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Success Stories
Financial Modeling –
Monte Carlo Analysis
Monte Carlo analysis is a cornerstone for implementing financial
models in the industry. These simulations have many advantages,
including the ease of implementation and applicability to multi-dimensional
problems commonly encountered in finance. Option pricing can
be represented as expectations and Monte Carlo analysis is very
attractive and commonly used. An example is an Asian Option
Call, which is a financial contract dependant on the average
security price over discrete dates in the future. The asset
price at some time, t, in the future follows the classic
Black-Scholes model as follows
where r is the risk-free rate of return, ó is the volatility
of the asset price and dWt is the increment of standard Brownian
motion. The price of this option is a function of the strike
price, K, and option maturity, T, shown as
follows
where the average asset price is
The combination of these equations does not have a closed form
solution. To solve this pricing problem, a Monte Carlo simulation
is used.
However, for risk management, hedging, and stress testing of
a portfolio, the price sensitivity as a function of changes
to model inputs, the "Greeks" as they are commonly known, becomes
quite valuable. One Greek of interest is vega; the option price
sensitivity to changes in the securities volatility, which is
as follows
The price and vega calculation using Monte Carlo techniques
is very time consuming for several reasons. For simulation
accuracy, many Brownian motion trajectories are used for price
determination. For each option simulation, there are several
contract dates during the option maturity; monthly dates for
an annual contract. In addition, an accurate picture of price
volatility is achieved by rerunning the simu lation w ith
many different values for the volatility, ó. The option trader
faced with minimizing risk to their client-base and portfolio
may look to have this price and volatility analysis before
making trades or in post-closing analysis. For very large,
multi-commodity portfolios, analysts frequently wait hours
for model simulations affecting their ability to respond in
real-time to a dynamic market or to complete risk analysis
before the next day of trading.
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With Star-P, analysts can quickly take advantage of the power and
speed of task parallel computation to implement Monte Carlo analyses.
By maintaining existing Monte Carlo MATLAB®® code, quantitative
analysts can parallelize their simulation with minimal code modification
avoiding the need to recode in C/C++ or FORTRAN with MPI extensions.
An example of Asian Option Pricing using MATLAB® is shown in Figure
1. This Monte Carlo function is used to analyze price volatility
by initializing the option maturity, T, the range for the
pricing volatility, ó (sigma in Fig. 1), the number of
contract dates within T, and the number of trajectories, Num_Traj,
to solve, which determines the approximation error. Larger Num_Traj
results in smaller Monte Carlo approximation errors.
To run the MATLAB® analysis in serial, the analyst can simply loop
over the dao function from Figure 1.
Simple Task-Parallel Computing
In the MATLAB® environment, the analyst gains ready access to task
parallel computation by calling the Monte Carlo function directly
by using the ppeval command. With the following simple change to
the serial MATLAB® code, replace the serial loop above, performance
gains approaching the number of parallel processors can typically
be achieved.
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 Summary
and Metrics
- Large multi-commodity portfolios can
be rapidly simulated, enabling real-time trading response to market
dynamics
- Minimal modification of MATLAB® code
to take advantage of powerful parallel computing
- High-speed parallel processing, scalability
approaching the number of processors
- No need to program in C/C++ with MPI
to take advantage of parallel computation
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