Accelerated Option Pricing
in
Multiple Scenarios
04.07.2008

Stefan Dirnstorfer (stefan@thetaris.com)
Andreas J. Grau (grau@thetaris.com)

1

Abstract
This paper covers a massive acceleration of Monte-Carlo based pricing method for financial
products and financial derivatives. The method is applicable in risk management settings, where a
financial product has to be priced under a number of potential future scenarios. Instead of starting
a separate nested Monte Carlo simulation for each scenario under consideration, the new method
covers the utilization of very few representative nested simulations and estimating the product
prices at each scenario by a smoothing method based on the state-space. This smoothing
technique can be e.g. non-parametric regression or kernel smoothing.

Background
The currently most resource intensive computational task in risk management is pricing of
financial products. Observable market parameters are simulated according to statistical properties
based on historical observations. The simulated scenarios represent potential future values for
which a financial institution has to prepare. Since each financial product in a large portfolio acts
differently to changes in market parameters each product's price has to be evaluated under each
scenario.
A number of regulatory provisions require financial institutions to perform a valuation of the
portfolio under each potential future scenario. The accuracy and speed with which such
evaluations can be performed is crucial to the companies financial success. As regulated by the
Basel II directive each bank has to back its risky investments with risk-less assets such as
government bonds. Since risky assets are generally expected to yield higher profits, banks are
interested in measuring each asset's risk as accurately as possible. If banks fail to produce accurate
risk figures, they are required to place large amounts of risk capital and consequently reducing the
banks ability to invest large amounts into a particular asset. While Basel II is focused on the
banking sector, similar regulations are planed for the insurance industry as proposed by Solvency
II.
The current Basel II regulation requires a portfolio to be evaluated under a number potential
future scenarios. For each scenario the development of all portfolio positions have to be
determined for several time steps. It is currently best practice to randomly simulate future market
parameters for each scenario and each time step. In a typical setting with 5000 paths and 250
time steps the total number of portfolio evaluations is given by 5000x250 = 1.25 million prices
for each financial product in the portfolio. Since banks have portfolios with thousands of financial
products, the computational challenge is enormous. There are a couple of financial products, such
as European options or futures for which extremely fast algebraic evaluations can be performed.
Other financial products can efficiently be priced by solving an associated partial differential
equation. Other products, especially those that depend on a large number of traded instruments,
e.g. basket options, can only be solved with a Monte Carlo approach. On prevailing computer
hardware with a single processor one single Monte Carlo evaluation can take several minutes to
compute and this is far to slow. The problem can only partially be mitigated through the use of
multi-processor machines, since this process has to be performed for each individual financial
asset in portfolio. The cost and size of today's computing hardware is already at a maximum in
many banks, while risks of many exotic derivatives are still largely overestimated. For banks with
a complete portfolio risk management system in place, the method of this paper will cut down the
required computing time dramatically and thus reduces costs for maintenance and energy.
2

The method proposed in this document yields a massive reduction in the computational effort
required to compute product prices in future scenarios as required e.g. for risk management. The
method is an amendment to Monte Carlo based pricing. As such it can be applied to the same
range of pricing problems, many of which can not be solved by other methods than Monte Carlo.
However the new method provides an unprecedented speed-up. A computation that previously
took months can be performed in a couple of seconds instead. Apart from being faster it is also
more accurate than a couple of previously suggested solutions.
The speed-up of the method of this paper stems from the fact that a nested Monte Carlo
simulation for each physical scenario for the risk assessment computes a lot of redundant
information. Creating a pricing function of the financial products by smoothing of the simulated
Monte Carlo values can be performed in a single step. Utilizing this pricing function instead of
nested Monte Carlo simulations avoids the computation of the redundant information and thus
accelerates the risk-management task.
Overall a bank will benefit from this method by the possibility to reduce its required risk capital.
Any financial institution doing risk analysis will benefit from this method. It provides higher
accuracy and reduces costs for performance computing hardware.

Previous solutions
Depending on the instrument type and the time horizon of the risk estimation, the risk
assessment of a product position in a portfolio can be conducted in several ways. In any case, risk
is measured by a characteristic number, e.g. value at risk (VaR), conditional value at risk (CVaR) or
standard deviation.
For a short time horizon, a risk estimate based on the sensitivities of the product with respect to
the underlying (“Delta” and “Gamma”) delivers fast and accurate results without the need of
simulation. However, this sensitivity-based approach fails to estimate risks accurately when the
remaining maturity time of the product is short or when the risk estimate for several weeks ahead
has to be computed.
Estimating the market risk for long time spans, a simulation of the risk factors has to be
conducted and at each time step of each scenario, the portfolio has to be evaluated. This is easy, if
a fast pricing method for the specific instrument type exists, e.g. an analytic solution for the price.
However, for many instrument types, only computational-expensive simulation methods exist,
especially for Basket or path-dependent options. The cost of a simulation of the risk factors and a
nested simulation for the product prices is in many realistic settings prohibitively high such that
different solutions have been proposed to mitigate this problem. The important solutions are:
1. Usage of variance reduction techniques in the nested Monte Carlo simulation. Many
variance reduction techniques have been proposed, e.g. control variables, low discrepancy
sequences and importance sampling [Glasserman 2003]. But, the speed-up using the
variance reduction techniques – typically between 2 and 10 - is by far not sufficient for a
nested simulation estimating market risk.
2. Portfolio compression, which creates a new portfolio with the same risk properties as the
considered portfolio but with fewer instruments [Dembo 1998]. This approach helps to
some extend by reducing the number of instruments to price, but this technique is often
not applicable to complex structured products.

3

3. Risk estimation by combinatorial scenario simulation, which effectively reduces the
number of physical scenarios in settings with many risk factors. For each of the s risk
factor only a small number n of physical scenarios is computed. Then the whole setting of
the physical simulation is created by computing all possible combinatorial combinations of
the riskfactors. Then, a Monte Carlo simulation along these precomputed risk factor
realizations is performed to estimate the portfolio risk. This reduces the required number
of option valuations considerably but shows slow convergence [Abken 2000].
4. Importance sampling, which computes scenario samples that are of particular importance
for the risk measure to estimate and shifts the scenario weights such that the estimate of
the risk measure is unbiased [Glassermann 2000]. This technique can improve the
accuracy of risk estimates considerably but the speed-up is often still not sufficient for
nested Monte Carlo simulations.
5. Usage of few paths is another method for mitigating the computational burden. It turns
out that if at each of the physical paths, a nested simulation with e.g. 100 paths is
conduced (where e.g. 10.000 would be needed for a precise option price estimate) can
already lead to sufficiently precise risk estimates. The reason for this is that the errors in
the option price estimations mutually annihilate almost completely and the few paths are
sufficient for estimates of the risk measure. However, the resulting risk-measure estimate
is biased and has to be corrected in a post process for precise estimates. [Gordy and Juneja
2008]

The new method
The task the method of this paper performs efficiently is defined as follows: Consider that a
financial product shall be priced under many different future market scenarios and at different
time-steps. Instead of starting a separate nested Monte Carlo simulation for each scenario under
consideration, the new method covers the utilization of one or more well selected nested
simulations. Based on the simulations, a function will be derived which maps the state of all
relevant variables onto a price estimate for the financial product. Such a function can be built
from e.g. non-parametric regression or kernel smoothing. In realistic settings, the overhead of
building this function is small and substantial performance improvements are gained: Simply
evaluating the estimated function is quicker than evaluation by nested Monte Carlo simulation,
even if the nested simulation uses only very few paths.

Overview
The algorithm for the pricing of a financial product in multiple scenarios comprises 7 separate
steps:
1. Importing scenarios
at which the product prices shall be computed.
Each element of
is associated with a scenario for risk-factors drawn at each time step
in
.
2. Generation of scenarios
for the product price estimation. is
computed by sampling of a stochastic process. Each element of
is associated with a
scenario for risk-factors drawn at each time step in .
3. Compute path-dependent product specific variables
corresponding to
4

4.

5.

6.

7.

the scenarios in step 1. These path-dependent variables include for example fixings or
exercises performed by the product's issuer or the holder.
For each scenario of step 2: Computation of product specific variables
corresponding to the scenario. These path-dependent variables include fixings and optimal
exercises.
Computation of the product's remaining discounted cash-flows
. At each
scenario and each time-step in the scenarios of step 2 all remaining cash-flows are
scenario-wise discounted and summed.
For each scenario of step 1: Computation of a price estimate function
. is obtained by a smoothing procedure on the scenarios of step
2 and the path-dependent variables of step 4.
Computation of the product prices for each scenario
and each time step
from step 1. This evaluation is performed efficiently by
.

Details of the Steps
In step 1, the scenarios consist of a realization of values for each risk factor which has to be taken
into account. Typical risk factors for a structured financial product are: prices of underlyings,
implied volatilities and long-term as well as short term interest-rates. In the following, we will
refer to the scenarios from step 1 as physical scenarios and all associated variables are denoted by
an index .
The origin of the scenarios in step 1 can be manifold: historical simulation, shifting of current
risk factor values and Monte Carlo simulation are possible choices. The scenarios can consist of a
single time step or multiple time steps. The particular choice depends on the specific result one
expects from the analysis. A multi-time step Monte Carlo simulation might be useful for the
computation of risk-measures such as Value at Risk while a single time step which a shift of the
risk factors is useful for stress testing and estimating the risk contribution of single instruments.
The physical scenarios are denoted by
numbering for the scenarios and
each time step, an -tuple of risk-factors is given.

, whereas

is a

is the set of time steps. At each scenario and

The scenarios of step 2 are used for the product valuation itself and it is useful to generate so
called risk-neutral scenarios for this task as defined by the option pricing theory. All associated
variables are denoted by an index . Examples for such scenarios are e.g. geometric Brownian
motion where the drift is set to the risk-free rate of interest and constant volatility as well as
geometric Brownian motion with Heston volatility [Heston 1993].
The scenarios of step 2 are denoted by

, whereas

is a

numbering for the scenarios and
is the set of time steps. At each scenario and
each time step, an -tuple of risk-factors is sampled from a stochastic model. Additionally, there
is a mapping
. For each scenario
, the scenario path
is called
active for
.
The set
contains all relevant time steps (fixings) for the evaluation of the financial product.
Furthermore, the algorithm works well when relevant physical scenario time steps are contained

5

as well, i.e.
product.

, where

is the maturity time of the financial

For step 2, an implementation to create the simulations with one or more of the following
properties can be beneficial:
a. the scenario paths start at the same time and the same value as the physical scenarios, i.e.
and
,
,
b. the scenario paths start at the same time and similar value as the physical scenarios, i.e.
and
,
,
c. each scenario path
forks a physical scenario at some time
, i.e.
and
d. each scenario path
forks at some time
in the proximity of a
physical scenario, i.e.
and
.
In step 3, for time step
and each scenario
the path dependent values
are computed. These -tuples together with the current risk factor values
must be sufficient to price the financial product at time step . The values at time
are given by an initialization function , i.e.
. These values are
part of the product's structural features and could be imported from a database. The successive
values at times
are computed from an update formula
, i.e.
for

.

Examples for path dependent variables
are
● information about knock-out of Barrier options,
● the current average of Asian options,
● exercise, conversion and calls of the financial product based on e.g. the investor's utility,
● measurable characteristic values w.r.t. the stochastic model in step 2,
● portfolio weights of dynamic strategies, e.g. Simulation-Based Hedging (Grau 2008),
● previous values of risk factors.
In step 4, similar to step 3, the path-dependent variables
for time step
and each scenario
. These -tuples together with the current risk factor values
must be sufficient to price the financial product at time step . The values at time
are initialized with appropriate values.
For an implementation to compute the initial path-dependent variables
use one of the following methods:

it can be beneficial to

a. If there exists at least one physical path that matches a scenario path at its first active time
step then we can use a physical path-dependent state as initial state, i.e,
,
then use
.
b. Alternatively, we can choose a physical path
which is similar to the scenario paths
at time
. Then, we initialize the path-dependent state
to be equal or
similar to
, i.e.
,
then use
where
is an artificial realization of the path-dependent
variables. Note that the new values should be consistent with the structure of the financial
6

product and possible path histories.
c. For each scenario path , we initialize the path-dependent variables from a synthetic path
with
. The synthetic path must have the same value as the
scenario path at time , i.e.
. Equivalent to the update formula in
step 3, we compute
through an iterative process.
In step 5, the product's remaining cash-flows are discounted to a cash value
.
Consider a discount factor
. For each scenario
returns the
function
the discount factor from time
to time . This function is constructed
knowing the full history of the path .
In each scenario the product-specific cash flows are given by
, i.e. in scenario
pays
at time . The cumulated and discounted remaining cash flows V are computed
by
.

The crucial part of the algorithm is step 6 where the product prices in each physical scenario and
each physical time-step are computed using the scenario paths from step 2. Consider subsets
and
. The set
is defined as a set of (X, Y)-pairs that can be used for
smoothing algorithms,

The operator computes a smoothing on a set of (X, Y)-pairs which results in a function
mapping risk factor tuples and path-dependent state tuples onto product prices, i.e
.
It is useful that creates an estimator for the conditional expected values
. Useful
smoothing algorithms for are:
a. Non-parametric regression sets the result function as a linear combination of basis
functions , i.e.
. The coefficients are determined by minimizing the
quadratic error
.
b. Kernel smoothing is defined by a sum of weighted
values, i.e.
, with a weight function
constructed from the location
of the

values.
7

Further information about the smoothing algorithms mentioned here and further smoothing
algorithms can be found at Härdle 2001. An interesting approach to non-parametric regression is
presented by Garcke et al 2001. Sometimes it is useful to select a subset of
before performing
one of the above smoothing algorithms. Furthermore, it also can be useful to use semi-parametric
regression, thin-plate splines or b-spline basis functions.
The function
allows the efficient evaluation of prices in all time-steps
and all physical scenarios. It can be constructed in one of the following ways:
a. The smoothing is done on all data at once, i.e.
b. The

smoothing

is

.

done

on
each
time
step
individually,
i.e.
. With only a single time step per smoothing,
regression methods benefit from decreasing the dimensionality by 1.

c. Other partitions of
and
might be useful to cut the large smoothing problem into a
set of smaller smoothing problem.
Finally, the product price is computed in Step 7. For each scenario
and each time step
, the evaluation is performed efficiently as
. The
price estimates
are computed within the stochastic model generating the scenario paths in step
2. Hence, this algorithm is an efficient way to compute product prices in physical scenarios based
on an arbitrary stochastic model.
It can be useful that the scenario paths
and the associated path-dependent variables
are
made persistent such that later computations of the smoothing function can be performed
efficiently. Another possibility for an improvement is make the smoothing function
persistent
itself such that later computations of product prices for new risk factor tuples
can be performed
efficiently. Then, is can be useful to refine the smoothing function
iteratively by computing
additional scenario paths on demand, based on an error estimate for the price generated at the
new risk factor tuples.

8

Numerical examples
The following section describes a detailed example of the computation of financial product prices
by Monte Carlo simulation in several physical scenarios using the method of this paper. In order
to present a concise and reproduce able example, we restrict the example to 3 physical and 5 riskneutral scenarios. It is easy to extend this small example to a realistic setting by adding more
scenarios and additional risk-factors.

Example
Consider a European call option with a strike price of 100 and a maturity time of 3 years. We use
the physical scenarios
and the time-steps
. Consider further the
possible values of the physical scenarios
for a stock price, which serves as the underlying of the
European option:

1

100

110

120

2

100

100

100

3

100

90

80

In each of these scenarios at each time-step the European option value shall be estimated by
Monte Carlo simulation. The 6 option prices for time steps
and
shall be computed as fast
and accurate as possible. Prior art would perform 6 completely separate pricing procedures. The
method of this paper only requires a single scenario set
of risk-neutral scenario paths
at time steps
starting at time
, i.e.
:

1

100.0000 211.7568

214.8651 106.2542

2

100.0000 112.9350

3

100.0000 154.1112

193.8189 221.6990

4

100.0000

90.2616

155.3396 121.7245

5

100.0000 174.4274

199.2726 258.4810

70.6952

70.8322

These risk-neutral scenarios are created using a stochastic model with geometric Brownian
motion for but other (risk-neutral) simulations are suitable, too. Now, we compute the payoff of
the option value at time ,
, which is equal to
for
all
because there is only a single cash-flow at maturity time and the risk-free rate of
interest is zero (
). Note that the option has no path-dependency, thus
is empty
and
. The values are:
9

1

6.2542

6.2542

2

0.0000

0.0000

3
4
5

121.6990 121.6990
21.7245

21.7245

158.4810 158.4810

In order to obtain estimates of the option prices at time

, we create the set

:

6.2542
0.0000
121.6990
21.7245
158.4810
Next, the smoothing operation
has to be applied to the data set
. Since option pricing is
performed in a Black-Scholes setting, option prices are given by conditional expected values
. Thus, estimates for the expected values are also estimates for the option price. Here, we
use a simple nonparametric regression in
.
is constant and will not be considered. The
smoothing functions is

where ,
and
are coefficients of polynomial basis functions. In a realistic setting, other
smoothing methods or other basis functions are useful, too. Now, we compute the coefficients as a
solution to the minimization of

for all risk-neutral scenarios
. This least-squares minimization is a standard problem
and can be solved by normal equations using the matrix
,

10

Now, the coefficients

,

and

can be obtained by

which leads to
=
=
=

-

651.7604
9.9033
0.0317

In order to obtain the approximations for the physical asset paths, we need to set-up the
corresponding matrix with one row for each physical scenario
.

This leads to

and the approximations of the smoothing function
by

are obtained for the first time step,

.

The result is the option price estimation

1
2
3

for each scenario

,

54.57
22.01
- 16.87

This is a very efficient way of estimating option prices. Note that 5 scenarios and 3 basis
functions are not sufficient for precise estimates. This simplified example results in a negative
price estimate for physical scenario 3.
11

Now, we can use the same scenarios to obtain the prices at time
of the physical scenarios. In
order to obtain estimates of the option prices at time , we create the set
:

6.2542
0.0000
121.6990
21.7245
158.4810
Again, we construct a matrix

for the regression

and solving

with the scenarios at time
=
=
=

-

leads to

137.9136
2.2651
0.0058

With the corresponding matrix for the physical scenarios

the option price estimates

1
2
3

evaluate to

49.77
30.17
5.90

12

Concluding this first example, we obtain for each physical scenario and each time-step, an option
price

1

110

54.57

120

49.77

2

100

22.01

100

30.17

3

90

-16.87

80

5.90

Extension 1
The example can be extended in several ways. First of all, we can change the example to utilizing
risk-neutral scenarios starting at different initial values, i. e.

1

80.0000

64.1116 115.4375

105.6911

2

90.0000

41.3639

72.6489

100.4893

3 100.0000

105.1411 103.5702

81.8529

4 110.0000

122.1953 137.8884

316.2593

5 120.0000

83.2175

87.7915

84.1920

These scenarios can be used in the exact same way as before. Using the method with such a riskneutral scenario set can lead to considerably higher precision of the option prices in extreme
physical scenarios.

13

Extension 2
Another extension of the example again considers the risk-neutral scenarios. In some settings it is
beneficial to create additional scenarios in the risk-neutral setting at the exact value of the
physical scenario, i. e.

1

100.0000

211.7568

214.8651 106.2542

2

100.0000

112.9350

3

100.0000

154.1112

193.8189 221.6990

4

100.0000

90.2616

155.3396 121.7245

5

100.0000

174.4274

199.2726 258.4810

6

110.0000

139.2342 149.1234

7

100.0000

78.9872

90.2324

8

90.0000

98.9079

78.2347

9

120.0000

98.8968

10

100.0000

76.2563

11

80.0000

87.2342

70.6952

70.8322

The scenarios 6 to 11 are added to the scenario set in order to fit the physical scenarios 1 to 3.
Similar to the first extension of this example, the utilization of the method of this paper can
ensure higher accuracy for extreme scenarios. Note that scenarios 9-11 are not utilized for the
pricing at time-step .

14

Extension 3
A third extension to the example is required for the pricing of a path dependent option. Consider
an Asian option, which has a payoff depending on the average asset price until maturity time of
the option. This means that the current average
must be computed for the physical as well as
the risk-neutral simulations. For the physical scenarios
is given by

1

100

105

110

2

100

100

100

3

100

95

90

and for the risk-neutral scenarios

is given by

1

100.0000

155.8784

175.5406

158.2190

2

100.0000

106.4675

94.5434

88.6156

3

100.0000

127.0556

149.3100

167.4073

4

100.0000

95.1308

115.2004

116.8314

5

100.0000

137.2137

157.9000

183.0452

6

100.0000

105.0000

116.4114

124.5894

7

100.0000

100.0000

92.9957

92.3049

8

100.0000

95.0000

96.3026

91.7857

9

100.0000

105.0000

110.0000

107.2242

10

100.0000

100.0000

100.0000

94.0641

11

100.0000

95.0000

90.0000

89.3085

Note that the values for
at time
in scenarios 9 to 11 can be obtained directly from the
physical scenarios. This ensures that the added scenarios are consistent with the other scenarios
and that they are still increasing the numerical accuracy of the prices in extreme scenarios.

15

Computing the payoff

1
2
3
4
5
6
7
8
9
10
11

58.2190
0
67.4073
16.8314
83.0452
24.5894
0
0
7.2242
0
0

In oder to create a simple regression method we set up our data set as
and
. We compute the regression

in order to estimate
At time step

, whereas

, i.e. the option prices in each scenario.

, this leads to the coefficients

= 0
= 0
= - 0.0510
= 0
= - 0.0739
= 0.1257
and thus to the option price estimates of

1
2
3

20.28
8.24
- 5.13

The option prices

can be obtained correspondingly.

16

Conclusion
This paper outlines a technique for estimating option prices for many scenarios by an efficient
function smoothing method on risk-neutral Monte Carlo simulations. The resulting technique
has much lower computational complexity by avoiding nested Monte-Carlo simulations.
Numerical examples show that the new technique can be applied in realistic settings and that the
gain in speed is substantial. Compared with previous techniques, this new technique can be
applied in almost any setting, where option pricing by Monte Carlo simulation is applicable. The
previously proposed solutions like nested Monte Carlo simulation with few paths or valuation by
solving partial differential solutions (PDEs) are applicable only in much more restrictive settings.

References
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http://www.gloriamundi.org/picsresources/pa.pdf
R. Dembo et al 2001, Computer-implemented method and apparatus for portfolio compression,
US Patent 6278981
S. Dirnstorfer, A. J. Grau, R. Zagst 2006,
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grau@ma.tum.de
J. Garcke, M. Griebel and M. Thess 2001, Data mining with sparse grids, Computing, 67,
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17

