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Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
Stochastic Differential Equations in Financeand Monte Carlo Simulations
Xuerong Mao
Department of Statistics and Modelling ScienceUniversity of Strathclyde
Glasgow, G1 1XH
China 2009
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
Outline
1 Stochastic Modelling in Asset Prices
2 The Black–Scholes World
3 Monte Carlo SimulationsEM methodEM method for financial quantities
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
Outline
1 Stochastic Modelling in Asset Prices
2 The Black–Scholes World
3 Monte Carlo SimulationsEM methodEM method for financial quantities
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
Outline
1 Stochastic Modelling in Asset Prices
2 The Black–Scholes World
3 Monte Carlo SimulationsEM methodEM method for financial quantities
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
One of the important problems in finance is the specification ofthe stochastic process governing the behaviour of an asset. Wehere use the term asset to describe any financial object whosevalue is known at present but is liable to change in the future.Typical examples are
shares in a company,commodities such as gold, oil or electricity,currencies, for example, the value of $100 US in UKpounds.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
One of the important problems in finance is the specification ofthe stochastic process governing the behaviour of an asset. Wehere use the term asset to describe any financial object whosevalue is known at present but is liable to change in the future.Typical examples are
shares in a company,commodities such as gold, oil or electricity,currencies, for example, the value of $100 US in UKpounds.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
One of the important problems in finance is the specification ofthe stochastic process governing the behaviour of an asset. Wehere use the term asset to describe any financial object whosevalue is known at present but is liable to change in the future.Typical examples are
shares in a company,commodities such as gold, oil or electricity,currencies, for example, the value of $100 US in UKpounds.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
Now suppose that at time t the asset price is S(t). Let usconsider a small subsequent time interval dt , during which S(t)changes to S(t) + dS(t). (We use the notation d · for the smallchange in any quantity over this time interval when we intend toconsider it as an infinitesimal change.) By definition, the returnof the asset price at time t is dS(t)/S(t). How might we modelthis return?
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
If the asset is a bank saving account then S(t) is the balance ofthe saving at time t . Suppose that the bank deposit interest rateis r . Thus
dS(t)S(t)
= rdt .
This ordinary differential equation can be solved exactly to giveexponential growth in the value of the saving, i.e.
S(t) = S0ert ,
where S0 is the initial deposit of the saving account at timet = 0.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
However asset prices do not move as money invested in arisk-free bank. It is often stated that asset prices must moverandomly because of the efficient market hypothesis. There areseveral different forms of this hypothesis with differentrestrictive assumptions, but they all basically say two things:
The past history is fully reflected in the present price,which does not hold any further information;Markets respond immediately to any new information aboutan asset.
With the two assumptions above, unanticipated changes in theasset price are a Markov process.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
However asset prices do not move as money invested in arisk-free bank. It is often stated that asset prices must moverandomly because of the efficient market hypothesis. There areseveral different forms of this hypothesis with differentrestrictive assumptions, but they all basically say two things:
The past history is fully reflected in the present price,which does not hold any further information;Markets respond immediately to any new information aboutan asset.
With the two assumptions above, unanticipated changes in theasset price are a Markov process.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
Under the assumptions, we may decompose
dS(t)S(t)
= deterministic return + random change.
The deterministic return is the same as the case of moneyinvested in a risk-free bank so it gives a contribution rdt .The random change represents the response to externaleffects, such as unexpected news. There are many externaleffects so by the well-known central limit theorem this secondcontribution can be represented by a normal distribution withmean zero and and variance v2dt . Hence
dS(t)S(t)
= rdt + N(0, v2dt) = rdt + vN(0,dt).
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
WriteN(0,dt) = B(t + dt)− B(t) = dB(t),
where B(t) is a standard Brownian motion. Then
dS(t)S(t)
= rdt + vdB(t).
In finance, v is known as the volatility rather than the standarddeviation.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
The Black–Scholes germetric Brownian motion
If the volatility v is independent of the underlying assert price,say v = σ = const ., then the asset price follows the well-knownBlack–Scholes geometric Brownian motion
dS(t) = rS(t)dt + σS(t)dB(t).
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
Theta process
However, in general, the volatility v depends on the underlyingassert price. There are various types of volatility functions usedin financial modelling. One of them assumes that
v = v(S) = σSθ−1,
where σ and θ are both positive numbers. Then the asset pricefollows
dS(t) = rS(t)dt + σSθ(t)dB(t),
which is known as the theta process.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
Square root process
To fit various asset prices, one could choose various values forθ. For example, θ = 1.3 or 0.5 have been used to fit certainasset prices. In particular, if θ = 0.5, we have the well-knownsquare root process
dS(t) = rS(t)dt + σ√
S(t)dB(t).
This makes the “variance" of the random change termproportional to S(t). Hence, if the asset price volatility does notincrease “too much" when S(t) increases (greater than 1, ofcourse), this model may be more appropriate.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
The asset price follows the geometric Brownian motion
dS(t) = rS(t)dt + σS(t)dB(t).
The risk-free interest rate r and the asset volatility σ areknown constants over the life of the option.There are no transaction costs associated with hedging aportfolio.The underlying asset pays no dividends during the life ofthe option.There are no arbitrage possibilities.Trading of the underlying asset can take placecontinuously.Short selling is permitted and the assets are divisible.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
The asset price follows the geometric Brownian motion
dS(t) = rS(t)dt + σS(t)dB(t).
The risk-free interest rate r and the asset volatility σ areknown constants over the life of the option.There are no transaction costs associated with hedging aportfolio.The underlying asset pays no dividends during the life ofthe option.There are no arbitrage possibilities.Trading of the underlying asset can take placecontinuously.Short selling is permitted and the assets are divisible.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
The asset price follows the geometric Brownian motion
dS(t) = rS(t)dt + σS(t)dB(t).
The risk-free interest rate r and the asset volatility σ areknown constants over the life of the option.There are no transaction costs associated with hedging aportfolio.The underlying asset pays no dividends during the life ofthe option.There are no arbitrage possibilities.Trading of the underlying asset can take placecontinuously.Short selling is permitted and the assets are divisible.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
The asset price follows the geometric Brownian motion
dS(t) = rS(t)dt + σS(t)dB(t).
The risk-free interest rate r and the asset volatility σ areknown constants over the life of the option.There are no transaction costs associated with hedging aportfolio.The underlying asset pays no dividends during the life ofthe option.There are no arbitrage possibilities.Trading of the underlying asset can take placecontinuously.Short selling is permitted and the assets are divisible.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
The asset price follows the geometric Brownian motion
dS(t) = rS(t)dt + σS(t)dB(t).
The risk-free interest rate r and the asset volatility σ areknown constants over the life of the option.There are no transaction costs associated with hedging aportfolio.The underlying asset pays no dividends during the life ofthe option.There are no arbitrage possibilities.Trading of the underlying asset can take placecontinuously.Short selling is permitted and the assets are divisible.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
The asset price follows the geometric Brownian motion
dS(t) = rS(t)dt + σS(t)dB(t).
The risk-free interest rate r and the asset volatility σ areknown constants over the life of the option.There are no transaction costs associated with hedging aportfolio.The underlying asset pays no dividends during the life ofthe option.There are no arbitrage possibilities.Trading of the underlying asset can take placecontinuously.Short selling is permitted and the assets are divisible.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
The asset price follows the geometric Brownian motion
dS(t) = rS(t)dt + σS(t)dB(t).
The risk-free interest rate r and the asset volatility σ areknown constants over the life of the option.There are no transaction costs associated with hedging aportfolio.The underlying asset pays no dividends during the life ofthe option.There are no arbitrage possibilities.Trading of the underlying asset can take placecontinuously.Short selling is permitted and the assets are divisible.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
European call option
Given the asset price S(t) = S at time t , a European call optionis signed with the exercise price K and the expiry date T . Thevalue of the option is denoted by C(S, t).
The payoff of the option at the expiry date is
C(S,T ) = (S − K )+ := max(S − K , 0).
The Black–Scholes PDF
∂V (S, t)∂t
+ 12σ
2S2∂2V (S, t)∂S2 + rS
∂V (S, t)∂S
− rV (S, t) = 0.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
European call option
Given the asset price S(t) = S at time t , a European call optionis signed with the exercise price K and the expiry date T . Thevalue of the option is denoted by C(S, t).
The payoff of the option at the expiry date is
C(S,T ) = (S − K )+ := max(S − K , 0).
The Black–Scholes PDF
∂V (S, t)∂t
+ 12σ
2S2∂2V (S, t)∂S2 + rS
∂V (S, t)∂S
− rV (S, t) = 0.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
European call option
Given the asset price S(t) = S at time t , a European call optionis signed with the exercise price K and the expiry date T . Thevalue of the option is denoted by C(S, t).
The payoff of the option at the expiry date is
C(S,T ) = (S − K )+ := max(S − K , 0).
The Black–Scholes PDF
∂V (S, t)∂t
+ 12σ
2S2∂2V (S, t)∂S2 + rS
∂V (S, t)∂S
− rV (S, t) = 0.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
Given the initial value S(t) = S at time t , write the SDE as
dS(u) = rS(u)du + σS(u)dB(u), t ≤ u ≤ T .
Hence the expected payoff at the expiry date T is
E(S(T )− K )+
Discounting this expected value in future gives
C(S, t) = e−r(T−t)E[max(S(T )− K ,0)],
which is known as the Cox formula.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
The SDE has the explicit solution
S(T ) = S exp[(r − 1
2σ2)(T − t) + σ(B(T )− B(t))
]= exp
[log(S) + (r − 1
2σ2)(T − t) + σ(B(T )− B(t))
]= eµ̂+σ̂Z ,
where Z ∼ N(0,1),
µ̂ = log(S) +(
r − 12σ2)
(T − t), σ̂ = σ√
T − t .
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
Noting that S(T )− K ≥ 0 iff
Z ≥ log(K )− µ̂σ̂
,
compute
E(S(T )− K )+ =
∫ 8
log(K )−µ̂σ̂
(eµ̂+σ̂z − K
) 1√2π
e−12 z2
dz
=1√2π
∫ ∞−d2
eµ̂+σ̂z−12 z2
dz − K√2π
∫ ∞−d2
e−12 z2
dz,
where
d2 := − log(K )− µ̂σ̂
=log(S/K ) +
(r − 1
2σ2)
(T − t)
σ√
T − t.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
But
1√2π
∫ ∞−d2
e−12 z2
dz =1√2π
∫ d2
−∞e−
12 z2
dz := N(d2),
and1√2π
∫ ∞−d2
eµ̂+σ̂z−12 z2
=eµ̂+
12 σ̂
2
√2π
N(d1),
where d1 = d2 + σ̂. Hence
E(S(T )− K + =eµ̂+
12 σ̂
2
√2π
N(d1)− KN(d2).
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
BS formula for call option
TheoremThe explicit formula for the value of the European call option is
C(S, t) = SN(d1)− Ke−r(T−t)N(d2),
where N(x) is the c.p.d. of the standard normal distribution,namely
N(x) =1√2π
∫ x
−∞e−
12 z2
dz,
while d1 = d2 + σ̂ and
d2 =log(S/K ) + (r − 1
2σ2)(T − t)
σ√
T − t.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
The Black–Scholes formula benefits from the explicit solution ofthe geometric Brownian motion. However, most of SDEs usedin finance do not have explicit solutions. Hence, numericalmethods and Monte Carlo simulations have become more andmore popular in option valuation. There are two mainmotivations for such simulations:
using a Monte Carlo approach to compute the expectedvalue of a function of the underlying asset price, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
The Black–Scholes formula benefits from the explicit solution ofthe geometric Brownian motion. However, most of SDEs usedin finance do not have explicit solutions. Hence, numericalmethods and Monte Carlo simulations have become more andmore popular in option valuation. There are two mainmotivations for such simulations:
using a Monte Carlo approach to compute the expectedvalue of a function of the underlying asset price, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
The Black–Scholes formula benefits from the explicit solution ofthe geometric Brownian motion. However, most of SDEs usedin finance do not have explicit solutions. Hence, numericalmethods and Monte Carlo simulations have become more andmore popular in option valuation. There are two mainmotivations for such simulations:
using a Monte Carlo approach to compute the expectedvalue of a function of the underlying asset price, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Typically, let us consider the square root process
dS(t) = rS(t)dt + σ√
S(t)dB(t), 0 ≤ t ≤ T .
A numerical method, e.g. the Euler–Maruyama (EM) methodapplied to it may break down due to negative values beingsupplied to the square root function. A natural fix is to replacethe SDE by the equivalent, but computationally safer, problem
dS(t) = rS(t)dt + σ√|S(t)|dB(t), 0 ≤ t ≤ T .
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Outline
1 Stochastic Modelling in Asset Prices
2 The Black–Scholes World
3 Monte Carlo SimulationsEM methodEM method for financial quantities
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Discrete EM approximation
Given a stepsize ∆ > 0, the EM method applied to the SDEsets s0 = S(0) and computes approximations sn ≈ S(tn), wheretn = n∆, according to
sn+1 = sn(1 + r∆) + σ√|sn|∆Bn,
where ∆Bn = B(tn+1)− B(tn).
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Continuous-time EM approximation
s(t) := s0 + r∫ t
0s̄(u))du + σ
∫ t
0
√|s̄(u)|dB(u),
where the “step function” s̄(t) is defined by
s̄(t) := sn, for t ∈ [tn, tn+1).
Note that s(t) and s̄(t) coincide with the discrete solution at thegridpoints; s̄(tn) = s(tn) = sn.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
The ability of the EM method to approximate the true solution isguaranteed by the ability of either s(t) or s̄(t) to approximateS(t) which is described by:
Theorem
lim∆→0
E(
sup0≤t≤T
|s(t)−S(t)|2)
= lim∆→0
E(
sup0≤t≤T
|s̄(t)−S(t)|2)
= 0.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
The ability of the EM method to approximate the true solution isguaranteed by the ability of either s(t) or s̄(t) to approximateS(t) which is described by:
Theorem
lim∆→0
E(
sup0≤t≤T
|s(t)−S(t)|2)
= lim∆→0
E(
sup0≤t≤T
|s̄(t)−S(t)|2)
= 0.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Outline
1 Stochastic Modelling in Asset Prices
2 The Black–Scholes World
3 Monte Carlo SimulationsEM methodEM method for financial quantities
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Bond
If S(t) models short-term interest rate dynamics, it is pertinentto consider the expected payoff
β := E exp
(−∫ T
0S(t)dt
)
from a bond. A natural approximation based on the EM methodis
β∆ := E exp
(−∆
N−1∑n=0
|sn|
), where N = T/∆.
Theorem
lim∆→0|β − β∆| = 0.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Bond
If S(t) models short-term interest rate dynamics, it is pertinentto consider the expected payoff
β := E exp
(−∫ T
0S(t)dt
)
from a bond. A natural approximation based on the EM methodis
β∆ := E exp
(−∆
N−1∑n=0
|sn|
), where N = T/∆.
Theorem
lim∆→0|β − β∆| = 0.
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Up-and-out call option
An up-and-out call option at expiry time T pays the Europeanvalue with the exercise price K if S(t) never exceeded the fixedbarrier, c, and pays zero otherwise.
Using the discrete numerical solution to approximate the truepath gives rise to two distinct sources of error:
a discretization error due to the fact that the path is notfollowed exactly—the numerical solution may cross thebarrier at time tn when the true solution stays below, or viceversa,a discretization error due to the fact that the path is onlyapproximated at discrete time points—for example, thetrue path may cross the barrier and then return within theinterval (tn, tn+1).
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Up-and-out call option
An up-and-out call option at expiry time T pays the Europeanvalue with the exercise price K if S(t) never exceeded the fixedbarrier, c, and pays zero otherwise.
Using the discrete numerical solution to approximate the truepath gives rise to two distinct sources of error:
a discretization error due to the fact that the path is notfollowed exactly—the numerical solution may cross thebarrier at time tn when the true solution stays below, or viceversa,a discretization error due to the fact that the path is onlyapproximated at discrete time points—for example, thetrue path may cross the barrier and then return within theinterval (tn, tn+1).
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Up-and-out call option
An up-and-out call option at expiry time T pays the Europeanvalue with the exercise price K if S(t) never exceeded the fixedbarrier, c, and pays zero otherwise.
Using the discrete numerical solution to approximate the truepath gives rise to two distinct sources of error:
a discretization error due to the fact that the path is notfollowed exactly—the numerical solution may cross thebarrier at time tn when the true solution stays below, or viceversa,a discretization error due to the fact that the path is onlyapproximated at discrete time points—for example, thetrue path may cross the barrier and then return within theinterval (tn, tn+1).
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
Up-and-out call option
An up-and-out call option at expiry time T pays the Europeanvalue with the exercise price K if S(t) never exceeded the fixedbarrier, c, and pays zero otherwise.
Using the discrete numerical solution to approximate the truepath gives rise to two distinct sources of error:
a discretization error due to the fact that the path is notfollowed exactly—the numerical solution may cross thebarrier at time tn when the true solution stays below, or viceversa,a discretization error due to the fact that the path is onlyapproximated at discrete time points—for example, thetrue path may cross the barrier and then return within theinterval (tn, tn+1).
Xuerong Mao SM and MC Simulations
Stochastic Modelling in Asset PricesThe Black–Scholes WorldMonte Carlo Simulations
EM methodEM method for financial quantities
TheoremDefine
V = E[(S(T )− K )+I{0≤S(t)≤c, 0≤t≤T}
],
V∆ = E[(s̄(T )− K )+I{0≤s̄(t)≤B, 0≤t≤T}
].
Thenlim
∆→0|V − V∆| = 0.
Xuerong Mao SM and MC Simulations