S(0)
s , s)V ij (S(0)
s , s)ds
ηi2,1(t) = (iξ)
∫ t
0
d∑j=1
N∑i′=1
V 1j (S(0)
t , t)∂
∂xi′V i
j (S(0)t , t)ηi′
1,1(s)ds
ηi,k2,2(t) = (iξ)
∫ t
0
d∑j=1
V 1
j (S(0)s , s)V i
j (Ss(0), s)ηk1,1(s) + V 1
j (S(0)s , s)V k
j (S(0)s , s)ηi
1,1(s)ds
+∫ t
0
d∑j=1
V ij (S(0)
s , s)V kj (S(0)
s , s)ds
ηi3,1(t) = (iξ)
∫ t
0
d∑j=1
3
N∑i′=1
V 1j (S(0)
s , s)∂
∂xi′V i
j (S(0)t , t)ηi′
2,1(s)
+3N∑
i′=1
N∑k′=1
V 1j (S(0)
s , s)∂2
∂xi′∂xk′V i
j (S(0)s , s)ηi′,k′
2,2 (s)ds
ηi,k3,2(t) = (iξ)
∫ t
0
d∑j=1
V 1
j (S(0)s , s)V i
j (S(0)s , s)ηk
2,1(s)
+2N∑
i′=1
V 1j (S(0)
s , s)∂
∂xi′V k
j (S(0)s , s)ηi,i′
2,2 (s)ds
+∫ t
0
d∑j=1
2
N∑i′=1
V ij (S(0)
s , s)∂
∂xi′V k
j (S(0)t , t)ηi′
1,1(s)ds
ηi,k,l3,3 (t) = (iξ)
∫ t
0
d∑j=1
V 1
j (S(0)s , s)V i
j (Ss(0), s)ηk,l2,2(s)
+V 1j (S(0)
s , s)V kj (S(0)
s , s)ηi,l2,2(s) + V 1
j (S(0)s , s)V l
j (S(0)s , s)ηi,k
2,2(s)ds
+∫ t
0
d∑j=1
V k
j (S(0)s , s)V l
j (Ss(0), s)ηi1,1(s)
+V ij (S(0)
s , s)V lj (Ss(0), s)ηk
1,1(s) + V ij (S(0)
s , s)V kj (Ss(0), s)ηl
1,1(s)ds
33
ηi4,1(t) = (iξ)
∫ t
0
d∑j=1
3
N∑i′=1
V 1j (S(0)
s , s)∂
∂xi′V i
j (S(0)t , t)ηi′
3,1(s)
+12N∑
i′=1
N∑k′=1
V 1j (S(0)
s , s)∂2
∂xi′∂xk′V i
j (S(0)s , s)ηi′,k′
3,2 (s)
+3N∑
i′=1
N∑k′=1
N∑l′=1
V 1j (S(0)
s , s)∂3
∂xi′∂xk′∂xl′V i
j (S(0)s , s)ηi′,k′,l′
3,3 (s)ds
ηi,k4,2,1(t) = (iξ)
∫ t
0
d∑j=1
V 1
j (S(0)s , s)V i
j (S(0)s , s)ηk
3,1(s) + 3N∑
i′=1
V 1j (S(0)
s , s)∂
∂xi′V k
j (S(0)s , s)ηi,i′
3,2 (s)
+3N∑
i′=1
N∑k′=1
V 1j (S(0)
s , s)∂2
∂xi′∂xk′V k
j (S(0)s , s)ηi,i′,k′
3,3 (s)ds
+∫ t
0
d∑j=1
3
N∑i′=1
V ij (S(0)
s , s)∂
∂xi′V k
j (S(0)s , s)ηi′
2,1(s)
+3N∑
i′=1
N∑k′=1
V ij (S(0)
s , s)∂2
∂xi′∂xk′V k
j (S(0)s , s)ηi′,k′
2,2 (s)ds
ηi,k4,2,2(t) = (iξ)
∫ t
0
d∑j=1
2
N∑i′=1
V 1j (S(0)
s , s)∂
∂xi′V i
j (S(0)s , s)ηi′,k
3,2 (s)
+2N∑
i′=1
V 1j (S(0)
s , s)∂
∂xi′V k
j (S(0)s , s)ηi′,i
3,2 (s)ds
+∫ t
0
d∑j=1
4
N∑i′=1
N∑k′=1
∂
∂xi′V i
j (S(0)s , s)
∂
∂xk′V k
j (S(0)s , s)ηi′,k′
2,2 (s)ds
ηi,k,l4,3 (t) = (iξ)
∫ t
0
d∑j=1
V 1
j (S(0)s , s)V k
j (S(0)s , s)ηi,l
3,2(s) + V 1j (S(0)
s , s)V ij (S(0)
s , s)ηk,l3,2(s)
+2N∑
i′=1
V 1j (S(0)
s , s)∂
∂xi′V l
j (S(0)s , s)ηi,k,i′
3,3 (s)ds
+∫ t
0
d∑j=1
V i
j (S(0)s , s)V k
j (S(0)s , s)ηl
2,1(s)
+2N∑
i′=1
V kj (S(0)
s , s)∂
∂xi′V l
j (S(0)s , s)ηi,i′
2,2 (s) + 2N∑
i′=1
V ij (S(0)
s , s)∂
∂xi′V l
j (S(0)s , s)ηk,i′
2,2 (s)ds
ηi,k5,2(t) = (iξ)
∫ t
0
d∑j=1
2
N∑i′=1
V 1j (S(0)
s , s)∂
∂xi′V i
j (S(0)s , s)ηk,i′
4,2,1(s)
+3N∑
i′=1
V 1j (S(0)
s , s)∂
∂xi′V k
j (S(0)s , s)ηi,i′
4,2,2(s)
+3N∑
i′=1
N∑k′=1
V 1j (S(0)
s , s)∂2
∂xi′∂xk′V k
j (S(0)s , s)ηi′,k′,i
4,3 (s)ds
+∫ t
0
d∑j=1
6
N∑i′=1
N∑k′=1
∂
∂xi′V i
j (S(0)s , s)
∂
∂xk′V k
j (S(0)s , s)ηi′,k′
3,2 (s)
34
+6N∑
i′=1
N∑k′=1
N∑l′=1
∂
∂xi′V i
j (S(0)s , s)
∂2
∂xk′∂xl′V k
j (S(0)s , s)ηi′,k′,l′
3,3 (s)ds
ηi,k,l5,3 (t) = (iξ)
∫ t
0
d∑j=1
V 1
j (S(0)s , s)V i
j (S(0)s , s)ηk,l
4,2(s)
+2N∑
i′=1
V 1j (S(0)
s , s)∂
∂xi′V l
j (S(0)s , s)ηi′,i,k
4,3 (s)
+2N∑
i′=1
V 1j (S(0)
s , s)∂
∂xi′V k
j (S(0)s , s)ηi′,i,l
4,3 (s)ds
+∫ t
0
d∑j=1
2
N∑i′=1
V ij (S(0)
s , s)∂
∂xi′V l
j (S(0)s , s)ηi′,k
3,2 (s)
+2N∑
i′=1
V ij (S(0)
s , s)∂
∂xi′V k
j (S(0)s , s)ηi′,l
3,2(s)
+4N∑
i′=1
N∑k′=1
∂
∂xi′V k
j (S(0)s , s)
∂
∂xk′V l
j (S(0)s , s)ηi,i′,k′
3,3 (s)ds
ηi,k,l6,3 (t) = (iξ)
∫ t
0
d∑j=1
2
N∑i′=1
V 1j (S(0)
s , s)∂
∂xi′V i
j (S(0)s , s)ηi′,k,l
5,3 (s)
+2N∑
i′=1
V 1j (S(0)
s , s)∂
∂xi′V k
j (S(0)s , s)ηi′,i,l
5,3 (s)
+2N∑
i′=1
V 1j (S(0)
s , s)∂
∂xi′V l
j (S(0)s , s)ηi′,i,k
5,3 (s)ds
+∫ t
0
d∑j=1
4
N∑i′=1
N∑k′=1
∂
∂xi′V k
j (S(0)s , s)
∂
∂xk′V l
j (S(0)s , s)ηi′,k′,i
4,3 (s)
+4N∑
i′=1
N∑k′=1
∂
∂xi′V i
j (S(0)s , s)
∂
∂xk′V l
j (S(0)s , s)ηi′,k′,k
4,3 (s)
+4N∑
i′=1
N∑k′=1
∂
∂xi′V i
j (S(0)s , s)
∂
∂xk′V k
j (S(0)s , s)ηi′,k′,l
4,3 (s)ds
35
• From the discussion above, it is easily shown that the asymptotic expan-sion of the characteristic function can be expressed as a product of theGaussian characteristic function and a polynomial of (iξ). Indeed, theasymptotic expansion of the characteristic function of X(ε)
T up to the ε3-order is given by
ψX
(ε)T
(ξ) = 1 + εC23(iξ)3+ε2C32(iξ)2 + C34(iξ)4 + C36(iξ)6+ε3C43(iξ)3 + C45(iξ)5 + C47(iξ)7 + C49(iξ)9 exp−ξ
2
2ΣT + o(ε3)
for some constants C23, C32, C34, C36, C43, C45, C47, C49. (Precisely, eachηn,. is a nth-order polynomial of (iξ).)
• To obtain the asymptotic expansion of density function, we need to invertψ
X(ε)T
using the inverse Fourier transformation:
F−1[(iξ)ne−ξ2
2 Σ](x) =1
ΣnHn(x; Σ)n[x; 0,Σ],
whereHn(x; Σ) := (−Σ)nex2/2Σ dn
dxne−x2/2Σ
and
n[x;µ,Σ] :=1√2πΣ
exp− (x− µ)2
2Σ.
– For example, for n = 0, 1, 2, 3, 4,
H0(x; Σ) = 1, H1(x; Σ) = x, H2(x; Σ) = x2 − Σ,H3(x; Σ) = x3 − 3Σx, H4(x; Σ) = x4 − 6Σx2 + 3Σ2.
• Using Hermite polynomials, we obtain the asymptotic expansion of thedensity function of X(ε)
T up to the ε3-order as
fX
(ε)T
(x) = n[x; 0,ΣT ]
+εC23
Σ3T
H3(x; ΣT )n[x; 0,ΣT ]
+ε2C32
Σ2T
H2(x; ΣT ) +C34
Σ4T
H4(x; ΣT ) +C36
Σ6T
H6(x; ΣT )n[x; 0,ΣT ]
+ε3C43
Σ3T
H3(x; ΣT ) +C45
Σ5T
H5(x; ΣT ) +C47
Σ7T
H7(x; ΣT ) +C49
Σ9T
H9(x; ΣT )n[x; 0,ΣT ]
+o(ε3).
36
On Computation of Average Options
• S(ε) is defined by
S(ε)t :=
∫ t
0
S(ε)s ds.
• The asymptotic expansion of S(ε) as ε ↓ 0 is given by:
S(ε)t = S
(0)t + εA1t +
ε2
2A2t +
ε3
6A3t +
ε4
24A4t + o(ε3),
where S(0)t , Ait, (i = 1, 2, 3, 4) are obained as follows:
S(0)t =
∫ t
0
S(0)s ds,
dAit = Aitdt.
• Fix T > 0 and apply Fubini’s theorem to A11T :
A11T =
d∑j=1
∫ T
0
∫ t
0
V 1j (S(0)
s , s)dW js dt
=d∑
j=1
∫ T
0
V Tj (S(0)
s , s)dW js ,
whereV T
j (S(0)t , t) := (T − t)V 1
j (S(0)t , t).
• Define AT1 as
dAT1t =
d∑j=1
V Tj (S(0)
t , t)dW jt , AT
10 = 0.
Note that A11T = AT
1T at t = T．
• Using AT1 , define Zt := Z
T,〈ξ〉t for ξ ∈ R as
ZT,〈ξ〉t := exp
(iξ)AT
1t − 12(iξ)2〈AT
1·〉t.
Then, ZT,〈ξ〉t is a martingale and satisfies the following SDE:
dZT,〈ξ〉t = (iξ)
d∑j=1
V Tj (S(0)
t , t)ZT,〈ξ〉t dW j
t .
• It also holds that
exp(iξ)A11T = Z
T,〈ξ〉T exp−ξ
2
2ΣT ,
where
ΣT :=∫ T
0
d∑j=1
V Tj (S(0)
t , t)2dt.
37
• The asymptotic expansion of the characteristic function of X(ε)T = S
(ε),1T
−S(0),1T
εis expressed as follows:
ψX
(ε)T
(ξ) =
1 + ε
(iξ)2
E[A12TZ
T,〈ξ〉T ]
+ε2
(iξ)6
E[A13TZ
T,〈ξ〉T ] +
(iξ)2
8E[(A1
2T )2ZT,〈ξ〉T ]
+ε3
(iξ)24
E[A14TZ
T,〈ξ〉T ] +
(iξ)2
12E[A1
2T A13TZ
T,〈ξ〉T ] +
(iξ)3
48E[(A1
2T )3ZT,〈ξ〉T ]
× exp−ξ2
2ΣT + o(ε3).
Each expectation on the right hand side can be obtained in the similarmanner as before.
38
Numerical Examples: λ-SABR model
• λ-SABR model:
dS(t) = σ(t)S(t)βdW 1(t),dσ(t) = λ(θ − σ(t))dt+ ν1σ(t)dW 1(t) + ν2σ(t)dW 2(t),
where ν1 = ρν，ν2 = (√
1 − ρ2)ν．（The correlation between S and σ isρ ∈ [−1, 1].）
• European plain-vanilla call and put prices(interest rate=0%):
Approximated prices by the asymptotic expansion method upto the fifth-order:
All the solutions to differential equations are obtained analytically.
Benchmark values are computed by Monte Carlo simulations:
Table 7:
Parameter λ σ(0) β ρ θ ν T
i 0.1 3.0 0.5 -0.7 3.0 0.3 10ii 0.1 3.0 0.5 -0.7 3.0 0.1 10iii 0.1 3.0 0.5 -0.7 3.0 0.3 1
Case i: Euler-Maruyama scheme, 1024 time stepsCase ii: Euler-Maruyama scheme, 1024 time stepsCase iii: Euler-Maruyama scheme, 512 time stepsThe number of trials: 108
39
Case Strike(C/P) MC 1st 2nd 3rd 4th 5th 1st 2nd 3rd 4th 5thi 50 Put 13.109 4.876 5.000 2.313 1.067 0.260 37.20% 38.14% 17.64% 8.14% 1.98%
60 Put 16.618 4.544 4.648 1.931 0.938 0.195 27.34% 27.97% 11.62% 5.65% 1.17%70 Put 20.482 4.241 4.322 1.585 0.844 0.149 20.71% 21.10% 7.74% 4.12% 0.73%80 Put 24.720 3.965 4.020 1.269 0.778 0.117 16.04% 16.26% 5.14% 3.15% 0.47%90 Put 29.347 3.710 3.738 0.980 0.735 0.094 12.64% 12.74% 3.34% 2.51% 0.32%
100 Call 34.375 3.472 3.472 0.712 0.712 0.077 10.10% 10.10% 2.07% 2.07% 0.22%110 Call 29.811 3.246 3.217 0.459 0.704 0.063 10.89% 10.79% 1.54% 2.36% 0.21%120 Call 25.659 3.026 2.971 0.220 0.711 0.050 11.79% 11.58% 0.86% 2.77% 0.19%130 Call 21.914 2.809 2.728 -0.010 0.731 0.035 12.82% 12.45% -0.04% 3.33% 0.16%140 Call 18.571 2.591 2.487 -0.230 0.762 0.018 13.95% 13.39% -1.24% 4.10% 0.10%150 Call 15.615 2.370 2.246 -0.441 0.804 -0.002 15.18% 14.38% -2.83% 5.15% -0.02%
ii 50 Put 1.682 -0.914 0.030 0.475 0.182 -0.016 -54.33% 1.81% 28.25% 10.84% -0.92%60 Put 2.607 -1.056 0.129 0.445 0.103 -0.003 -40.52% 4.94% 17.06% 3.95% -0.13%70 Put 3.950 -1.047 0.214 0.364 0.061 0.008 -26.51% 5.41% 9.22% 1.55% 0.20%80 Put 5.883 -0.825 0.254 0.258 0.048 0.013 -14.03% 4.32% 4.39% 0.82% 0.23%90 Put 8.631 -0.390 0.237 0.150 0.047 0.016 -4.52% 2.75% 1.74% 0.54% 0.18%
100 Call 12.450 0.166 0.166 0.048 0.048 0.018 1.33% 1.33% 0.39% 0.39% 0.14%110 Call 7.577 0.664 0.037 -0.050 0.053 0.022 8.76% 0.49% -0.67% 0.70% 0.29%120 Call 4.131 0.927 -0.153 -0.149 0.062 0.027 22.43% -3.70% -3.60% 1.49% 0.65%130 Call 2.008 0.894 -0.367 -0.217 0.086 0.033 44.52% -18.27% -10.79% 4.30% 1.64%140 Call 0.887 0.663 -0.522 -0.205 0.136 0.030 74.77% -58.78% -23.16% 15.35% 3.36%150 Call 0.372 0.396 -0.548 -0.104 0.189 -0.009 106.35% -147.29% -27.82% 50.82% -2.34%
iii 50 Put 0.633 -0.038 0.094 0.061 0.015 0.005 -6.05% 14.84% 9.64% 2.33% 0.85%60 Put 1.335 -0.063 0.111 0.058 0.013 0.006 -4.74% 8.32% 4.34% 0.97% 0.42%70 Put 2.571 -0.072 0.121 0.048 0.011 0.006 -2.79% 4.72% 1.87% 0.45% 0.22%80 Put 4.579 -0.046 0.124 0.034 0.010 0.005 -1.00% 2.71% 0.75% 0.22% 0.12%90 Put 7.608 0.019 0.119 0.019 0.008 0.004 0.25% 1.57% 0.26% 0.11% 0.05%
100 Call 11.857 0.111 0.111 0.008 0.008 0.004 0.94% 0.94% 0.07% 0.07% 0.03%110 Call 7.430 0.197 0.096 -0.004 0.008 0.003 2.65% 1.29% -0.05% 0.10% 0.05%120 Call 4.289 0.244 0.074 -0.015 0.009 0.004 5.70% 1.74% -0.36% 0.20% 0.09%130 Call 2.260 0.239 0.046 -0.027 0.009 0.003 10.57% 2.03% -1.21% 0.40% 0.14%140 Call 1.080 0.192 0.017 -0.036 0.009 0.002 17.77% 1.62% -3.30% 0.88% 0.19%150 Call 0.466 0.129 -0.004 -0.036 0.010 0.001 27.62% -0.75% -7.81% 2.13% 0.13%
A.E.(Relative Difference)A.E.(Difference)
6.2 Case of Random Limit(Expansion around a Log-normal Distribution)
Takahashi-Toda-Yamada(2008)
• The underlying asset price S(ε) follows:
dS(ε)t = g(X(ε)
t )S(ε)t σdWt; S
(ε)0 = s0
dX(ε)t = V0(X
(ε)t , ε)dt+ εV (X(ε)
t )dWt; X(ε)0 = x0 ∈ Rd,
where W denotes a m-dimensional standard Wiener process, ε ∈ [0, 1] isa known parameter, and σ ∈ Rm is a constant vector.
Suppose that coefficients h : Rd → R, V0 : Rd × [0, 1] → Rd, V : Rd →Rd × Rm are smooth and satisfy regularity conditions.
• Define X(ε) as
X(ε)t = log
(S
(ε)t
s0
).
Then,
X(ε)t = −|σ|2
2
∫ t
0
g(X(ε)u )2du+
∫ t
0
g(X(ε)u )σdWu,
and note thatX
(0)T ∼ N(µT ,ΣT ),
where
µT = −|σ|22
∫ T
0
g(X(0)u )2du = −1
2ΣT
ΣT = |σ|2∫ T
0
g(X(0)u )2du.
• Then, an asymptotic expansion of X(ε)T upto ε2 is expressed as
X(ε)T = εDT +
ε2
2ET + o(ε2),
where Dt and Et are defined as
Dt =∂X
(ε)t
∂ε|ε=0,
Et =∂2X
(ε)t
∂ε2|ε=0.
That is, D and E are given by
Dt = −|σ|2∫ t
0
∂xg(0)u Dug
(0)u du+
∫ t
0
∂xg(0)u DuσdWu
Et = −|σ|2∫ t
0
d∑j,k=1
∂j∂kg(0)u Dj
uDkug
(0)u du− |σ|2
∫ t
0
∂xg(0)u Euh
(0)u du− |σ|2
∫ t
0
∂xg(0)u Du2du
+∫ t
0
d∑j,k=1
∂j∂kg(0)u Dj
uDkuσdWu +
∫ t
0
∂xg(0)u EuσdWu,
where g(0)t = g(X(0)
t ).
40
• An asymptotic expansion of the characteristic function of X(ε)T upto ε2 is
obtained as
ψX
(ε)T
(ξ) = E[exp(iξX(ε)T )]
= exp(iξµ− ξ2Σ
2
)+ ε(iξ)eiξµE[eiξZT E[DT |ZT ]]
+ε2
2(iξ)eiξµE[eiξZT E[ET |ZT ]] +
ε2
2(iξ)2eiξµE[eiξZT E[(DT )2|ZT ]] + o(ε2),
where
ZT =∫ T
0
g(X(0)t )σdWt.
Note that E[DT |ZT = x] = p2(x), E[ET |ZT = x] = p3(x) andE[(DT )2|ZT = x] = p22(x) are polynomial functions of x.
• Hence, an asymptotic expansion of the density function of X(ε)T upto ε2 is
obtained as
fX
(ε)T
(x) = n[x;µT ,ΣT ] + ε
[− ∂
∂xp2(x− µT )n[x;µT ,ΣT ]
]
+ ε2[− ∂
∂xp3(x− µT )n[x;µT ,ΣT ]
]+
12ε2[∂2
∂x2p22(x− µT )n[x;µT ,ΣT ]
]+ o(ε2).
• Call option price with strike price K and maturity T :
C(0)P (0, T )
=∫ ∞
k
(s0ex −K)fX
(ε)T
(x)dx,
where k = log Ks0
, and P (0, T ) denotes the price at time 0 of a zero couponbond with maturity T .
41
An Log-Normal Asymptotic Expansion for Stochastic Volatil-ity Models
• The underlying process under :
dS(ε)t = g(σ(ε)
t , t)S(ε)t dW 1
t ,
dσ(ε)t = a(σ(ε)
t , t)dt+ εν1(σ(ε)t , t)dW 1
t + εν2(σ(ε)t , t)dW 2
t ,
S(ε)0 = s0(> 0), σ
(ε)0 = σ0(> 0).
where W = (W 1,W 2) is a two dimensional standard Wiener process andg, a, ν1, ν2 ∈ C∞(R+ × R+;R+).
• Define X(ε)t := log S
(ε)t
s0, then
dX(ε)t = −1
2g(σ(ε)
t , t)2dt+ g(σ(ε)t , t)dW 1
t ,
X(ε)0 = 0.
• The asymptotic expansion of X(ε)t and σ(ε)
t :
X(ε)t = X
(0)t + εA1t +
ε2
2A2t +
ε3
6A3t + o(ε3),
σ(ε)t = σ
(0)t + εB1t +
ε2
2B2t +
ε3
6B3t + o(ε3).
where
X(0)t = −1
2
∫ t
0
g(σ(0)s , s)2ds+
∫ t
0
g(σ(0)s , s)dW 1
s ,
dA1t = −g(σ(0)t , t)
∂
∂σg(σ(0)
t , t)B1tdt+∂
∂σg(σ(0)
t , t)B1tdW1t ,
dA2t =
−g(σ(0)
t , t)∂
∂σg(σ(0)
t , t)B2t −g(σ(0)
t , t)∂2
∂σ2g(σ(0)
t , t) +(∂
∂σg(σ(0)
t , t))2B2
1t
dt
+∂
∂σg(σ(0)
t , t)B2t +∂2
∂σ2g(σ(0)
t , t)B21t
dW 1
t ,
and
dσ(0)t = a(σ(0)
t , t)dt
dB1t =∂
∂σa(σ(0)
t , t)B1tdt+ ν1(σ(0)t , t)dW 1
t + ν2(σ(0)t , t)dW 2
t ,
dB2t =∂
∂σa(σ(0)
t , t)B2t +∂2
∂σ2a(σ(0)
t , t)B21t
dt
+2∂
∂σν1(σ
(0)t , t)B1tdW
1t + 2
∂
∂σν2(σ
(0)t , t)B1tdW
2t ,
42
• The asymptotic expansion of the characteristic function of X(ε)T up to the
ε2-order:
ψX
(ε)T
(ξ) := E[exp(iξ)X(ε)T ]
= E[exp(iξ)X(0)T ]
+ε
(iξ)E[A1T exp(iξ)X(0)T ]
+ε2
(iξ)2E[A2T exp(iξ)X(0)
T ] +(iξ)2
2E[A2
1T exp(iξ)X(0)T ]
+o(ε2)
• For ξ ∈ R, we define a complex-valued stochastic process Zt := Z〈ξ〉t as
Z〈ξ〉t := exp
(iξ)∫ t
0
g(σ(0)s , s)dW 1
s − 12(iξ)2
∫ t
0
g(σ(0)s , s)2ds
then, Z〈ξ〉t is a martingale and
dZ〈ξ〉t = (iξ)g(σ(0)
t , t)Z〈ξ〉t dW 1
t
and we have
exp(iξ)X(0)T = Z
〈ξ〉T exp (iξ)2 − (iξ)
2ΣT
where
ΣT :=∫ T
0
g(σ(0)t , t)2dt
• Then, ψX
(ε)T
can be expressed as
ψX
(ε)T
(ξ) =
1 + ε
(iξ)E[A1TZ
〈ξ〉T ]
+ε2
(iξ)2E[A2TZ
〈ξ〉T ] +
(iξ)2
2E[A2
1TZ〈ξ〉T ]
× exp (iξ)2 − (iξ)2
ΣT + o(ε2).
43
• Define η as
η1,1,1(t) := E[A1tZt]η1,1,2(t) := E[B1tZt]η2,1,1(t) := E[A2tZt]η2,1,2(t) := E[B2tZt]η2,2,1(t) := E[A2
1tZt]η2,2,2(t) := E[A1tB1tZt]η2,2,3(t) := E[B2
1tZt]
• Then, we can derive ordinary differential equations as follows:
d(A1tZt) = A1tdZt + ZtdA1t + dA1tdZt
=
(iξ)g(σ(0)t , t)
∂
∂σg(σ(0)
t , t)B1tZt − g(σ(0)t , t)
∂
∂σg(σ(0)
t , t)B1tZt
dt
+
(iξ)g(σ(0)t , t)A1tZt +
∂
∂σg(σ(0)
t , t)B1tZt
dW 1
t
Taking the expectation on both sides, we have the following ordinarydifferential equation of η1,1,1:
d
dtη1,1,1(t) = (iξ)g(σ(0)
t , t)∂
∂σg(σ(0)
t , t)η1,1,2(t)−g(σ(0)t , t)
∂
∂σg(σ(0)
t , t)η1,1,2(t).
In the similar manner, we have
d
dtη1,1,2(t) = (iξ)g(σ(0)
t , t)ν1(σ(0)t , t) +
∂
∂σa(σ(0)
t , t)η1,1,2(t).
44
• Proposition 2 The asymptotic expansion of the characteristic functionof X(ε)
T up to the ε2-order is expressed as
ψX
(ε)T
(ξ) =
1 + ε (iξ)η1,1,1(T )
+ε2
(iξ)2η2,1,1(T ) +
(iξ)2
2η2,2,1(T )
× exp (iξ)2 − (iξ)2
ΣT + o(ε2).
where η1,1,1(T ), η2,1,1(T ) and η2,2,1(T ) are obtained by the following sys-tem of ordinary differential equations:
d
dtη1,1,1(t) = (iξ)g(σ(0)
t , t)∂
∂σg(σ(0)
t , t)η1,1,2(t) − g(σ(0)t , t)
∂
∂σg(σ(0)
t , t)η1,1,2(t)
d
dtη1,1,2(t) = (iξ)g(σ(0)
t , t)ν1(σ(0)t , t) +
∂
∂σa(σ(0)
t , t)η1,1,2(t)
d
dtη2,1,1(t) = (iξ)
g(σ(0)
t , t)∂
∂σg(σ(0)
t , t)η2,1,2(t) + g(σ(0)t , t)
∂2
∂σ2g(σ(0)
t , t)η2,2,3(t)
−g(σ(0)t , t)
∂
∂σg(σ(0)
t , t)η2,1,2(t)
−g(σ(0)
t , t)∂2
∂σ2g(σ(0)
t , t) +(∂
∂σg(σ(0)
t , t))2η2,1,2(t)
d
dtη2,1,2(t) = 2(iξ)g(σ(0)
t , t)∂
∂σν1(σ
(0)t , t)η1,1,2(t)
+∂
∂σa(σ(0)
t , t)η2,1,2(t) +∂2
∂σ2a(σ(0)
t , t)η2,2,3(t)
d
dtη2,2,1(t) = 2(iξ)g(σ(0)
t , t)∂
∂σg(σ(0)
t , t)η2,2,2(t)
−2g(σ(0)t , t)
∂
∂σg(σ(0)
t , t)η2,2,2(t) +(∂
∂σg(σ(0)
t , t))2
η2,2,3(t)
d
dtη2,2,2(t) = (iξ)
g(σ(0)
t , t)∂
∂σg(σ(0)
t , t)η2,2,3(t) + g(σ(0)t , t)ν1(σ
(0)t , t)η1,1,1(t)
+∂
∂σa(σ(0)
t , t)η2,2,2(t) + g(σ(0)t , t)
∂
∂σg(σ(0)
t , t)η2,2,3(t)
+g(σ(0)t , t)ν1(σ
(0)t , t)η1,1,1(t) +
∂
∂σg(σ(0)
t , t)ν1(σ(0)t , t)η1,1,2(t)
d
dtη2,2,3(t) = 2(iξ)g(σ(0)
t , t)ν1(σ(0)t , t)η1,1,2(t)
+2∂
∂σa(σ(0)
t , t)η2,2,3(t) + ν1(σ(0)t , t)2 + ν1(σ
(0)t , t)2
45
• It is easily shown that the asymptotic expansion of the characteristic func-tion can be expressed as a product of the Gaussian characteristic functionand a polynomial of (iξ).
• To obtain the asymptotic expansion of density function, we need to invertψ
X(ε)T
using the inverse Fourier transformation:
F−1[(iξ)neµ(iξ)− ξ2
2 Σ](x) =1
ΣnHn(x− µ; Σ)n[x− µ; 0,Σ],
where Σ ≡ ΣT , µ = − 12Σ and
Hn(x; Σ) := (−Σ)nex2/2Σ dn
dxne−x2/2Σ
and
n[x;µ,Σ] :=1√2πΣ
exp− (x− µ)2
2Σ.
• The asymptotic expansion of the density function of X(ε)T up to the ε2-
order is obtained as:
fX
(ε)T
(x) = n[x;µ,Σ]
+ ε
η1,1,1(T )
H1(x− µ; Σ)Σ
n[x;µ,Σ]
+ε2
2
η2,1,1(T )
H1(x− µ; Σ)Σ
+ η2,2,1(T )H2(x− µ; Σ)
Σ2
n[x;µ,Σ] + o(ε2).
• Call option price with strike price K and maturity T :
C(0)P (0, T )
=∫ ∞
k
(s0ex −K)fX
(ε)T
(x)dx,
where k = log Ks0
, and P (0, T ) denotes the price at time 0 of a zero couponbond with maturity T .
46
Model cf. BERESTYCKI-BUSCA-FLORENT(2004) S0 100 v 0.30dS = σSdW1 sigma0 0.30 rho -0.70dσ = κ(θ-σ) dt + vσdW2 kappa 0.10 T 30
theta 0.30OTM Call/Put Options (using Put-Call parity)
moneyness 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0Strike 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200Call/Put Put Put Put Put Put Put Put Put Put Call Call Call Call Call Call Call Call Call Call Call
true valueNV-256-10^8 2.174 5.594 9.775 14.545 19.801 25.471 31.500 37.847 44.476 51.357 48.465 45.780 43.281 40.954 38.782 36.753 34.856 33.080 31.415 29.854
Log-Normal Asymptotic Expansionsdiffmoneyness 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0Log-Normal -0.674 -0.434 0.281 1.227 2.280 3.371 4.459 5.520 6.541 7.512 8.430 9.291 10.097 10.848 11.545 12.191 12.788 13.338 13.844 14.308log-AE1 0.167 0.102 -0.175 -0.524 -0.889 -1.248 -1.594 -1.927 -2.246 -2.555 -2.856 -3.150 -3.439 -3.724 -4.007 -4.289 -4.569 -4.848 -5.127 -5.405log-AE2 0.466 0.675 0.857 1.012 1.143 1.254 1.351 1.437 1.515 1.586 1.652 1.715 1.774 1.831 1.886 1.939 1.992 2.043 2.094 2.145log-AE3 -0.277 -0.478 -0.560 -0.589 -0.592 -0.581 -0.560 -0.535 -0.505 -0.474 -0.442 -0.409 -0.376 -0.342 -0.309 -0.276 -0.243 -0.210 -0.178 -0.146log-AE4 0.006 0.152 0.196 0.199 0.182 0.154 0.120 0.081 0.039 -0.005 -0.051 -0.098 -0.147 -0.197 -0.248 -0.299 -0.351 -0.404 -0.457 -0.511
relative diffmoneyness 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0Log-Normal -31.0% -7.7% 2.9% 8.4% 11.5% 13.2% 14.2% 14.6% 14.7% 14.6% 17.4% 20.3% 23.3% 26.5% 29.8% 33.2% 36.7% 40.3% 44.1% 47.9%log-AE1 7.7% 1.8% -1.8% -3.6% -4.5% -4.9% -5.1% -5.1% -5.1% -5.0% -5.9% -6.9% -7.9% -9.1% -10.3% -11.7% -13.1% -14.7% -16.3% -18.1%log-AE2 21.4% 12.1% 8.8% 7.0% 5.8% 4.9% 4.3% 3.8% 3.4% 3.1% 3.4% 3.7% 4.1% 4.5% 4.9% 5.3% 5.7% 6.2% 6.7% 7.2%log-AE3 -12.8% -8.5% -5.7% -4.1% -3.0% -2.3% -1.8% -1.4% -1.1% -0.9% -0.9% -0.9% -0.9% -0.8% -0.8% -0.8% -0.7% -0.6% -0.6% -0.5%log-AE4 0.3% 2.7% 2.0% 1.4% 0.9% 0.6% 0.4% 0.2% 0.1% 0.0% -0.1% -0.2% -0.3% -0.5% -0.6% -0.8% -1.0% -1.2% -1.5% -1.7%
An Expansion of Implied Volatility
I follows Fouque-Papanicolaou-Sircar(2000) using the result in the previous sub-section.
• An implied volatility σI is determined so that:
CBS(K,T, σI) = C(K,T ).
Here, CBS(K,T, σI) denotes the call price with strike K, maturity T andvolatility σI under the Black-Scholes model; C(K,T ) denotes the call pricewith strike K and maturity T under a stochastic volatility model in theprevious section.
• Define σ as σ = Σ12T .
• Suppose that an implied volatility is expanded around σ:
σI = σ + εσ1 + ε2σ2 + o(ε2)
• Then, CBS(K,T, σI) is expanded as:
CBS(K,T, σI) = CBS(K,T, σ) + ε
∂CBS(K,T, σ)
∂σ|σ=σ
σ1
+ ε2∂CBS(K,T, σ)
∂σ|σ=σ
σ2 +
ε2
2
∂2CBS(K,T, σ)
∂σ2|σ=σ
(σ1)2.
• On the other hand, by an asymptotic expansion,
C(K,T ) = CBS(K,T, σ) + εC1 + ε2C2.
• Hence, σ1 and σ2 are obtained as
σ1 = C1/
∂CBS(K,T, σ)
∂σ|σ=σ
σ2 =C2 − 1
2
(∂2CBS(K,T, σ)
∂σ2|σ=σ
)σ2
1
/
∂CBS(K,T, σ)
∂σ|σ=σ
.
• The higher order expansion is obtained in the similar way.
47
• 4-th order expansion:
C = C0 + εC1 + ε2C2 + ε3C3 + ε4C4 +O(ε5).
σimplied.vol = σ + εσ1 + ε2σ2 + ε3σ3 + ε4σ4 +O(ε5)
σ1 =C1
CBSσ (σ)
,
σ2 =C2
CBSσ (σ)
− 12
C21
CBSσ (σ)3
CBSσσ (σ),
σ3 =C3
CBSσ (σ)
−(
C1
CBSσ (σ)2
)(C2
CBSσ (σ)
− 12
C21
CBSσ (σ)3
CBSσσ (σ)
)CBS
σσ (σ) − 13!
C31
CBSσ (σ)4
CBSσσσ(σ).
σ4 = 1CBS
σ (σ)C4− 1
2CBSσσ (σ)σ2
2−CBSσσ (σ)σ1σ3− 1
2CBSσσσ(σ)σ2
1σ2− 14!C
BSσσσσ(σ)σ4
1.
CBSσ = f
√Tn(d1),
CBSσσ =
f√T
σn(d1)d1d2,
CBSσσσ =
f√T
σ2n(d1)d2
1d22 − d1d2 − d2
1 − d22
CBSσσσσ =
f√T
σ3n(d1)d3
1d32 − 3d1d
32 − 3d3
1d2 − 3d21d
22 + 3d2
1 + 3d22 + 6d1d2
n(x) =1√2πe−
x22 ,
d1 =log(
f0K
)√σ2T
+12
√σ2T ,
d2 =log(
f0K
)√σ2T
− 12
√σ2T .
48
30.0%
32.0%
34.0%
36.0%
implie
d v
ol
T=30 4th Order Asymptotic Expansion of Implied Volatility
ImpVol
20.0%
22.0%
24.0%
26.0%
28.0%
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
implie
d v
ol
moneyness
AE-
ImpVol4
7 Future Research
• Detailed examination of approximations
• Comparison with other methods
• Dynamic optimal portfolio in incomplete markets
• Barrier type derivatives
• Levy processes
49
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