N
HPCFinance: New Thinking in Finance
Calculating Variable Annuity Liability
‘Greeks’ Using Monte Carlo Simulation
Dr. Mark Cathcart, Standard Life
February 14, 2014
Monte Carlo Estimates of Variable Annuity ‘Greeks’
0 / 58
N
Outline
Outline of Presentation
1
Introduction
2
Main Monte Carlo Greek Approaches
3
Main MC Greek Approaches: B-S Tests
4
Monte Carlo Greeks for More Advanced Models
5
Monte Carlo Greeks for Variable Annuities
Monte Carlo Estimates of Variable Annuity ‘Greeks’
0 / 58
Introduction
Overview of Variable Annuity contracts
Variable Annuities
What are Variable Annuity (VA) Products?
Types of VA
GMDB,GMAB,GMIB,GMWB
Nature of Liabilities of Guarantee
complex
multi-dimensional
path-dependent
Even valuing the liabilities is fairly involved - simulation.
Can we construct more sophisticated estimators of the VA
Greeks to aid insurer’s hedging strategies?
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
1 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
There are three main approaches...
1. Bump and Revalue
2. Pathwise Approach
3. Likelihood Ratio Method
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
2 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
1. Bump and Revalue
Idea is to simulate for the price of the option under some base
scenarios and then again under some ‘bumped’ scenarios
e.g., delta: perturb the initial stock price by ∆S(0):
C (S(0) + ∆S(0), K , σ, r , T ) − C (S(0), K , σ, r , T )
∆S(0)
i.e., a Forward Difference estimate
Common Random Numbers
Good practice to use same random number generator with
same initial seed in base and bumped scenarios
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
3 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
2. Pathwise Approach
Let α(θ) = E[Y (θ)] then we can estimate the derivative of α(θ)
using:
Y (θ + h) − Y (θ)
Y 0 (θ) = lim
h→0
h
This estimator has expectation E[Y 0 (θ)] and is an unbiased
estimator of α0 (θ) if interchanging differentiation and taking
expectations is justified, i.e.,
d
d
E
Y (θ) =
E[Y (θ)]
dθ
dθ
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
4 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
2. Pathwise Approach
Let us consider a simple, illustrative example...
European option under B-S model: Delta
Y
= e −rT max(S(T ) − K , 0) Payoff
1
S(T ) = S(0)e (r − 2 σ
2 )T +σ
√
TZ
Dynamics
Applying the chain rule gives...
dY
dY dS(T )
=
dS(0)
dS(T ) dS(0)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
5 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
2. Pathwise Approach
Let us consider a simple, illustrative example...
European option under B-S model: Delta
Y
= e −rT max(S(T ) − K , 0) Payoff
1
S(T ) = S(0)e (r − 2 σ
2 )T +σ
√
TZ
Dynamics
Applying the chain rule gives...
dY
dY dS(T )
=
dS(0)
dS(T ) dS(0)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
5 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
2. Pathwise Approach
Let us consider a simple, illustrative example...
European option under B-S model: Delta
Y
= e −rT max(S(T ) − K , 0) Payoff
1
S(T ) = S(0)e (r − 2 σ
2 )T +σ
√
TZ
Dynamics
Applying the chain rule gives...
dY
dS(T )
= e −rT I{S(T ) > K }
dS(0)
dS(0)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
5 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
2. Pathwise Approach
Let us consider a simple, illustrative example...
European option under B-S model: Delta
Y
= e −rT max(S(T ) − K , 0) Payoff
1
S(T ) = S(0)e (r − 2 σ
2 )T +σ
√
TZ
Dynamics
Applying the chain rule gives...
dY
dS(T )
= e −rT I{S(T ) > K }
dS(0)
dS(0)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
5 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
2. Pathwise Approach
Let us consider a simple, illustrative example...
European option under B-S model: Delta
Y
= e −rT max(S(T ) − K , 0) Payoff
1
S(T ) = S(0)e (r − 2 σ
2 )T +σ
√
TZ
Dynamics
Applying the chain rule gives...
dY
dS(T )
= e −rT I{S(T ) > K }
dS(0)
dS(0)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
5 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
2. Pathwise Approach
Let us consider a simple, illustrative example...
European option under B-S model: Delta
Y
= e −rT max(S(T ) − K , 0) Payoff
1
S(T ) = S(0)e (r − 2 σ
2 )T +σ
√
TZ
Dynamics
Applying the chain rule gives...
dY
S(T )
= e −rT I{S(T ) > K }
dS(0)
S(0)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
5 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
2. Pathwise Approach
Let us consider a simple, illustrative example...
European option under B-S model: Delta
Thus, the Pathwise Estimator for a European call option
(under the B-S model underlying dynamics) is given as:
dY
S(T )
= e −rT I{S(T ) > K }
dS(0)
S(0)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
5 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
2. Pathwise Approach
What about estimating the Gamma option ‘Greek’ ?
The Pathwise is inapplicable in this case!
&
d
dS(T )
%
&
d
dS(T )
%
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
6 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
2. Pathwise Approach
What about estimating the Gamma option ‘Greek’ ?
The Pathwise is inapplicable in this case!
More formally, the requirement for using the PW method
d
d
Y (θ) =
E[Y (θ)]
E
dθ
dθ
is not satisfied in this instance. Indeed,
d
dY
0=E
6=
E[Y ]
dS(0)
dS(0)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
7 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
3. Likelihood Ratio Method
Relies on differentiating prob.density rather than payoff funct.,
thus does not require smoothness in the payoff function
Suppose we have a discounted payoff Y expressed as a function
f (X), where X is a m-dimensional vector of different asset prices
(or alternatively, one asset price at multiple valuation dates).
Then assuming that X has a probability density g with
parameter θ, and that Eθ denotes an expectation taken with
respect to the density gθ , taking the expected discounted payoff
with respect to this density gives
Z
Eθ [Y ] = Eθ [f (X1 , . . . , Xm )] =
f (X)gθ (X)dX
Rm
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
8 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
3. Likelihood Ratio Method
Now, similarly to the pathwise derivative approach, we assume
the order of differentiation and integration can be interchanged.
Here, however, this is not so strong an assumption, as typically
densities are smooth functions, whereas payoff functions are not.
This gives
Z
d
gθ (X)dX
dθ
Rm
Z
d
gθ (X)
=
f (X) dθ
gθ (X)dX
gθ (X)
Rm
d
dθ gθ (X)
= Eθ f (X)
gθ (X)
d
Eθ [Y ] =
dθ
f (X)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
9 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
3. Likelihood Ratio Method
d
g (X)
θ
Then f (X) dθgθ (X)
gives the Likelihood-Ratio estimator for the
sensitivity with respect to the parameter θ, and this estimator is
unbiased
d
g (X)
θ
The term dθgθ (X)
is often known in the statistics literature as
the “score function”
The Likelihood-Ratio method will still be applicable and robust
in the case of options with discontinuous payoff functions (and
estimating second-order sensitivities)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
10 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
3. Likelihood Ratio Method
Let us again consider our simple, illustrative example...
European option under B-S model: Delta
The log-normal density used to calculate S(T ) is given by
ln x − (r − 1 σ 2 )T 1
2
S(0)
√ φ
√
g (x) =
xσ T
σ T
where φ(·) represents the standard normal density function.
In this case the score function is:
dg (x)
x
ln S(0)
− (r − 12 σ 2 )T
dS(0)
=
g (x)
S(0)σ 2 T
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
11 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
3. Likelihood Ratio Method
Let us again consider our simple, illustrative example...
European option under B-S model: Delta
Evaluating this at S(T ) and multiplying by the option payoff
gives the unbiased estimator of the Black-Scholes delta as:
ln S(T )/S(0) − (r − σ 2 /2)T
−rT
e
max(S(T ) − K , 0)
S(0)σ 2 T
1
Using: S(T ) = S(0)e (r − 2 σ
2 )T +σ
√
TZ
, where Z ∼ N(0,1). . .
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
11 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
3. Likelihood Ratio Method
Let us again consider our simple, illustrative example...
European option under B-S model: Delta
Evaluating this at S(T ) and multiplying by the option payoff
gives the unbiased estimator of the Black-Scholes delta as:
ln S(T )/S(0) − (r − σ 2 /2)T
−rT
e
max(S(T ) − K , 0)
S(0)σ 2 T
1
Using: S(T ) = S(0)e (r − 2 σ
2 )T +σ
√
TZ
, where Z ∼ N(0,1). . .
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
11 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
3. Likelihood Ratio Method
Let us again consider our simple, illustrative example...
European option under B-S model: Delta
Evaluating this at S(T ) and multiplying by the option payoff
gives the unbiased estimator of the Black-Scholes delta as:
e −rT max(S(T ) − K , 0)
1
Using: S(T ) = S(0)e (r − 2 σ
2 )T +σ
√
TZ
Z
√
S(0)σ T
, where Z ∼ N(0,1). . .
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
11 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
3. Likelihood Ratio Method
Let us again consider our simple, illustrative example...
European option under B-S model: Delta
Evaluating this at S(T ) and multiplying by the option payoff
gives the unbiased estimator of the Black-Scholes delta as:
e −rT max(S(T ) − K , 0)
Z
√
S(0)σ T
Giving the simplified unbiased LRM estimate of Delta, for a
European call option under the B-S model for the underlying
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
11 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
3. Likelihood Ratio Method
And what about the case of discontinuous payoff functions...
Binary (cash-or-nothing) option under B-S model
Payoff =
1 if S(T ) ≥ K
= I{S(T ) > K }
0 if S(T ) < K
Digital call:
e −rT I{S(T ) > K }
Z
√
S(0)σ T
European call: e −rT max(S(T ) − K , 0)
Z
√
S(0)σ T
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
12 / 58
Main Monte Carlo Greek Approaches
Estimating ‘Greeks’ by MC simulation
3. Likelihood Ratio Method
And finally what about estimating the gamma sensitivity...
European option under B-S model: Gamma
As the gamma is the second-order sensitivity of the payoff
wrt to S0 the likelihood ratio weight will be different to that
calculated for the delta. However, it is easy to show that the
LRM estimator for the gamma is given by:
√
Z2 − Zσ T − 1
−rT
e
max(S(T ) − K , 0)
S(0)2 σ 2 T
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
13 / 58
Main MC Greek Approaches: B-S Tests
B-S Model Test Cases for analysis
Let us look at applying the three approaches for the Delta and
Gamma sensitivities of a simple European put option
Case
S
K
σ
r
T
A
B
C
D
E
F
100
100
100
100
100
100
90
100
110
100
100
100
20%
20%
20%
10%
30%
20%
4%
4%
4%
4%
4%
1.5%
1
1
1
1
1
1
Table: Different put option set-ups considered in the tests. In all the
above cases 100,000 paths and S(0) perturbation of 0.2% used.
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
14 / 58
Main MC Greek Approaches: B-S Tests
European put option delta estimators
0
-0.1
Delta Estimates
-0.2
-0.3
Case A
95%
Confidence
Intervals
-0.4
Case D
Case B
Case E
-0.5
Case F
-0.6
Case C
-0.7
B'nR Delta Est's
LRM Delta Est's
Pathwise Delta Est's
Analytical Values
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
15 / 58
Main MC Greek Approaches: B-S Tests
European put option delta estimators
-0.375
Delta Estimates
-0.38
-0.385
-0.39
Case B
-0.395
95%
Confidence
Intervals
Case E
-0.4
B'nR Delta Est's
LRM Delta Est's
Pathwise Delta Est's
Analytical Values
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
16 / 58
Main MC Greek Approaches: B-S Tests
European put option gamma estimators
0.045
0.040
0.035
Gamma Estimates
Case D
95% Confidence
Intervals
0.030
0.025
Case F
0.020
Case E
0.015
Case B
0.010
Case C
Case A
0.005
0.000
B'nR Gamma Est's
LRM Gamm Est's
Analytical Gamma Values
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
17 / 58
Main MC Greek Approaches: B-S Tests
European put option gamma estimators
0.0215
Gamma Estimates
0.0205
0.0195
0.0185
95% Confidence
Intervals
Case C
Case F
Case B
0.0175
0.0165
B'nR Gamma Est's
LRM Gamm Est's
Analytical Gamma Values
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
18 / 58
Main MC Greek Approaches: B-S Tests
Conclusions from B-S analysis
1. B&R and PW give very similar estimates and standard errors
This is expected as the Pathwise Estimator is the small perturbation limit of the Bump and Revalue method
2. When the Pathwise estimator is available it generally gives
estimators with smaller standard errors than the LRM estimator
This is because Pathwise estimator is specific to a given payoff
function, whereas the LRM determines a weight for a given
density which can then be multiplied by any payoff function
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
19 / 58
Main MC Greek Approaches: B-S Tests
Conclusions from B-S analysis
3. The LRM provides an estimate of Gamma with smaller standard error than the corresponding Bump and Revalue estimate.
If the Pathwise method is inapplicable, then, essentially, the
Bump&Revalue approach cannot converge to the true value
Note: Although these observations have been inferred from analysis on simple European options, they generally hold for other
options and models (see, e.g., Glasserman).
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
20 / 58
Main MC Greek Approaches: B-S Tests
Estimating ‘Greeks’ by MC simulation
Mixed estimator for second-order Greeks
Let us again consider our simple, illustrative example...
European option under B-S model: Gamma
Can form a hybrid estimator for call option Gamma by applying the PW method to the LRM estimator for Delta. . .
d
Z
−rT
√
e
max(S(T ) − K , 0)
dS(0)
S(0)σ T
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
21 / 58
Main MC Greek Approaches: B-S Tests
Estimating ‘Greeks’ by MC simulation
Mixed estimator for second-order Greeks
Let us again consider our simple, illustrative example...
European option under B-S model: Gamma
. . . resulting in the LR-PW mixed estimator for Gamma:
e −rT
Z
√ I{S(T ) > K }K
S(0)2 σ T
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
22 / 58
Main MC Greek Approaches: B-S Tests
Estimating ‘Greeks’ by MC simulation
Mixed estimator for second-order Greeks
Let us again consider our simple, illustrative example...
European option under B-S model: Gamma
Can also form a PW-LR
hybrid estimator for Gamma. . .
)
Z√
−rT
e
I{S(T ) > K } S(T
S(0) · S(0)σ T
d
1
+I{S(T ) > K }S(T ) dS(0)
S(0)
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
23 / 58
Main MC Greek Approaches: B-S Tests
Estimating ‘Greeks’ by MC simulation
Mixed estimator for second-order Greeks
Let us again consider our simple, illustrative example...
European option under B-S model: Gamma
. . . and after a couple of lines of maths, the PW-LR hybrid
estimator for the call option Gamma is given as:
S(T )
Z
√
e −rT I{S(T ) > K }
−
1
S(0)2
σ T
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
24 / 58
Main MC Greek Approaches: B-S Tests
European put gamma mixed estimators
0.045
0.040
0.035
Gamma Estimates
0.030
Case D
95% Confidence
Intervals
0.025
Case F
0.020
Case E
0.015
0.010
Case B
Case C
Case A
0.005
0.000
B'nR Gamma Est's
LR-PW Mixed Gamma Est's
Analytical Gamma Values
LRM Gamm Est's
PW-LR Mixed Gamma Est's
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
25 / 58
Main MC Greek Approaches: B-S Tests
European put gamma mixed estimators
0.0215
Gamma Estimates
0.0205
0.0195
0.0185
95% Confidence
Intervals
Case C
Case F
Case B
0.0175
0.0165
B'nR Gamma Est's
LR-PW Mixed Gamma Est's
Analytical Gamma Values
LRM Gamm Est's
PW-LR Mixed Gamma Est's
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
26 / 58
Main MC Greek Approaches: B-S Tests
Conclusions of mixed estimator analysis
1. Both mixed estimators provide estimates with significantly
lower standard errors than the LRM estimator (which gave
lower standard errors than the Bump&Revalue)
2. Mixed estimators seem to utilise the respective advantages of
the PW and LRM estimators
3. Given that they are constructed from the PW and LRM
approaches, the mixed estimators are also unbiased
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
27 / 58
Main MC Greek Approaches: B-S Tests
Estimating ‘Greeks’ by MC simulation
Estimators for Greeks on path-dependent payoffs
Path-dependent payoff under B-S: LRM Estimator
We can see that S(0) is a parameter of the first factor,
g1 (x1 |S(0)), only. Then the score function is:
∂ log g (S(t1 ), . . . , S(tm ))
∂S(0)
=
=
∂ log g1 (S(t1 )|S(0))
∂S(0)
ζ1 (S(t1 )|S(0))
Z1
√
√
=
S(0)σ t1
S(0)σ t1
where Z1 = ζ1 (S(t1 )|S(0)) is the Gaussian increment which
takes us from time zero to time t1 , the first monitor point.
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
28 / 58
Monte Carlo Greeks for More Advanced Models
Accommodating more advanced models
The problems with Black-Scholes
Black-Scholes model presents many problems for use with
risk-management of Variable Annuity products:
Underestimates the probability of extremely low equity returns
Assumes volatility and risk-free rate are constant in time
Assumes stock price movements are always continuous
However, B-S model tractable and widely used/assumed
Thus we wish to address the problems without losing too much
of the B-S structure. . .
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
29 / 58
Monte Carlo Greeks for More Advanced Models
Accommodating more advanced models
Alternative models considered
1. Heston Stochastic Volatility model (Hest):
p
dSt = rSt dt + Vt St dWtS
p
dVt = κ(θ − Vt )dt + σV Vt dWtV
where κ > 0 represents the mean reversion speed, θ > 0 denotes
the mean reversion level and σV > 0 is the volatility of the
variance process.
Brownian motions have correlation ρ, i.e., dWtS dWtV = ρdt
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
30 / 58
Monte Carlo Greeks for More Advanced Models
Accommodating more advanced models
Alternative models considered
2. Heston SV with C-I-R stochastic rates (Hest-CIR):
p
dSt = rt St dt + Vt St dWtS
p
dVt = κV (θV − Vt )dt + σV Vt dWtV
√
drt = κr (θr − rt )dt + σr rt dWtr
corr(WtS , WtV ) = ρS,V ,
i.e.,
dWtS dWtV = ρS,V dt
corr(WtS , Wtr ) = ρS,r ,
i.e.,
dWtS dWtr = ρS,r dt
corr(WtV , Wtr ) = ρV ,r ,
i.e.,
dWtV dWtr = ρV ,r dt
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
31 / 58
Monte Carlo Greeks for More Advanced Models
Accommodating more advanced models
How do the three approaches change?
With the B&R and PW approaches, the estimators just require
the generated equity returns (mechanics same if SV or S-IR)
On the other hand, LRM uses the score function in determining
estimate and for the B-S model this score function can be found
With Hest or Hest-CIR the returns are no longer lognormal
and thus the score function cannot be determined
However, if we condition on a realisation of the variance (and
interest-rate) process, the returns are (conditionally) lognormal,
allowing the LRM to be applied
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
32 / 58
Monte Carlo Greeks for More Advanced Models
How does the LRM change with SV?
Building in equity-volatility correlation
Cholesky Decomposition
Brownian motions for the asset and variance processes have
correlation ρ. This can be achieved by setting:
p
dWtS = ρdWtV + 1 − ρ2 dWtInd
Get expression for asset price process (with corr. structure):
p
p
dSt = rSt dt + Vt St ρdWtV + 1 − ρ2 dWtInd
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
33 / 58
Monte Carlo Greeks for More Advanced Models
How does the LRM change with SV?
Itô’s calculus and Geometric Brownian Motion
Itô’s Lemma:
Given a r.v y satisfying the s.d.e
dy = a(y , t)dt + b(y , t)dWt
where Wt is a Brownian motion and a funct. f (y , t) differentiable wrt t and twice so wrt y , then f itself satisfies:
2
∂f
∂f
1
∂f
2∂ f
df =
+ a(y , t)
+ b(y , t)
dt+b(y , t) dWt
2
∂t
∂y
2
∂y
∂y
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
34 / 58
Monte Carlo Greeks for More Advanced Models
How does the LRM change with SV?
Itô’s calculus and Geometric Brownian Motion
Geometric Brownian motion:
dy = dSt = St (rdt + σdWt )
Then, f (y , t) = ln y =⇒
d ln St = dXt
∂f
∂t
Black-Scholes
= 0,
∂f
∂y
= y1 ,
∂2f
∂y 2
=
−1
y2
and
∂2f
∂f
∂f
1
dt +
+
(dy 2 )
∂t
∂y
2 ∂y 2
= 0 + dSt /St + (1/2)(−1/St2 ) dSt2
=
(dW )2 = dt → = rdt + σdWt + (1/2)(−1/St2 )St2 σ 2 dt
= (r − σ 2 /2)dt + σdWt
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
35 / 58
Monte Carlo Greeks for More Advanced Models
How does the LRM change with SV?
Itô’s calculus and Geometric Brownian Motion
Geometric Brownian motion:
d ln St = (r − σ 2 /2)dt + σdWt
w

1 2
ST = S0 exp (r − σ )T + σdWT
2
i.e., lognormal returns in the Black-Scholes model
Same approach used to obtain the conditional lognormal returns from conditioning on the Heston & CIR processes
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
36 / 58
Monte Carlo Greeks for More Advanced Models
How does LRM change with SV & S-IR?
Heston-CIR Model:
p dSt
V
r
Ind
= rt dt + Vt C1 dWt + C2 dWt + C3 dWt
St
where
C1 = ρS,V
ρS,r − ρS,V ρV ,r
q
C2 =
1 − ρ2V ,r
s
(ρS,r − ρS,V ρV ,r )2
1 − ρ2S,V −
C3 =
1 − ρ2V ,r
Applying Itô’s Formula and collecting terms gives. . .
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
37 / 58
Monte Carlo Greeks for More Advanced Models
How does LRM change with SV & S-IR?
Heston-CIR Model:
ST
= S0 exp(YT ) exp
Z
1
1 T
rt dt − C32
Vt dt
2 T 0
0
s
Z
1 T
+C3
Vt dWtInd
T 0
1
T
Z
T
where
YT
1
= − C12
2
Z
+C1
Z T
1
Vt dt − C22
Vt dt
2
0
0
Z Tp
T p
V
Vt dWt + C2
Vt dWtr
Z
T
0
0
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
38 / 58
Monte Carlo Greeks for More Advanced Models
How does LRM change with SV & S-IR?
Heston-CIR Model:
ST
= S0 exp(YT ) exp
Z
1 T
1
rt dt − C32
Vt dt
2 T 0
0
s
Z
1 T
Ind
Vt dWt
+C3
T 0
1
T
Z
T
Therefore, the equity returns are conditionally lognormal:
S0 → S0 e YT
Z T
2
2 1
Vt dt
σ → C3
T 0
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
39 / 58
Monte Carlo Greeks for More Advanced Models
Hest-CIR Test Cases for analysis
Case
κV
θV
A
B
C
D
E
2
1
2
1
2
0.04
0.04
0.04
0.04
0.04
V
0.15
0.3
0.15
0.3
0.15
κIR
4
4
2
2
4
θIR
0.04
0.04
0.04
0.04
0.04
IR
0.06
0.06
0.2
0.2
0.06
ρS-V
-0.7
-0.7
-0.7
-0.7
-0.9
ρS-IR
-0.6
-0.6
-0.6
-0.6
-0.6
ρV-IR
0.7
0.7
0.7
0.7
0.7
Table: Different model settings considered in tests. Heston SV
parameters are denoted by a V subscript and the CIR parameters are
given by an IR subscript. The correlations denoted by obvious subscripts.
Asian put option: S0 = 1, K = 1.25, T = 30. Annual monitor points.
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
40 / 58
Monte Carlo Greeks for More Advanced Models
Hest-CIR Asian Option Delta
-0.08
-0.09
Case B
Delta Estimates
-0.1
-0.11
Case A
Case E
-0.12
Case D
-0.13
-0.14
95%
Confidence
Intervals
Case C
-0.15
-0.16
B'nR Delta Est's
Pathwise Delta Est's
LRM Delta Est's
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
41 / 58
Monte Carlo Greeks for More Advanced Models
Hest-CIR Asian Option Gamma
0.500
0.400
Gamma Estimates
0.300
0.200
0.100
Case C
Case A
Case D
Case B
0.000
-0.100
95% Confidence
Intervals
Case E
-0.200
B'nR Gamma Est's
LRM Gamm Est's
LR-PW Mixed Gamma Est's
PW-LR Mixed Gamma Est's
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
42 / 58
Monte Carlo Greeks for More Advanced Models
Conclusions from analysis
1. The Likelihood Ratio Method can be extended to price Asian
options under the Hest-CIR model
2. For the delta sensitivity, the PW/B&R method produced
estimates with smaller standard errors than the CLRM.
3. For the Gamma sensitivity, a mixed estimator will generally
be preferable to the B&R and CLRM for path-dependent payoff
functions. In certain cases the CLRM estimator can become very
inefficient.
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
43 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
Variable Annuity Assumptions
We now introduce a fairly simple VA contract to demonstrate
that the three MC Greek approaches can be extended to
estimate sensitivities of unit-linked insurance products
Assumptions in VA Example:
Policyholder is male and 65 years of age at annuitisation (and
static mortality rates assumed)
The policyholder has saved an amount £P which constitutes
the initial premium of the VA contract
Policyholder will take (full) withdrawals from end of year one
Policyholder lapsation behaviour is ‘static’
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
44 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product Structure
Fund/Account Value begins at the level of the policyholder
premium and grows/falls at the same rate as the equity index
with which the product is linked (at annual rebalancing dates)
At each rebalancing date the policyholder withdraws some
amount of Income from this Fund Value
This Income amount can grow from one year to the next if the
FV grows above its ‘high watermark level’. This maximum
lookback level will be denoted the Guarantee Base
Yearly withdrawal level given as fixed % of this GB level
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
45 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product Structure: Mathematically
Fund Value t years after annuitisation → Ft
Guarantee Base t years after annuitisation → Gt
Income Level t years after annuitisation → It
Fixed percentage of GB taken as Income → w
Equity Index, return from year t − 1 to t → Rt [= St /St−1 ]
Ft = max{(Ft−1 − It−1 )(1 + Rt ), 0} with F0 = P · S0
Gt = max{Ft , Gt−1 }
and
It = wGt
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
46 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product Structure: Mathematically
Fund Value t years after annuitisation → Ft
Guarantee Base t years after annuitisation → Gt
Income Level t years after annuitisation → It
Fixed percentage of GB taken as Income → w
Equity Index, return from year t − 1 to t → Rt [= St /St−1 ]
Then Value of Liability of this particular VA contract is:
" T
#
X
surv
L=E
Dt pt max(It − Ft , 0)
t=1
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
46 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product: Pathwise Estimator
Now a Pathwise method for this VA product will be developed
To demonstrate this consider the problem of estimating the
Delta Greek
" T
#
X
∂L
∂
surv
∆PW =
=
E
Dt pt max(It − Ft , 0)
∂S0
∂S0
t=0
With static policyholder behaviour, the derivative will only act
on third term above. . .
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
47 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product: Pathwise Estimator
Now a Pathwise method for this VA product will be developed
To demonstrate this consider the problem of estimating the
Delta Greek
" T
#
X
∂L
surv ∂
∆PW =
=E
Dt pt
max(It − Ft , 0)
∂S0
∂S0
t=0
With static policyholder behaviour, the derivative will only act
on third term above. . .
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
47 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product: Pathwise Estimator
This derivative is given by. . .
∂I
∂
∂Ft t
max(It − Ft , 0) = I{It > Ft } ·
−
∂S0
∂S0
∂S0
. . . but how are
∂It
∂S0
and
∂Ft
∂S0
calculated?
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
48 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product: Pathwise Estimator
These are given recursively by:
∂F
∂It−1 ∂Ft
t−1
=
−
(1 + Rt )
∂S0
∂S0
∂S0
∂It
∂Gt
=w·
∂S0
∂S0
with
∂I0
=0
∂S0
with
∂F0
=P
∂S0
∂
max(Ft , Gt−1 )
∂S0
h
∂Ft
∂Gt−1 i
= w · I{Ft >Gt−1 }
+ I{Ft ≤Gt−1 }
∂S0
∂S0
= w·
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
49 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product: Likelihood Ratio Method Estimator
Generality of LRM means it can be applied to VA product
liability Greeks without much extra effort
Similar structure to Asian option est’r. Uses implied shock
out to first valuation date, in our case the end of year 1
log(S1 /ξ¯1 S0 ) − (¯
r − σ̄12 /2) ∗ 1
√1
σ̄1 1
" "
! T
##
Imp.
X
Z
∆LRM = E E
Dt ptsurv max(It − Ft , 0)v1 , r1
S0 σ̄1 ∗ 1
Z Imp. =
t=0
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
50 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product: Likelihood Ratio Method Estimator
Generality of LRM means it can be applied to VA product
liability Greeks without much extra effort
Similar structure to Asian option est’r. Uses implied shock
out to first valuation date, in our case the end of year 1
log(S1 /ξ¯1 S0 ) − (¯
r − σ̄12 /2) ∗ 1
√1
σ̄1 1
" "
!
##
Imp.
Z
∆LRM = E E
× Liabilityv1 , r1
S0 σ̄1 ∗ 1
Z Imp. =
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
50 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product: Gamma Mixed PW-LR Estimator
The Pathwise and CLRM can be combined to give a PW-LR
estimator for the Gamma Greek. Along each path:
∂ LRM
∆
ΓLR-PW =
∂S0
"
# T
!
∂
Z Imp. X
surv
√
=
Dt pt max(It − Ft , 0)
∂S0
S0 σ̄1 1 t=1
# T
"
Z Imp. X
∂
√
=
Dt ptsurv
max(It − Ft , 0)
∂S
S0 σ̄1 1 t=1
0
"
# T
Z Imp. X
− 2 √
Dt ptsurv max(It − Ft , 0)
S0 σ̄1 1 t=1
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
51 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Example
VA Product: Gamma Mixed PW-LR Estimator
# T
"
Imp. X
∂
Z
√
Dt ptsurv
ΓLR-PW =
max(It − Ft , 0)
∂S0
S0 σ̄1 1 t=1
# T
"
Z Imp. X
Dt ptsurv max(It − Ft , 0)
− 2 √
S0 σ̄1 1 t=1
The two sums are already calculated in the intermediate steps to
find the Pathwise Delta estimator
Thus we just take the result of these sums along each path and
multiply by the respective factors above
A PW-LR estimator can be constructed in a similar way
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
52 / 58
Monte Carlo Greeks for Variable Annuities
Hest-CIR Test Cases for analysis
Case
κV
θV
A
B
C
D
E
2
1
2
1
1
0.04
0.04
0.04
0.04
0.04
V
0.15
0.3
0.15
0.3
0.3
κIR
4
4
2
2
2
θIR
0.04
0.04
0.04
0.04
0.04
IR
0.06
0.06
0.2
0.2
0.2
ρS-V
-0.7
-0.7
-0.7
-0.7
-0.9
ρS-IR
-0.6
-0.6
-0.6
-0.6
-0.6
ρV-IR
0.7
0.7
0.7
0.7
0.7
Table: Different model settings considered in tests. Heston SV
parameters are denoted by a V subscript and the CIR parameters are
given by an IR subscript. The correlations denoted by obvious subscripts.
V0 = θV , r0 = θIR . VA contract has T = 30 and Ratchet Term = 10.
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
53 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Delta Estimators
0
-0.001
-0.002
Delta Estimates
-0.003
-0.004
Case E
-0.005
Case D
Case B
-0.006
-0.007
95%
Confidence
Intervals
-0.008
-0.009
Case A
Case C
-0.01
B'nR Delta Est's
Pathwise Delta Est's
LRM Delta Est's
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
54 / 58
Monte Carlo Greeks for Variable Annuities
Variable Annuity Gamma Estimators
1.E-05
8.E-06
6.E-06
Gamma Estimates
4.E-06
2.E-06
0.E+00
Case C
Case A
Case B
-2.E-06
-4.E-06
-6.E-06
Case D
95% Confidence
Intervals
Case E
-8.E-06
B'nR Gamma Est's
LRM Gamm Est's
LR-PW Mixed Gamma Est's
PW-LR Mixed Gamma Est's
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
55 / 58
Monte Carlo Greeks for Variable Annuities
Conclusions from VA analysis
1. PW est’r has lower st.err. than CLRM est’r for VA Delta
This is expected, as the series of options in time which make
up the liability has a simple, continuous payoff
2. B&R & the CLRM estimators are both relatively inefficient
estimators for the VA Gamma. However, mixed estimators have
much lower variance than both of these estimators.
Similarly to Asian option, the CLRM estimator for Gamma
does not give as low st.err’s as for European options. (Uses
limited part of path in shock.) However, mixed estimator
again seems to provide the benefits from both approaches
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
56 / 58
Monte Carlo Greeks for Variable Annuities
Conclusions/Further Research
Conclusions from research to date
VA Greek estimators constructed are successful, in particular
the mixed second-order sensitivity estimators
Future Research
Variance Reduction. Control Variate, Importance
Sampling, Low-Discrepancy MC
Interest-rate sensitivities. wrt CIR param’s
More complex VA product. Two Equity Indices,
Equity/Bond mix, dynamic policyholder behaviour
Computational framework. GPU, Automatic Diff. ?
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
57 / 58
Monte Carlo Greeks for Variable Annuities
Bibliography
Glasserman, P., Monte Carlo Methods in Financial Engineering, 2003.
Springer-Verlag, New York.
Heston, S., A Closed-Form Solution for Options with Stochastic Volatility
with Applications to Bond and Currency Options, The Review of Financial
Studies, Volume 6 Number 2 pp. 327-343 (1993).
Cox, J.C., Ingersoll, J.E. and Ross, S.A., A Theory of the Term Structure of
Interest Rates. Econometrica 53 pp. 385-407 (1985).
Broadie, M. and Kaya, O., Exact Simulation of Option Greeks under
Stochastic Volatility and Jump Diffusion Models, Proceedings of the 2004
Winter Simulation Conference, The Society for Computer Simulation,
1607-1615.
Hobbs, C., et. al., Calculation of Variable Annuity Market Sensitivities using
a Pathwise Methodology, Life & Pensions, September 2009.
Monte Carlo Estimates of Variable Annuity ‘Greeks’
N
58 / 58