Download Bridge Between Real World Simulations and Risk Neutral

Deflator: Bridge Between Real
World Simulations and Risk
Neutral Valuation
The importance of market consistent valuation has risen in recent years throughout the
global financial industry. This is due to the new regulatory landscape and because banks and
insurers acknowledge the need to better understand the uncertainty in the market value of
their balance sheet.
The balance sheet of banks and insurers often include products with embedded options,
which can be properly valued with standard risk neutral valuation techniques. Determining
the uncertainty in the future value of such products (for example needed for regulatory or
economic capital calculations) is more difficult, because when using risk neutral valuation,
future outcomes are not simulated based on their historical return. For example, when using
risk neutral simulations, stock prices are assumed to grow with the risk-free interest rate,
which is not realistic.
an interest rate is used to show some results
based on this framework and using real market
data.
Pieter de Boer
is
HWBS model
Using real-world simulations, variables are simulated based on their historical return, stock
prices are chosen to grow at the actual expected return (the risk free rate combined with a
risk premium). The valuation of a product using
a ‘standard’ risk-neutral discount factor is inconsistent, since the returns are not risk-neutral in this case.
This article discusses the combination of these
two methods in order to simulate future outcomes based on the actual expected return and
still valuate products market consistently. Real
world simulations are needed to simulate future
values of the variables based on their historical
return and a stochastic discount factor (SDF),
called the ‘deflator’, is needed to calculate the
market value of these products. The uncertainty in future market value is estimated by combining these methods.
In the next two sections a Hull White Black
Scholes (HWBS) model is used to demonstrate
how a deflator can be determined and incorporated in a HWBS framework. An example product with a payout based on a stock return and
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In this article, the one-factor Hull White (HW)
model is used to simulate interest rates. The
HW model is chosen because it incorporates
mean-reverting features and, with proper calibration, fits the current interest rate term structure without arbitrage opportunities (Rebonato,
2000). Furthermore, an appealing future of the
HW model is its analytical tractability (Hull &
White, 1990).
Stock prices are simulated with a Black and
Scholes (Black & Scholes, 1973) based Brownian
motion that is correlated with the HW process
using a Cholesky decomposition.
Assume a probability space (Ω, F, F, Q), where
Ω is the sample space, Q is the risk neutral probability measure, F is the sigma field and F is
the natural filtration {Ft}0≤t≤T. Suppose the interest rate is also an F-adapted random process.
The HW model for the process of the short rate
under a risk neutral probability measure can be
expressed as in equation 1, where a and σr are
constants, WrQ is a Wiener process for the interest rate and θ(t) is a deterministic function,
chosen in such a way that it fits the current
term structure of interest rates. The process for
the stock price is shown in equation 2, where ρ
indicates the correlation between both processes and WsQ is a Wiener process for the stock
price.
GUW = LJW − DUW GW + ı U G:WU4 (1)
G6W = UW 6W GW + ı V 6W ǏG: U4 + ı V 6W − Ǐ G: V4 (2)
When simulating these processes under Q, the
present value of a product can be determined,
since the proper discount factor is known to be
the risk free interest rate.
Under the assumption of a different probability
space (Ω, F, F, P), where Ω is the sample space,
P is the real world probability measure and F
is the natural filtration {Ft}0≤t≤T, the process
for the interest rate and the stock price can be
written as.
U3
W
U3
(3)
GUW = NjU − DUW GW + ı U G: G6W = NjW 6W GW + ı V 6W ǏG: + ı V 6W − Ǐ G: V3 (4)
Where μr is the historical mean for the interest
rate and μs is the expected return of the stock
price, which is equal to the expected return under a risk neutral probability measure plus a
market risk premium (πs).
Stochastic discount factor
When simulating these processes under the
real world probability measure P, the value of
a product is more difficult to determine, since
the risk free interest rate is not the proper discount factor anymore. Discounting with the risk
free interest rate under actual expected returns
would not lead to a market consistent value.
To find a proper stochastic discount factor under the real world probability measure P, suppose X is a F-measurable random variable and
the risk neutral probability measure is Q. L, the
Radon-Nikodym derivative of Q with respect to
P (Etheridge, 2002), equals
L = dQ/dP
(5)
and
P
Lt = E [L|Ft]
(6)
For equivalent probability measures1 Q and
P, given the Radon-Nikodym derivative from
equation 5, the following equation holds for the
random variable X (Duffie, 1996)
Q
P
E (X) = E (LX)
(7)
and
EQ[Xt|Ft] = EP[XtLT/Lt|Ft]
(8)
It can be seen from the above equation that the
expectation of X under the probability measure
Q is equal to the expectation of L times X under
the probability measure P.
Furthermore, suppose {Wt} is a Q-Brownian
motion with the natural filtration that was given
above as {Ft}. Define:
W
/W = H[S− ³ LJV
G:V3 −
W LJVLJVGV
³
(9)
and assume that the following equation holds
ƪ>H[S
7 LJW GW @ < ∞ ³
(10)
where the probability measure P is defined in
such a way that Lt is the Radon-Nikodym derivative of Q with respect to P. Now, it is possible
to use the preceding to rewrite equations 7 and
8 to link risk neutral valuation and valuation
under a real world probability measure:
7
ƪ4 >H[S− ³ UVGV; 7 _ )W @
W
7
= ƪ3 >H[S− ³ UVGV −
W
³
7
W
LJV
G:V −
³
7
W
( 11)
LJV
LJVGV; W _ )W @
Combining the above equations and using
Girsanov’s theorem (Girsanov, 1960) states
that the process
:W4 = :W3 +
³
W
LJV GV
(12)
is a standard Brownian motion under the probability measure P. A useful feature of this
theorem is that when changing the probability
measure from real world to risk neutral, the volatility of the random variable X is invariant to
the process. In changing from a risk neutral to
a real world probability measure, it is essential
to make WtP a standard Brownian motion.
SDF in HWBS model
Now, according to the above theory, it is possible to change from probability measure P to
probability measure Q. For this, it is sufficient
to find θs from equation 12. This leaves the following two equations:
:WU4 = :WU3 +
³
V
G:WȺ4 = :WV3 +
LJVU GV
³
V
LJVVGV
(13)
By choosing a proper value for LJVU the substitution of the first part of equation 13 into equation
1 should be equal to equation 3. By solving this
inequality, LJVU is found to be:
LJVU = (μr-θ(t))/σr
(14)
1
Q and P are equivalent probability measures when it is provided that Q(A) > 0 if and only if P(A) > 0, for any
event A (Duffie, 1996).
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Something similar can be done to compute LJVV .
With this knowledge, substituting the second
part of equation 13 into equation 4 and solving
yields:
ȺV
G:WV4 = G:WV3 +
−
ı V − Ǐ
Ǐ
GW
(15)
G:WU4 − G:WU3 − Ǐ
Figure 1: Development of the AEX-index under
both probability measures
Which results in:
ª
ªLJVU º «
« V» = «
¬LJV ¼ « − Ǐ
¬
º
»
»
− Ǐ ¼»
−Ǐ
§ NjU
¨
¨
¨
¨¨
©
− LJW ·
¸
ıU
¸
¸
ȺV
¸¸
ıV
¹
(16)
Assuming that equation 10 holds, which is a requirement, the stochastic discount factor in the
BSHW model can be written as:
7
6') W 7 = H[S− ³ UVGV −
W
− ³
7
W
LJVV LJVV GV −
³
7
W
³
LJVU G: U3 − 7
LJVVG: V3
W
³
7
W
LJVU LJVU GV
(17)
Example
Using the theory described in the previous section, the value and the uncertainty in the future
value of a theoretical product are calculated.
The following guaranteed product is chosen;
the client receives the return on the AEX-index
unless the return is below the 1 month Euribor
interest rate, in that case the payout is equal to
the 1 month Euribor interest rate. These types
of products are common on the balance sheet of
insurers and due to the complex payout structure, a simulation model is needed to evaluate the value of such a product. Therefore, the
HWBS framework using a stochastic discount
factor is suitable to value this product and calculate risk figures for this product.
First, the value of the product on two different
dates is calculated in a standard risk neutral
setting. This value is compared with the value
resulting from the real world simulations and
the use of the stochastic discount factor. See
the insert for the expectations and variances
that where used for the risk neutral processes.
For the stock price, the volatility was based on at
the money (ATM) options with a time to maturity of one year. The mean reversion parameter
and the volatility in the HW model were calibrated using a set of ATM swaptions. The average
1-month interest rate μr is chosen to be 4,27%
based on historical data. Furthermore, the risk
premium, πs, is fixed at 3%.
In figure 1, the result of running 10,000 simulations of the (1-month) interest rate and the
stock price is shown. The history and a forecast
for the next 3 years, including the boundaries of
a 98% confidence interval (CI) of the AEX-index
are shown, under both probability measures.
As can be seen, the average predicted value
Risk neutral expectations and variances
Interest rates
ƪ4 >U W _ )V @ = U V H − DW − V + ĮV H − DW − V ı
9DU 4 >U W _ )V @ = U − H − DW − V D
(18)
(19)
Stock
(4 >OQ
67 − H − D ƩW
I 0 7 _ )V @ =
[W − ı V ƩW + OQ 0 W + 9DU 4 >U W _ )V @
6W D
I
ı
= U >ƩW − H − D7 − H − DW −
H − D7 − H − DW @
D
D
D
where:
[W UWI 0 W −
9DU 4 >OQ
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(20)
ı U
− H − DW D
ı
Ǐı V ı U
67 − DƩ W _ )V @ = U >ƩW + H − DƩW −
H
−
@ + ı V ƩW +
ƩW − − H − DƩW 6W D
D
D
D
D
D
April 2009
(21)
Figure 3: Implied volatility of the AEX-index
Figure 2: Results of the backtest
rence should be corrected for the actual expected change, since the time to maturity of the
product has declined at t=1. The 95% VaR is
defined as the difference between the average
market value at t=1 and the 5% boundary of
the 90% confidence interval of the market value at t=1. The results are given in table 2.
Whether the value of the product in one year is
estimated correctly can be tested by using the
method of backtesting.
of the AEX-index has a smooth course, but the
width of the confidence interval shows that the
predicted values of the index are in fact rather
volatile. As expected, under the real world probability measure the index increases faster on
average.
The market value of the product can be estimated by calculating the future payout in each
scenario and calculating the average of the discounted value over all scenarios. The market
"Quote"
value of the product and the boundaries of the
90% confidence interval are shown in table 1.
As expected, the market value is similar under
both probability measures on both calculation
dates. The (minor) differences can first be explained by the fact that a different set of simulations is run for both methods. Second, a discrete approximation for a part of the stochastic
discount factor had to be made in order to use
it in the stochastic simulation model.
The higher average value of the product, when
valued at the 29th of August in 2008, results
from the rise of the volatility of the stock price.
The recent increase in the implied volatility can
be related to the ‘credit crisis’.
Next to this, it is interesting to examine the risk
an insurer runs by holding this product on its
balance sheet. Since the insurer sold the product, the risk arises from a value increase of this
product. As a measure for this risk, the Value at
Risk (VaR) of the product is estimated.
The 95% Value at Risk (VaR) of the product can
be calculated by examining the difference between the market value of the product and the
market value of the product at =1. This diffeDate
Backtest
To examine the forecast capabilities of the model, the results can be tested by performing a
backtest. Both models are used to predict the
value of the product in one year. However, it is
difficult to collect enough observations and therefore, a one year rolling window is used.
The dataset starts in May 2003, which leaves 51
observations available for the backtest. In all of
these 51 observations, it will be tested whether
the actual value of the product lies outside the
90% confidence intervals of the predicted value, generated by both models. The results of
the backtest are shown in figure 2.
What can be concluded from figure 2, is that in
particular the observations in the last year of
the dataset fall outside the predicted confidence intervals. In total 15 of the 51 observations,
lie outside the predicted 90% confidence interval of the real world model. These results can
be mainly attributed to the rise in the implied
volatility due to the turbulent market conditions
from May 2007 on, which can be seen in figure
3.
Risk Neutral
Real World
Market value
5% LB
95% UB
Market value
5% LB
95% UB
30/6/2006
28.2
-30.2
-131.0
27.5
-30.5
127.2
29/8/2008
57.2
-19.0
188.1
56.4
-17.8
181.6
Table 1: Average market value and the boundaries of the 90% CI under both measures
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Date
Risk Neutral
Real World
Expected
market
value in
1 year
5% LB
95% UB
VaR
Expected
market
value in
1 year
5% LB
95% UB
VaR
30/6/2006
28.1
-34.5
-20.2
6.4
28.4
35.6
21.2
7.2
29/8/2008
57.4
-67.2
44.1
9.8
55.7
-66.5
45.0
10.7
Table 2: Risk figures for the product under both measures
Whether the model passes the backtest can be
calculated in a likelihood ratio testing framework (Christoffersen, 1998). In this framework,
suppose that ^,W `7W = is the indicator variable for
the interval forecast given by one of either models, which means that whenever It=1 the actual value lies in the interval. The conditional
coverage can be tested by comparing the null
hypothesis that E[It]=p with the alternative hypothesis that E[It]≠p.
The likelihoods under the null hypothesis and
under the alternative hypothesis are given by:
/ S , , , = S [ − SQ − [
/Ⱥ , , , = Ⱥ [ − ȺQ − [
(22)
On the other hand, when these tests are performed for the model under a risk neutral probability measure, both tests result in a rejection of
the model, see table 3.
So, even when the data until May 2007 are
used to backtest the model under a risk neutral
probability measure, it is rejected as accurate.
This is unlike the model under the real world
probability measure. This evidence suggests
that the risk of the guaranteed product might
be estimated better using the model under the
real world probability measure using the stochastic discount factor.
Conclusions
The objective of the this article was to link real
world simulation to risk neutral valuation and
thereby investigating if it is possible to improve
the estimation of uncertainty in future market
value. To be able to determine this, a HWBS
framework in combination with a stochastic
discount factor (SDF) was used. The SDF, also
called deflator, was needed for proper valuation
using real world simulations. In an example
based on real market date using this framework
this method was tested.
The most important conclusions that can be
drawn from the results and the backtest are:
l
Where the maximum likelihood estimate of Ⱥ
is
x/n, the number of values outside the interval
forecast divided by n, the total number of observations. Using these likelihoods, a likelihood
ratio test for the test of the conditional coverage can be formulated
/5 FF = − ORJ
/ S , , , a ǒ l , , , /Ⱥ
(23)
Where the test statistic is actually asymptotically Chi-Squared distributed with s(s-1) degrees
of freedom, with s=2 as the number of possible outcomes. It is difficult to take the autocorrelation (due to the rolling window) into account. Therefore, the resulting conclusions are
less reliable. In this case, the LR-test statistic
is 14,4, significantly higher than the 0,10 from
the (5%) confidence level of the Chi-squared
distribution, what justifies the conclusion that
the model is inaccurate.
However, the recent crisis is a very unexpected
event. If data from May 2003 until May 2007
are only taken into account, the backtest would
have a totally different outcome. The LR test
statistic for this dataset is 0,05, which would
lead to not rejecting the model, as opposed to
a rejection taking the data from May 2007 until
August 2008 into account.
Date
Real world
Risk neutral
1% critical value
5% critical value
Until August 2008
14.4
23.6
0.02
0.10
Until May 2007
0.05
1.67
0.02
0.10
Table 3: Results of the backtest for both models
36
• Valuation under the real world probability
using a stochastic discount factor results in a
market value that is consistent with the risk
neutral value. The main advantage of using
real world simulations is that the simulations
can also be used for a ‘realistic’ simulation of
random variables.
• Combining the real world simulations with a
stochastic discount factor is very useful for
banks and insurers. They can use this method to estimate the current value of their
products and, more importantly, estimate the
uncertainty in this value in one year in a consistent way. This can be used in regulatory
(e.g. Basel II or Solvency II) and economic
capital calculations.
• Capital calculations are typically based on a
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one year 99% VaR. When using real world
simulations and a standard discount factor,
estimated average values are inaccurate,
therefore, resulting VaR calculations can be
as well. When using risk neutral valuation to
estimate the VaR, only current market conditions are taken into account. Current market
conditions are not necessarily a good measure for future outcomes, which could also lead
to inaccurate VaR estimations.
However, some drawbacks of the model must
be noted.
• The model under the real world probability
measure, using the SDF, did not pass the
backtest. The null hypothesis that the model
correctly predicts the uncertainty in the future value is rejected. The failure of the model
in the backtest needs to be taken seriously.
However, as already mentioned, the market
conditions in the last period of the sample,
are quite unusual. Whenever the dataset is
cut off at May 2007, the model passes the
backtest unlike the model under a risk neutral probability measure. Of course, doing this
would be a case of data mining, but it does
not alter the fact that the current market
conditions are difficult to take into account. It
could be defined as an outlier, some theories
state that the recent crisis is comparable to
the crisis in the twenties.
• Two variables, the stock price and interest
rate, are modelled stochastically. When more
variables are modelled stochastically, the
SDF becomes more complicated. For banks
and insurers, who also model variables like
exchange rates and volatility stochastically,
several more random variables enter the
model. As the results have shown, the value
of the product greatly depends on this input
and modelling this input as a random variable
could help to improve the forecasting qualities of the model. However, this would make
the model and the SDF more complicated and
less practical.
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