Monte Carlo Simulations: Autumn 2 0 0 9
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A U T U M N 2 0 0 9
MONTE CARLO SIMULATIONS
MSc
COMPUTATIONAL FINANCE
Numerical Methods in Finance (Implementing Market Models)
©Finbarr Murphy 2007
Lecture Objectives Hedge Sensitivities
MSc
COMPUTATIONAL FINANCE
Multiple Stochastic Factors
©Finbarr Murphy 2007
Agenda Page
Computing Hedge Sensitivities using MC Multiple Stochastic Factors
1
2 2
9
MSc
COMPUTATIONAL FINANCE
3
3
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
By now, we know that Delta is the rate of change of the option relative to the underlying asset price.
From our Finite Difference lectures
∂C C ( S + ∆S ) − C ( S − ∆S ) delta = ∆ = ≈ ∂S 2∆S
In Monte Carlo Simulations, a better approach is to take the discounted expectations approach
4
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
For a standard european call, we can express delta as
∂C ∂ − rT delta = ∆ = = e E [ ST − K ]1ST > K ∂S ∂S
(
where and
ST = Se
)
( vT +σzT )
1ST > K
is the indicator function which equals one if ST>K or zero otherwise
5
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
This gives us;
∆=e
− rT
[
Ee
( vT +σzT )
1ST > K
]
Now, we can simulate this easily in MatLab
6
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
euroMCDeltaCalc.m
7
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
Gamma is given by
∂∆ ∂ C C ( S + ∆S ) − 2C ( S ) + C ( S − ∆S ) gamma = Γ = = 2 ≈ ∂S ∂S ∆S 2 2
We can’t (easily) differentiate delta
[
∆ = e − rT E e ( vT +σzT ) 1ST > K
]
To give an analytical solution so we rely on finite difference methods to calculate gamma using Monte Carlo methods 8
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
Gamma by finite difference:
∂∆ ∂ C delta( S + ∆S ) − delta( S − ∆S ) gamma = Γ = = 2 ≈ ∂S ∂S 2∆S 2
9
©Finbarr Murphy 2007
Agenda Page
Computing Hedge Sensitivities using MC Multiple Stochastic Factors
1
2 2
9
MSc
COMPUTATIONAL FINANCE
3
10
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC
To now, we have assumed that the asset price changes have been driven by one stochastic factor, the Brownian motion in the GBM process
MC is very effective when we price options on asset prices that are assumed to be driven by two or more stochastic factors.
Typically, we want to simulate stock price movements, volatility movements and interest rates
11
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC
We could also include any number of other factors such as
Macro-economic variables Correlation with broad based indices etc
We will begin by considering a GBM as before with a stochastic volatility
12
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC Firstly, we should note that volatility moves in a
random manner but tends towards a mean value We therefore need to estimate The mean volatility value The rate at which the volatility moves around this mean value
MSc
COMPUTATIONAL FINANCE
The volatility of the volatility
We also need to consider the correlation between the
stock price movement and the volatility movement
13
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC Starting again with our familiar GBM, it is governed by
the Stochastic Differential Equation (SDE)
dS = Sµdt + σSdz S
The variance† SDE is given by
(
)
MSc
COMPUTATIONAL FINANCE
dV = α V − V dt + ξ V dzV Where V is the mean volatility ξ is the volatility of the variance α is the mean reversion rate Note that the wiener processes are different! 14
† Remember, variance = σ
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC
MSc
COMPUTATIONAL FINANCE
5Y NASDAQ Volatility versus index value
V
15
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC The interest in volatility is directly related to several
of its basic characteristics. In particular implied volatility is considered to be a gauge for uncertainty. It tends to grow during times of economic or political crises
and thus reflects the general sentiment of market participants. Another important statistical property is its trend to revert to
MSc
COMPUTATIONAL FINANCE
its historical mean. This means volatility ‘recovers’ from shocks or spikes over time and reverts to its typical levels. In addition, implied volatility tends to be higher than realised
volatility. This reflects the general aversion of investors to carry a short option position. Therefore, a risk premium has to be paid to the investor in reward for selling volatility.
16
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC One of the most important characteristics of volatility
is its negative correlation with respective to the underlying equity index. If the equity index rises, volatility tends to fall and vice versa.
The negative correlation between an equity index and
MSc
COMPUTATIONAL FINANCE
its volatility assures that a volatility index derivative is a cheap hedge for equity market exposure. In the hedged position, the gain in the volatility index
derivative covers the loss out of the equity exposure. By adding volatility as a hedge a portfolio is
diversified with a new asset class, generating excess returns and lowering risk. 17
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC Using some statistical tools, we can estimate V, ξ and
α from historical data
The movement in volatility is correlated with stock
price movements
MSc
COMPUTATIONAL FINANCE
We can do this by sampling zS and zV from the standard
normal bivariate distribution with correlation ρ
18
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC We do this by letting zS = ε1, a draw from a standard
normal (univariate) distribution function and setting zV as
zV = ρ ε1 + 1 − ρ ε 2 2
Where ε2 is a second independent draw from a
MSc
COMPUTATIONAL FINANCE
standard normal (univariate) distribution function
The subsequent MatLab code is fairly routine at this
stage
19
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC
MSc
COMPUTATIONAL FINANCE
euroMCDoubleStochastic.m
20
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC Here is a plot of the average of 100 simulated
volatilities over 1000 time steps
MSc
COMPUTATIONAL FINANCE
Notice the mean reversion characteristic
21
©Finbarr Murphy 2007
Recommended Texts Required/Recommended Clewlow, L. and Strickland, C. (1996) Implementing derivative
models, 1st ed., John Wiley and Sons Ltd. — Chapter 4
MSc
COMPUTATIONAL FINANCE
Additional/Useful
22
MONTE CARLO SIMULATIONS
MSc
COMPUTATIONAL FINANCE
Numerical Methods in Finance (Implementing Market Models)
©Finbarr Murphy 2007
Lecture Objectives Hedge Sensitivities
MSc
COMPUTATIONAL FINANCE
Multiple Stochastic Factors
©Finbarr Murphy 2007
Agenda Page
Computing Hedge Sensitivities using MC Multiple Stochastic Factors
1
2 2
9
MSc
COMPUTATIONAL FINANCE
3
3
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
By now, we know that Delta is the rate of change of the option relative to the underlying asset price.
From our Finite Difference lectures
∂C C ( S + ∆S ) − C ( S − ∆S ) delta = ∆ = ≈ ∂S 2∆S
In Monte Carlo Simulations, a better approach is to take the discounted expectations approach
4
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
For a standard european call, we can express delta as
∂C ∂ − rT delta = ∆ = = e E [ ST − K ]1ST > K ∂S ∂S
(
where and
ST = Se
)
( vT +σzT )
1ST > K
is the indicator function which equals one if ST>K or zero otherwise
5
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
This gives us;
∆=e
− rT
[
Ee
( vT +σzT )
1ST > K
]
Now, we can simulate this easily in MatLab
6
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
euroMCDeltaCalc.m
7
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
Gamma is given by
∂∆ ∂ C C ( S + ∆S ) − 2C ( S ) + C ( S − ∆S ) gamma = Γ = = 2 ≈ ∂S ∂S ∆S 2 2
We can’t (easily) differentiate delta
[
∆ = e − rT E e ( vT +σzT ) 1ST > K
]
To give an analytical solution so we rely on finite difference methods to calculate gamma using Monte Carlo methods 8
©Finbarr Murphy 2007
Computing Hedge Sensitivities with MC
Gamma by finite difference:
∂∆ ∂ C delta( S + ∆S ) − delta( S − ∆S ) gamma = Γ = = 2 ≈ ∂S ∂S 2∆S 2
9
©Finbarr Murphy 2007
Agenda Page
Computing Hedge Sensitivities using MC Multiple Stochastic Factors
1
2 2
9
MSc
COMPUTATIONAL FINANCE
3
10
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC
To now, we have assumed that the asset price changes have been driven by one stochastic factor, the Brownian motion in the GBM process
MC is very effective when we price options on asset prices that are assumed to be driven by two or more stochastic factors.
Typically, we want to simulate stock price movements, volatility movements and interest rates
11
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC
We could also include any number of other factors such as
Macro-economic variables Correlation with broad based indices etc
We will begin by considering a GBM as before with a stochastic volatility
12
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC Firstly, we should note that volatility moves in a
random manner but tends towards a mean value We therefore need to estimate The mean volatility value The rate at which the volatility moves around this mean value
MSc
COMPUTATIONAL FINANCE
The volatility of the volatility
We also need to consider the correlation between the
stock price movement and the volatility movement
13
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC Starting again with our familiar GBM, it is governed by
the Stochastic Differential Equation (SDE)
dS = Sµdt + σSdz S
The variance† SDE is given by
(
)
MSc
COMPUTATIONAL FINANCE
dV = α V − V dt + ξ V dzV Where V is the mean volatility ξ is the volatility of the variance α is the mean reversion rate Note that the wiener processes are different! 14
† Remember, variance = σ
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC
MSc
COMPUTATIONAL FINANCE
5Y NASDAQ Volatility versus index value
V
15
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC The interest in volatility is directly related to several
of its basic characteristics. In particular implied volatility is considered to be a gauge for uncertainty. It tends to grow during times of economic or political crises
and thus reflects the general sentiment of market participants. Another important statistical property is its trend to revert to
MSc
COMPUTATIONAL FINANCE
its historical mean. This means volatility ‘recovers’ from shocks or spikes over time and reverts to its typical levels. In addition, implied volatility tends to be higher than realised
volatility. This reflects the general aversion of investors to carry a short option position. Therefore, a risk premium has to be paid to the investor in reward for selling volatility.
16
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC One of the most important characteristics of volatility
is its negative correlation with respective to the underlying equity index. If the equity index rises, volatility tends to fall and vice versa.
The negative correlation between an equity index and
MSc
COMPUTATIONAL FINANCE
its volatility assures that a volatility index derivative is a cheap hedge for equity market exposure. In the hedged position, the gain in the volatility index
derivative covers the loss out of the equity exposure. By adding volatility as a hedge a portfolio is
diversified with a new asset class, generating excess returns and lowering risk. 17
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC Using some statistical tools, we can estimate V, ξ and
α from historical data
The movement in volatility is correlated with stock
price movements
MSc
COMPUTATIONAL FINANCE
We can do this by sampling zS and zV from the standard
normal bivariate distribution with correlation ρ
18
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC We do this by letting zS = ε1, a draw from a standard
normal (univariate) distribution function and setting zV as
zV = ρ ε1 + 1 − ρ ε 2 2
Where ε2 is a second independent draw from a
MSc
COMPUTATIONAL FINANCE
standard normal (univariate) distribution function
The subsequent MatLab code is fairly routine at this
stage
19
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC
MSc
COMPUTATIONAL FINANCE
euroMCDoubleStochastic.m
20
©Finbarr Murphy 2007
Multiple Stochastic Factors with MC Here is a plot of the average of 100 simulated
volatilities over 1000 time steps
MSc
COMPUTATIONAL FINANCE
Notice the mean reversion characteristic
21
©Finbarr Murphy 2007
Recommended Texts Required/Recommended Clewlow, L. and Strickland, C. (1996) Implementing derivative
models, 1st ed., John Wiley and Sons Ltd. — Chapter 4
MSc
COMPUTATIONAL FINANCE
Additional/Useful
22
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