By J Robert Buchanan
This textbook presents an creation to monetary arithmetic and fiscal engineering for undergraduate scholars who've accomplished a 3 or 4 semester series of calculus classes. It introduces the idea of curiosity, random variables and likelihood, stochastic techniques, arbitrage, alternative pricing, hedging, and portfolio optimization. the coed progresses from realizing in simple terms straightforward calculus to realizing the derivation and resolution of the Black–Scholes partial differential equation and its strategies. this is often one of many few books with regards to monetary arithmetic that is available to undergraduates having just a thorough grounding in uncomplicated calculus. It explains the subject material with no “hand waving” arguments and comprises quite a few examples. each bankruptcy concludes with a suite of workouts which try the chapter’s techniques and fill in information of derivations.
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Additional info for An Undergraduate Introduction to Financial Mathematics
The proofs of these theorems are beyond the scope of this work. The interested reader should consult a book on the mathematical underpinnings of statistics for proofs of the various versions of the Central Limit Theorem (for example [DeGroot (1975)]). Nevertheless we can observe the consequences of the Central Limit Theorem on data from a random number simulation. The reader should keep the following points in mind. Given a random variable X which can be either discrete or continuous and which may have any probability distribution, we may collect a sample of size n and denote the mean of that sample Xn.
We will make use of the result that limAa;^o(l + aAx)l/^x = ea. Let us define f(x,t) as f(x,t) 1 X /, A \ 5Z^ = — 7 = lim 1 + ^ - A x 2\fkrt Ax^o V 2kt J = 2~7kTt^^(e_5ll)f X /-, A \ 22^ 1 - —-Ace \ 2kt J (e"^) / \ 1 f4-Axl I2kt J 2 We must verify that this expression satisfies the properties of a probability distribution function. For k, t > 0 the expression is non-negative. Since for \x\ > 1 it is true that 0 < e~x < e - ^ , then the integral of f(x, t) over the entire real number line converges, suppose we write: f°° 1 0< / — J-oo e _ikt^ i dx = S < oo.
Thus we say that if F is 26 An Undergraduate Introduction to Financial Mathematics a function applied to X, then E[F(X)]=Y,F(X)P(X). x When the function F is merely multiplication by a constant then the expected value takes on a simple form. 3 If X is a random variable and a is a constant, E[aX] =aE[X}. Proof. then By the definition of expected value E [aX] = £ daX) • P PO) =aJ2(X-P (X)) = aE [X]. a Later in this work sums of random variables will become important. Thus some attention must be given to the expected value of the sum of random variables.
An Undergraduate Introduction to Financial Mathematics by J Robert Buchanan