Option price as a path integral

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Lecture 2 Monte Carlo method in finance

• Option price as a path integral
• The Monte Carlo method
• The Monte Carlo method applied to option pricing
• Random numbers generation
• Monte Carlo Variance reduction techniques
• Low-discrepancy sequence

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The option pricing as path integral (I)

• The price estimate of an exotic option can be
  reformulated as a path integral (where the paths
  consist of all possible future evolution of
  underlying equity prices).

• Within risk-neutral probability measure, the
  option fair value, f, can be expressed as
• where fp is the option pay-off for a given path,
  Ê is the expectation value in a risk neutral
  world, T is the time to maturity and r is the
  risk neutral interest rate.

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The option pricing as path integral II

• The path integral formulation of option pricing
  problem shows up its intrinsic probabilistic
  nature.
• In Physics there are many powerful computational
  methodologies to compute path integrals (e.g.
  among the others the Monte Carlo method).
• The application of Monte Carlo method to
  finance/option pricing, consists to generate a
  sufficiently high number of estimate of fp in
  order to compute its mean value (and also
  standard deviation).

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The Monte Carlo method

• The Monte Carlo method consists to
  formulate/solve the problem as a series of
  sampling, generated according to some random
  numbers generator, from which ones can extract an
  estimate of the expected value. The standard
  deviation can be used to evaluate the error.
• In other words Monte Carlo is a stochastic
  methodology based on random numbers generation.

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Problems that can be solved by resorting to Monte
Carlo

• 1. Problems with an intrinsic probabilistic
  nature, i.e. problems which involve stochastic
  phenomenon. E.g.
• the option pricing
• the Value at Risk (VaR) estimation of a financial
  portfolio.
• 2. Problems with an intrinsic deterministic
  nature, i.e. problems which do not involve any
  stochastic variables but where the solution can
  be equivalently rewritten in term of an expected
  value of a function of some non deterministic
  variables.
• Integral calculation in D dimension

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Example Monte Carlo and the calculation of p (I)

• An extraction from a sample of stochastic random
  numbers can be used to estimate an integral

This integral can be viewed as the expectation
value of a function (f) of two random variables,
uniformly distributed in the interval -1, 1x
-1, 1
Uniform probability density function.
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Example Monte Carlo and the calculation of p (II)

• The integral value con be estimated as the
  arithmetic average of N values of f(xi yi), where
  each pair (xi yi) is randomly sampled according
  to a uniform distribution in -1, 1x -1, 1.
  I.e.

is an estimator of Ip/4.

• An estimation of error can be derived by
  resorting to the central limit theorem, which
  states that the sum S_n of n independent and
  identically distributed (i.i.d.) random variables
  (having mean m and finite variance s2) is well
  approximated by a gaussian distribution with mean
  m and variance s2/n as n (the sample size)
  increases.

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Error scaling in Monte Carlo simulations

• In Monte Carlo simulations the error scales as
• This behavior is independent from problem
  dimension.

• The error involved by applying lattice method on
  integral calculation in D dimension, scales as

• The Monte Carlo error is independent from
  dimension, making Monte Carlo one of the most
  powerful numerical methods for high dimensional
  problems (typically more than four).

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Monte Carlo method for pricing an option

• STEP 1 Define a stochastic process for the
  underlying
• STEP 2 Calculate multiple scenarios by
  repeatedly sampling the possible path evolution
  for the underlying equity.
• First of all divide the option life into m time
  interval of length ?t.
• Compute the future (stochastic) values Si at time
  ti i ?t until the option maturity T is reached.

• The simulation process of a single path, requires
  the generation of m independent random numbers
  sampled from a gaussian distribution.

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Monte Carlo method for pricing an option (II)

• STEP 3 Evaluate the pay-off (i.e. the premium
  paid at maturity) under each scenario (equity
  path).
• STEP 4 Compute the (discounted) mean value
  (i.e. the option price!) and its error, basing on
  the above distribution.

Stochastic process for the underlying
Probabilistic distribution of discounted
pay-offs. Compute mean (option price) and
standard deviation (gt error).
Scenarios
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Monte Carlo method for pricing an option (III)
Indeed, for a log-normal process, instead to use
Eulero discretization procedure one can
resort to a closed form solution to simulate the
stock evolution within an arbitrary long finite
time interval Dt
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Problems to face in random number generation

• The Monte Carlo method is based on sampling
  random numbers.
• Is it possible for a computer (i.e. a
  deterministic machine) to generate true random
  numbers (that is real stochastic objects)?
• The answer is negative!
• Indeed there are only three possible solutions
• True random numbers
• they are numbers that lack any pattern and are
  generated by resorting to physical phenomenon
  that is expected to be intrinsically
  unpredictable (e.g. radioactive decay, thermal
  noise etc.). In such a case a specialized
  hardware or some database are required, being the
  use of truly random numbers not practical.
• Pseudo-random numbers
• they are numbers produced by a computer using
  computational (deterministic) algorithms. Being
  such algorithms deterministic, the sequences they
  produced are only apparently random, which in
  fact are completely determined by an initial
  value (known as the seed).
• Quasi-random numbers
• they are numbers generated according to a
  deterministic algorithm. In such a case, however,
  the sequences produced have a definite pattern
  that fills in gaps uniformly. As a result, the
  statistical error is lower than that obtained
  with pseudo random numbers.

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Algorithms to generate pseudo random numbers.

• The pseudo random number generators are usually
  based on linear congruential generator
• This generator produces integer numbers in the
  interval 0, m. In order to obtain random
  floating point numbers uniformly distributed
  between 0 and 1, we set

The parameters a, b and m characterize the
generator quality. a multiplier,
b increment, m module.
Pros It is very fast Cons The random number
sequence will repeat the pattern after some
trials (the so called period). The period is
always lower than m.