106 (a) - Martingales

hi guys so today let's talk about martingales and martingale are extremely important in finance because they are the foundations for no arbitrage theory I'm just neutral pricing okay so in this class what I want to do is I want to describe what a martingale is and later we're going to use this multi-kill property when we do English neutral valuation so let's start for the adaptive stochastic process and by adaptive stochastic process I mean we have sequence of random variables m0 m1 m2 dadada and n where each one of the random variables MN depends only on the first endpoint OSes okay this is called adaptive stochastic process now we say adaptive stochastic process is a martingale if if we take expectation at any time n of a random variable MN plus 1 then this is equal to M of n for all n from 0 to n minus 1 okay so as you can see this basically is a one step ahead property so we basically are taking expectation at time n of a random variable M of n plus 1 okay now we can also make it a two-step property so M of n plus 1 we can write as expected value of n plus 1 M of n plus 2 right and this comes from our martingale property right so if you can write it like this and you know you can see we can take iterated conditioning here and that would make it M of n plus 2 equal to MN now this has become a two step ahead property right now the random variable is two steps forward now likewise you can actually you know do it for MN plus three like Y and we can make a generic expectation what we could say is expectation at time n of a random variable M is equal to so this is called a multi step ahead martingale property okay so it follows from here that m0 is equal to expectation at time zero so I don't have to write a subscript here of m1 okay and from one key set by a property I can also write expectation of m2 expectation of m3 da-da-da-dah expectation of MN so as we can see from here for a martingale process the expectation remains constant okay as we go forward so Marty of the expectation doesn't change now what I would like to do is you know let's actually apply the swap see to our simple model okay we basically had to find a stock process where at time 0 the stock was at 4 if we toss the head then the stock went up to 8 if we toss the tail if we toss the tail then the stock went down to 2 okay now let's define a probability measure under which probability of getting a head is 1 by 3 and probability of getting a tail is 2 by 3 let's calculate what I would like to calculate the expected value at time 0 off random variable s1 okay and what would that be at time 1 random variable can be either 8 or 2 right with probability 1 by 3 and 2 by 3 so expectation is 8 times 1 by 3 plus 2 times 2 by 3 equals 12 by 3 equals 4 right and for the same as where we started so this can be written as s0 so from here we can see the expected value of random variable s1 at time 0 is equal to a 0 and this pretty much is the same thing as how we define a martingale right so we said for a martingale expected value at time n of em n plus one equal to MN so likewise here okay so this basically become the martingale under this probability measure but if the if we had defined the probabilities to be probability of a head to be 1/2 and probability of a tail also to be 1/2 then the expected value would have become 8 times 1 by 2 plus 2 times 1 by 2 equals 5 right then the expectation would not have been equal to a 0 so in this case in this probability measure we can see as 0 is less than expectation at time 1 and such processes are called sub martingale okay so martingale the expectation of 0 submartingale basically have a tendency to rise so we can see expectation of s 1 at time 0 is higher than a 0 so they tend to rise okay if on the other hand in another probability measure if if it was like this if the expectation was had a tendency to go down then this would be called the supermartingale okay so martingale sukumaran you submartingale have a tendency to rise supermartingale have a tendency to fall it but martingale basically the expectation remains constant okay they you know the expectation doesn't change and this is actually extremely important martingales are extremely important in in in mathematical finance as we will see later so at this point in time did that basically what i wanted to talk about in this particular lecture so we've described to be defining what a martingale is and you know I showed you you know under this measure of stock process for the martingale and this measure our stock process was a submartingale okay so in the next lecture now we're going to try to come up with our you know risk-neutral pricing formula for a binomial asset pricing model okay and there you will again see martingales and hopefully the concept is going to get here okay all right guys thank you

Channel: FinMath Simplified

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Keywords: random variable, conditional expectations, stochastic calculus for finance, steven shreve, CMU, carnegie mellon, MSCF, computational finance, fin math, Financial mathematics, Martingales, Sub martingale, super martingale, risk neutral pricing

Id: cq65eYVMOt4

Length: 6min 47sec (407 seconds)

Published: Fri Dec 09 2016