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Price path generation by
Hybrid Monte Carlo algorithm
Tetsuya Takaishi
Hiroshima University of Economics
IASC2008
•In the Black-Sholes model asset prices are assumed to follow the geometric Brownian motion and in the simplest case of European option an analytical solution is obtained.
•For options with path-dependent payoff functions like Asian option, an analytic formula is not known.
•We generate price paths by Markov Chain Monte Carlo (MCMC) methods and evaluate Asian options.
•Price paths are generated by two MCMC methods, the Hybrid Monte Carlo (HMC) and Metropolis method. We find that the HMC algorithm reduces correlations between the generated price paths effectively.
Introduction
Options
•European Option
•American option
•Asian option
An option that can only be exercised at the end of its life.
An option that can be exercised anytime during its life.
An option whose payoff depends on the average price of the underlying
asset over a certain period of time as opposed to at maturity.
payoff function for an Asian option:
Max(m-K,0) or Max(K-m,0) where m is the average price and K is the strike price
time
K
S0
Sn
strike price
m
price path
average
SnーK
m-K
payoff function
Max(m-K,0)Max(Sn-K,0)European call option Asian call option
Evolution of price path
SdWrSdtdS
dtAdt
dWAdtSd
)(ln
Sz ln
dt
Adtzz
dtzzp
2
2
2 2
))((exp
2
1)|(
Asset prices are assumed to follow a geometric Brownian motion.
2
2 rA
Transition probability from z to z’
)|()|()|()|( 012110 zzpzzpzzpdzdzzzp nnnn
dt
Adtzz
dtdzdzzzp kk
n
kn
nn 2
2
11
112
1012
))((exp
)2(
1)|(
Chapman-Kolmogorov equation
dt
Adtzz
dtzzp kk
n
kn
n 2
2
11
112
012
))((exp
)2(
1),,(
probability of finding the price path ),,( 01 zzz n
dt
Adtzz
dtzzp kk
n
kn
n 2
2
11
112
012
))((exp
)2(
1),,(
rt
nnn ezzpzzOdzdzZ
O
),,(),,(1
010111
We generated price paths with this probability by k
n
k zzz ),,( 01
rtM
k
k ezOM
O
1
)(1
average over M price paths
Option price ),,( 01 zzO n
payoff function
An option price is given by an expectation value with
1. Choose a new candidate
2. Calculate
3. We accept the candidate with
Otherwise keep the old value
Metropolis method
))(exp()( xfxp
)()( xfxfdh
))exp(,1min( dh
)5.0( dxx
Local update uniform random number in [0,1]
generate x with
price
time
x
x
))exp(,1min( dhaccept the new path with
candidate path
Hybrid Monte Carlo
•Molecular dynamics simulations
•Metropolis accept/reject step
HMC is a global algorithm that can update all the parameters at once.
S. Duane , A.D. Kennedy , B.J. Pendleton, D. Roweth (1987)
HMC consists of two steps:
HMC is useful for the models which have global interactions.
It is expected that HMC reduces the correlation between variables generated by MCMC.
))(exp( fdZ
))(exp()(1
fOdZ
O
)exp(
))(2
1exp( 2
Hdpd
fpdpdZ
)(2
1 2 fpH
Partition function
we introduce momenta p
conjugate to θ.
define
Hamiltonian
Hybrid Monte Carlo
Expectation value of O(θ)
This partition function unchanges the value of <O(θ)>.
Momenta have no dynamics.
This does not change the results.
•Solve the Hamilton's equations of motion (Molecular
dynamics simulation)
),(),( pHpH
HHdH
•Metropolis accept/reject step
)1),min(exp( dH
In general, dh is not zero in
the numerical integration.
Hybrid Monte Carlo
Hamiltonian is conserved.
p
H
Hp
pp
Accept new theta with
This can be small.
Hgg , , Poisson bracket
HggHL ,)(
)())(exp()( tgHtLttg
VT
xfLpLHL
))((2/)( 2
)()2/exp()exp()2/exp())(exp( 3ttTtVtTHtL
Leapfrog integrator
In general, we cannot solve this.
operator
g is x or p.
Simplectic integrator
Hamilton's equations of motion
2/)()2/()( :3
)2/()()( :2
2/)()()2/( :1
tttptttt
tttH
tpttp
ttpttt
θ
p
2/t
t
Δt is chosen such that the acceptance ratio takes 60~70%.
elementary step
repeat this step
Leapfrog integrator
t
t
optimal acceptance
interest rate r=0.05
volatility σ=0.01
expiration day T=1 (year)
dt=T/100=1/100
parameters
initial price S0=100
MCMC simulations
10 price paths generated every 1000sweeps
10 paths generated by HMC every 100sweeps
Acceptance ratio=約50%
d=0.4
Log(S100)
Autocorrelation is small
Log(S100)
distribution of Log(S100)
Asian option
rT
nnn
rT ezzpzzOdzdzZ
eO
),,(),,(1
010111
1
0
1
0
01
0,1
1max
0,)exp(1
1max),,(
n
k
k
n
k
kn
KSn
Kzn
zzO
payoff function depends on the average value over a price path
interest rate r=0.095
volatitiliy σ=0.2
T=1 (year)
initial price S0=100
dt=0.01
parameters
strike price K K=60 K=100 K=150
Metropolis
HMC
MCRW
40.923(18) 7.147(14) 0.0051(3)
40.86(2)
40.830(25)
6.923(13)
6.899(19)
0.0063(4)
0.0054(5)
Metropolis: 400k (every 100 sweeps)
HMC:400k(every 10 sweeps)
MCRW:200k
statistics
RCRW=Monte Carlo random walk
Summary
We have generated price paths by HMC
and Metropolis methods.
The price paths are generated effectively
by the HMC method.
Asian option prices are calculated based on
the price paths generated by the HMC.
182
A-2 , B-1
Option
あることが選択できる権利(一般)
ある資産を一定の価格で購入または売却できる権利(ファイナンス)
権利は購入する必要がある
権利は行使してもしなくても良い
オプションの例
1年後に1株300円で1000株購入できる権利
満期日の株価 S
300円 500円
200円の得
コール・オプション
株価が300円以下の場合は買う必要なし。
200円
この権利の価格は?
満期日
行使価格 K
)0,max( KS
Option prices
)|( 0Sqp
ヨーロピアンオプションの場合、解析解が得られている
rtrt eSqpqdqOZ
eO
)|()(1
0 r:利子率
満期日に株価がqの場合、どれだけ得するか(ペイオフ)
株価が(幾何)ブラウン運動をすると仮定すると
ブラック・ショールズ式
平均したらどれだけ得するか
満期日の株価のみに依存