Rare Event Simulation in Finance - Brown University...Portfolio risk measurement – Simulate...
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Rare Event Simulation in Finance Brown Summer School on Rare Event Simulation June 13-17, 2016 Paul Glasserman Columbia Business School
Transcript of Rare Event Simulation in Finance - Brown University...Portfolio risk measurement – Simulate...
Rare Event Simulation in Finance
Brown Summer School on Rare Event SimulationJune 13-17, 2016
Paul GlassermanColumbia Business School
Simulation in Finance
• Valuation of options and other derivative securities– Simulate paths of underlying assets (stocks, interest rates, etc.)– Calculate payoff on each path– Main challenge is fast, precise pricing consistent with market prices,
and calculation of price sensitivities for hedging
• Portfolio risk measurement– Simulate relevant scenarios; evaluate portfolio loss in each scenario– Often summarize through a quantile or other tail measure– Main challenges are modeling scenarios, revaluing complex portfolio,
sampling tails
• Systemic Risk and Financial Crises– Rare events but no good models 2
Topics
• Portfolio value-at-risk– Delta-gamma normal– Heavy-tailed setting
• Portfolio credit risk with dependent defaults– Gaussian copula model– Mixed Poisson model
• Other topics– Stress scenario selection– Conditional and unconditional margin levels– Path-dependent options
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Value-at-Risk and Tail Probabilities
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Value-at-Risk and Tail Probabilities
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Normal ∆S
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Normal ∆S
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Delta-Gamma Approximation
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Price sensitivities are calculated anyway, hence available
Using the Approximation
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Using the Approximation
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Voilà: Exponential Tilt
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CGF and Parameter Choice
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Asymptotic Optimality if Approximation is Exact…
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Increasing number of factors
Further Variance Reduction Through Stratification
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We use brute-force rejection sampling to generate Z conditional on bin for Q
Interesting problem: How to sample Z | Q efficiently. (Spherical case is easy)
This is like integrating out Q numerically, using simulation conditional on Q
Market Data Exhibits Heavy Tails
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Market Data Exhibits Heavy Tails
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Indirect Delta-Gamma
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Transform Result
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Importance Sampling: Twist Y then Z, conditionally
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• Numerator and denominator become dependent under IS distribution• Achieves bounded relative error (when delta-gamma approximation holds)
Topics
• Portfolio value-at-risk– Delta-gamma normal– Heavy-tailed setting
• Portfolio credit risk with dependent defaults– Gaussian copula model– Mixed Poisson model
• Other topics– Stress scenario selection– Conditional and unconditional margin levels– Path-dependent options
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Credit Risk Framework
•Loss L
•Loss given default ck assumed known for simplicity•Default indicators Yk linked through common factors
xxLPkY
kcm
cYL
k
k
m
kkk
large for Findobligor th of ) or ( indicator default
obligor th of default given lossobligors of number
)(01
1
>===
= ∑=
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Linking Default Indicators• Gaussian copula
xk
pk=marginal default probability
{ }, default indicatorsnormals standard tindependen
1,,...,1
kkk
kd
xXYZZ
>=ε
kkdkdkk bZaZaX ε+++= 11
factors and loadings specific risk
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Gaussian Copula: Bivariate Illustration
Obligor 1 defaults
Obligor 2 defaults
X1
X2
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Importance Sampling: Independent Defaults
• Loss L=Y1c1+ … +Ymcm
• IS: increase default probabilities pk to qk
• LR:
• Estimate of P(L>x):
{ }xL >1
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Exponential Twist of Ykck
pk
1-pk
ck
0
ck
0
kc
k peq kθ∝
kc
k
ck
k pepepq
k
k
−+=
1θ
θ
kk pq −∝− 11
More weight on higher default probabilities and on larger losses
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Independent Defaults (continued)• With
• Likelihood ratio
• Exponentially twisting the Yk ck= Exponentially twisting L
kc
k
ck
k pepepq
k
k
−+=
1θ
θ
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Independent Defaults: Parameter Choice
( )x
pepLE
L
m
kk
ckL
k
=
−+== ∑=
)('
1log)][exp(log)(1
θψ
θθψ θ
θmx /)(' =θψ
)(θψ
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Dependent Case: Twist Conditional on Z
New default probabilities:
)1)((1)(),(
−+=
k
k
ck
ck
k ezpezpzp θ
θ
θ
xczpczp mxmx
x
=++=
),(),( 11 θθθθ
so chosen with
xzZLE == ]|[i.e.,
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Importance Sampling for Normal Factors
• Shift mean from 0 to µ• Weight by likelihood ratio =
0 µ
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Combined Procedure
• Choose a factor mean µ• Repeat for each replication:
– Generate factors from shifted distribution– Calculate θ and apply IS to default probabilities conditional
on factors– Estimate:
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How to Choose New Factor Mean?
• Laplace approximation suggests choosing µ to solve
• This approximates optimal IS density using a normal density with the same mode
• Optimal z is most likely factor outcome leading to large losses
• Need to calculate or approximate P(L>x|z); use
)2/'exp()|(max zzzxLPz
−>
))(exp())),(()(exp()|( zFzzxzzxLP x=+−≤> θψθ
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))(exp()|( zFzZxLP x≤=>
Shifting the Factor Mean
Upper bound:Large losses become “certain”
Optimal shift
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10-Factor Model, 1000 Obligors
The Influence of Dependence
• In the dependent case, the model becomes very “stiff,” and almost all the variance reduction needs to come from shifting the factor mean, not twisting the default probabilities. The following holds without the shift:
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Combined Estimator: Two Regimes
• Tail probability for loss will be small if (i) threshold is large or (ii) individual default probabilities are small.
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Combined Estimator: Two Regimes
• Tail probability for loss will be small if (i) threshold is large or (ii) individual default probabilities are small.
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Mixed Poisson Model (CreditRisk+)
),Y(Y
RRaRaaS
SY
kk
j
dkdkkk
kk
1min
110
with replace could
variables random gamma e.g.,variables random positive tindependen
intensity variable, random Poisson
=+++=
=
• Replace 0/1 default indicators with conditionally Poisson variables
• Introduce dependence through conditioning variables
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IS Strategy
• Apply IS to factors• Apply IS to intensity conditional on factors
• Twisting a Gamma(α,β) by τ produces Gamma(α,β(τ)) with β(τ) = β/(1−βτ)
• Twisting a Poisson(λ) by θ produces Poisson(λeθ)
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IS Strategy (continued)
• LR from factors
• LR from Poisson variables
• Combined LR is product of the two• To simplify use
( )∑=
−+−d
jjjjjj R
1)1log(exp( τβατ
( )
−−− ∑
=
m
kkkkk
keScY1
)1(exp θθ
dkdkkk RaRaaS ++= 110
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IS Strategy (continued)
• Choose twisting parameters to satisfy
• Combined LR reduces to
• Cancel everything stochastic except for L• (This works because gamma mixture of Poissons is
negative binomial)
( ) }
)(
))(1log()1(exp{1 1
0
θψ
θτβαθ θ
L
m
k
d
jjjj
ck
keaL ∑ ∑= =
−−−+−
∑=
−=m
k
ckjj
kea1
)1( θτ
Topics
• Portfolio value-at-risk– Delta-gamma normal– Heavy-tailed setting
• Portfolio credit risk with dependent defaults– Gaussian copula model– Mixed Poisson model
• Other topics– Stress scenario selection– Conditional and unconditional margin levels– Path-dependent options
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Stress Scenario Selection
• Since the financial crisis, regulators have been more skeptical about stochastic models of risk. Emphasis has shifted toward “stress testing:” evaluating losses in extreme but plausible scenarios
• Ok, but how do we pick the scenarios?– Repeat history– Make things up– Use a stochastic model…
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Fed Scenarios: Paths Over Nine Quarters of 20+ Variables
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General Formulation
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Most Likely Scenario, With Some Data Available…
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Most Likely Scenario, With Some Data Available…
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Topics
• Portfolio value-at-risk– Delta-gamma normal– Heavy-tailed setting
• Portfolio credit risk with dependent defaults– Gaussian copula model– Mixed Poisson model
• Other topics– Stress scenario selection– Conditional and unconditional margin levels– Path-dependent options
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Daily Trading Profit and Loss and 99% VaR
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Bank of America Daily Trading P&L (red) and VaR (black)
2006 2007
Conditional Margin
• Each curve shows distribution of loss/gain over risk horizon
• Given current market conditions, set the required margin at level that covers losses with 99% confidence
• Lower margin when market is quiet
• Spike in margin when volatility spikes
49t t + 1 t + 2 …
Unconditional Margin
• Set margin high enough to cover losses over time with 99% confidence
• Eliminates spikes• But margin feels
unnecessarily high in quiet periods
• How much higher is unconditional margin than the average conditional margin?
50t t + 1 t + 2 …
A GARCH Lens On The Question
• GARCH model provides a simple setting that– Captures volatility clustering– Contrasts conditional and unconditional margin
• GARCH(1,1)
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Conditional And Unconditional Margin
• Conditional margin at confidence level 1-p
• Long-run average conditional (procyclical) margin
• Unconditional (stable) margin
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How Much Larger Does Stable Margin Need To Be?
• This is bad news if κ is small, particularly at high confidence levels
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Numerator grows much faster than the denominator at high confidence levels, particularly when kappa is small
Illustration
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Topics
• Portfolio value-at-risk– Delta-gamma normal– Heavy-tailed setting
• Portfolio credit risk with dependent defaults– Gaussian copula model– Mixed Poisson model
• Other topics– Stress scenario selection– Conditional and unconditional margin levels– Path-dependent options
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Down-and-In Barrier Option
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Importance Sampling
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Importance Sampling
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Some References
• Glasserman, P., Heidelberger, P., & Shahabuddin, P. (1999). Asymptotically Optimal Importance Sampling and Stratification for Pricing Path-Dependent Options. Mathematical finance, 9(2), 117-152.
• Glasserman, P., Heidelberger, P., & Shahabuddin, P. (2000). Variance reduction techniques for estimating value-at-risk. Management Science, 46(10), 1349-1364.
• Glasserman, P., Heidelberger, P., & Shahabuddin, P. (2002). Portfolio Value-at-Risk with Heavy-Tailed Risk Factors. Mathematical Finance, 12(3), 239-269.
• Glasserman, P., & Li, J. (2005). Importance sampling for portfolio credit risk. Management science, 51(11), 1643-1656.• Glynn, P. W. (1996). Importance sampling for Monte Carlo estimation of quantiles. In Mathematical Methods in Stochastic
Simulation and Experimental Design: Proceedings of the 2nd St. Petersburg Workshop on Simulation (pp. 180-185).• Guasoni, P., & Robertson, S. (2008). Optimal importance sampling with explicit formulas in continuous time. Finance and
Stochastics, 12(1), 1-19.• Kang, W., & Shahabuddin, P. (2005, December). Fast simulation for multifactor portfolio credit risk in the t-copula model. In
Proceedings of the 37th conference on Winter simulation (pp. 1859-1868). Winter Simulation Conference.• Newton, N. J. (1994). Variance reduction for simulated diffusions. SIAM journal on applied mathematics, 54(6), 1780-1805.
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