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What is the Monte Carlo method? | Monte Carlo Simulation in Finance | Pricing Options - YouTube
Channel: Patrick Boyle
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Hi my name is Patrick Boyle. Welcome back
to my YouTube channel where we learn all
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about derivatives and quantitative
finance. If this is the first video
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you're watching make sure that you click
the subscribe button to see more content
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like this. By the end of this video
you'll know what the Monte Carlo method
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is, how it works, how it differs from the
traditional methods for pricing options
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and when should you use it and when
should you not. Monte Carlo methods are
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used in finance to value and analyze
complex financial instruments by
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simulating the various sources of
uncertainty affecting their value and
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then determining their average value
over a large range of resultant outcomes
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or paths. Okay so what does that mean? it
means that rather than using a lot of
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financial theory to price complex
financial instruments we instead build a
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computer simulation of the moving parts
and see with enough runs of the model
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what the fair value of the financial
instrument is. The Monte Carlo options
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pricing method was developed by Phelim
Boyle in 1977. It's often considered a
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method of last resort by market
practitioners. The approach is
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particularly useful in the valuation of
options with multiple sources of
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uncertainty or with complicated features
which would make them difficult or
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impossible to value through a black
Scholes style partial differential
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equation or through lattice based
approaches like the binomial tree. The
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technique is widely used in valuing
path-dependent structures like look back
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and Asian options and in what is known
as real options analysis. I'll put a link
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above to my video explaining what real
options are above. Historically Monte Carlo
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methods were considered to be too slow
to be competitive, but with the faster
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computing capability today this
constraint is way less of a concern. Okay
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so how does it work well let's start
with an idea taken from gambling and
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probability. If you had two dice that you
were going to roll, the range of results
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that you could get would be any
number between 2 and 12. The lowest
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number 2 is what you get if you roll two
ones and the highest number 12 comes
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from rolling two sixes.
Now obviously with one dice you can get
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anything between 1 and 6 and the
probability of any of those outcomes is
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equal, it's 1 over 6 which is a 16.7% chance, however with the two dice
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you are most likely to roll a 7. Let me
explain. This is because there is only
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one way to roll a 2 and one way to roll
a 12 where in order to roll a 2 you have
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to roll two ones and in order to roll a
12 you have to roll two sixes. Now in
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order to get a score of four for example
with two dices can be achieved by
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rolling a 1 and a 3 a 2 and a 2 or a 3
and a 1. You have to consider the two dice
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separately, so even though the result is
the same a 1 on the first die and a 3 on
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the second die is actually a different
outcome from a 3 on the first die and a
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1 on the second die. For two dice seven is the
most likely result with 6 different ways
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that we are able to achieve that. In this
case probability equals 6 divided by 36
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giving a 16.7% chance. You can see a
histogram on screen right now showing
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the probabilities of each roll. So why am
I telling you all of this? Well, because
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there's two ways that we could work out
these probabilities. One is the
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mathematical way that I just described,
where we analyzed the problem and worked
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out numerically what the answer was, but
the other simpler and maybe more
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brute-force approach would be quite
simply to get two dice and roll them 5000
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times for example and note down the
numbers that we've rolled each time. The
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Monte Carlo model uses that type of a
pproach the brute-force approach and uses
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it to price derivatives the Monte Carlo
method involves simulate
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the underlying process followed by the
various risk factors affecting the price
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of the derivative that we are trying to
price first you generate a price path
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for the underlying based upon the random
movements of the various risk factors
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and calculate the payoff from the
derivative based on that path the image
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you see on screen right now shows you
what the simulated price paths might
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look like then you repeat these steps
generating numerous sample values of the
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payoff from the derivative in the future
once you have done that you simply
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calculate the average of the sample
payoffs giving an estimate of the
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derivatives expected payoff in a risk
neutral world finally you discount the
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payoff at the risk-free rate this result
is the fair value of the option today
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the number of iterations carried out is
at the discretion of the operator and
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depends on the required accuracy it is
usual to calculate the standard
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deviation of the discounted payoffs
generated by the simulation the
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uncertainty about the value of the
derivative is inversely proportional to
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the square root of the number of
iterations that you run so when do you
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use the Monte Carlo method the Monte
Carlo method can have great flexibility
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complex stochastic processes including
jumps mean reversion or both can be
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accommodated and different distributions
including changing distributions can be
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assumed the Monte Carlo method is
generally used when there are three or
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more stochastic variables which make
using a PDE or a lattice based approach
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extremely difficult or impossible Monte
Carlo in these situations can be more
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efficient than other approaches as the
time taken to run a Monte Carlo
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simulation increases in a linear manner
with the number of variables or for most
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other methods the time taken increases
exponentially with the number of
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variables the Monte Carlo method is a
brute-force approach
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pricing options it does not rely on a
lot of financial theory it simply uses
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computer power to simulate thousands of
possible price paths for the underlying
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the Monte Carlo method is by no means
free of assumptions though you always
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have to assume a distribution for the underlying asset as well as structure for
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its volatility and the absence or
existence of jumps the beauty of the
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Black-Scholes and lattice based
approaches is that they not only give
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you a fair value for the option but they
also specify a trading strategy like
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Delta hedging which allows you to hedge
your risk exposures if you have not
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watched my videos explaining dynamic
replication you should probably watch
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those to understand how important that
is to an options trader most of the
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options that require the Monte Carlo
method to price are close to impossible
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to hedge or at least very difficult to
hedge accurately and for this reason the
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option is usually sold only when the
buyer will pay well above fair value as
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the seller usually has to keep that
option on their books for its entire
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lifespan and can only roughly hedge it
advantages of the Monte Carlo method are
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that it can be used not only to price
options where the payoff depends on the
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final price of the underlying on the
expiration date but also when the payoff
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depends on the price path followed by
the underlying the Monte Carlo method
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can similarly be used to value options
where the payoff depends on the value of
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multiple underlying assets such as
basket options or rainbow options take a
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look at my videos on exotic options and
structured products to understand those
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in pricing those derivatives correlation
between asset returns is also
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incorporated the Monte Carlo method
allows for compounding in the
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uncertainties such as where a joint
probability distribution is used in the
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case of only two random variables this
is called a bivariate distribution but
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the concept generalizes to any number of
random variables
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both giving a multivariate distribution
an example would be pricing an option on
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a stock in a foreign currency where the
paths followed by the underlying stock
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and the exchange rate has to be modeled
but also the correlation between these
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two sources of risk must be incorporated.
The Monte Carlo method cannot easily
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handle situations where there is early
exercise in these situations a least
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square Monte Carlo method with a
backward induction approach is used. So
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how do we calculate the Greeks when
using the Monte Carlo method? Calculating
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the Greeks using the Monte Carlo method
is usually done by first pricing the
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derivative and then recalculating the
price of the derivative after making a
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small change in the input such as spot
price to calculate Delta or volatility
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if we are calculating Vega whose price
sensitivity it is we are trying to find
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the same number of iterations should be
run in calculating the new price as were
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used when initially pricing the
derivative. All of these videos are based
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on my book - Trading and Pricing Financial
Derivatives which is available on
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amazon.com I've put a link to it in the
description below. Hit the like button to
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let me know if you found this video
helpful and hit the subscribe button and
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the Bell button next to it in order to
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have a great day
bye
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you
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