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