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Submitted by plusadmin on September 1, September If I tell someone I am a financial mathematician, they often think I am an accountant with pretensions. Since accountants do not like using negative numbers, one of the oldest Financial mathematics essay technologies, I find this irritating.
A roll of the dice I was drawn into financial maths not because I was interested in finance, but because I was interested in making good decisions in the face of uncertainty.
Mathematicians have been interested in the topic of decision-making since Girolamo Cardano explored the ethics of gambling in his Liber de Ludo Aleae ofwhich contains the first discussion of the idea of mathematical probability.
Cardano, famously, commented that knowing that the chance of a fair dice coming up with a six is one in six is of no use to the gambler since probability does not predict the future. But it is of interest if you are trying to establish whether a gamble is fair or not; it helps in making good decisions.
The average value of rolling a dice converges to the expected value of 3. These ideas about probability were collected by Jacob Bernoulli Daniel's unclein his work Ars Conjectandi. He introduced the law of large numbers, proving that if you repeat the same experiment say rolling a dice a large number of times, then the observed mean the average of the scores you have rolled will converge to the expected mean.
In conjunction with statistics, probability theory became an essential tool of the scientist. For the first third of the twentieth century, probability was associated with inferring results, such as the life expectancy of a person, from observed data.
But as an inductive science i. Kolmogorov defined probability to be any measure on a collection of events — not necessarily based on the frequency of events. What is it worth?
This idea is counter-intuitive if you have been taught to calculate probabilities by counting events, but can be explained with a simple example. If I want to measure the value of a painting, I can do this by measuring the area that the painting occupies, base it on the price an auctioneer gives the painting or base it on my own subjective assessment.
For Kolmogorov, these are all acceptable measures which could be transformed into probability measures. The measure you choose to help you make decisions will depend on the problem you are addressing: Kolmogorov formulated the axioms of probability that we now take for granted.
Secondly, that you know all the possible outcomes, and the probability of one of these outcomes occurring is 1 e.
And finally, that you can sum the probability of mutually exclusive events e. You can read more about probability and its development on the Understanding Uncertainty siteand the Plus article Measure for measure is an excellent introduction to measure theory.
Deciding a fair price Why is the measure theoretic approach so important in finance? Financial mathematicians investigate markets on the basis of a simple premise; when you price an asset it should be impossible to make money without the risk of losing money, and by symmetry, it should be impossible to lose money without the chance of making money.
If you stop and think about this premise you should quickly realise it has little to do with the practicalities of business, where the objective is to make money without the risk of losing it, which is called an arbitrage, and financial institutions invest millions in technology that helps them identify arbitrage opportunities.
An asset should be priced so as to prevent such arbitrages. The explanation of risk-neutral measures is pretty straightforward and is described here. You can also read a general introduction to arbitrage and pricing in the Plus article Rogue Trading.An Introduction to Financial Mathematics Sandeep Juneja Tata Institute of Fundamental Research, Mumbai [email protected] 1 Introduction A wealthy acquaintance when recently asked about his profession reluctantly answered that he is a middleman in drug trade and has made a fortune helping drugs reach European.
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Financial Mathematics Essay. Topics: Call option, The Mathematics for Essay 2 The purpose of these notes are to explain some of the mathematics behind Essay 2.
Your own essay should not just repeat these arguments but have a more geometric ﬂavor. Write about how you can physically place the blocks. Do my financial mathematics assignment.
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