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A function $f : \mathbb R^n\backslash\{0\} →\mathbb R$ is called (positive) homogeneous of degree $k$ if $$f(\lambda \mathbf x) = \lambda^k f(\mathbf x) \,$$ for all $\lambda > 0$. Here $k$ can be any complex number. The homogeneous functions are characterized by Euler's Homogeneous Function Theorem. Suppose that the function $f : \mathbb R^n ... 11 My understanding is because the Ito's integration definition keeps the martingale property. With Brownian motion$W(t, \omega)$defined, to define stochastic integration in a Riemann–Stieltjes style: $$\int_0^t f(t, \omega) d W(t, \omega) = \lim_{\| \Delta_n\| \to 0 } \sum_{i=1}^{n} f(\tau_i,\omega) \left ( W(t_i, \omega) - W(t_{i-1}, \omega) \right )$$ , ... 9 In fact Ito and Stratonovich calculus are both mathematically equivalent. In the following paper you can e.g. see that both derivations lead to the same result, i.e. the Black-Scholes equation: Black-Scholes option pricing within Ito and Stratonovich conventions by J. Perello, J. M. Porra, M. Montero and J. Masoliver From the abstract: Options ... 5 The general idea is to bootstrap the discount factors in the correct order, based on the data you have given. I'm going to make some assumptions that your bonds are paying annual coupons. The longest maturity is 2.5 years, meaning you need discount factors for 6M, 1.5Y and 2.5Y. The 6M deposit has a rate of 5%, this tells you that you should use the 5% rate ... 3 Note that $$P(X_i >s)= \exp\Big(-\int_0^s \lambda_i(u) du \Big),$$ for$i=1, 2$. Then, $$P(\min(X_1, X_2) >s) = P((X_1>s)\cap (X_2>s)) = P(X_1>s)P(X_2>s) = \exp\Big(-\int_0^s (\lambda_1(u)+\lambda_2(u)) du \Big).$$ That is, the hazard function for$\min(X_1, X_2)$is$\lambda_1(s)+\lambda_2(s)$. Alternatively, note that $$\lambda_i(s) = ... 3 The initial condition for the backward Kolmogorov PDE is that$$ u(0,x) = g(x) $$for all x in the relevant domain and not just at a particular point. So if your functions f and g agree only at a single point the initial conditions are in fact different. 3 A very good book covering such fundamentals with no or only a minimal amount of maths — highly recommended! Puzzles of Finance: Six Practical Problems and Their Remarkable Solutions by Mark P. Kritzman The topics that are covered here are: Siegel's Paradox Likelihood of Loss Time Diversification Why the Expected Return Is Not To Be Expected Half Stocks ... 3 The essence of discounting is that now is less risky than later. So a contract to deliver £1 in 1 year is more risky than one to deliver £1 tomorrow, (the counterparty could suffer a credit event) so it is worth less. Discount factors multiply; if I know that £1 at 1y is worth £0.98 today, and £1 at 2y is worth £0.98 at 1y (i.e. equal rates for both ... 2 Look at randommatrixportfolios.com 2 The state price vector are the prices of securities which pay \1 if and only if that state of the world occurs. This is just a question of being able to replicate the payoffs$$ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$$with payoff vectors$\vec{b} = [1,1,1]^T$and ... 2 Some more concrete sources on Barrier option in the B&S setting and PDEs PDE methods for pricing barrier options (quite technical) Pricing Europ ean Barrier Options More of a general remark to PDE approaches in finance Ilya as far as I know the literature on that topic is quite limited. Solving a PDE means solving a PDE - it does not matter in ... 1 In many cases, clients want to be fully invested and don't want their assets lying around in cash. Hence the budget constraint$\sum_i w_i = 1$is fairly common in practice. By the way, there are also cases where the constraint$\sum_i w_i = 0\$ is applied: the result is a dollar neutral portfolio with long and short positions, but no net investment (short ...

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I see your argument with the math. "1" is an arbitrary choice of positive numbers, and you could choose anything. In the end, you're going to scale the whole thing to fit your capital anyway. If you are using a numerical optimizer, it will be happier with something noticeably away from 0 and away from infinity, so I recommend choosing a specific positive ...

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Portfolio management is about solving problems in the real world. In the real world, it is highly unlikely that EVERY asset has a negative expected return. If all the assets in your universe have negative returns, expand your universe to include a short-term fixed income security that is bound to produce a return greater than (or at a minimum equal to) ...

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Paul Wilmott on Quantitative Finance.

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I recommend to you : "Market Risk Analysis" by Alexander Carol for the "finance" part and "Time Series Analysis" by Hamilton for the "maths/stats" part;

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Some more references. Here are three starting books: for generic knowledge: Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management, by Bouchaud and Potters; for risk + statistical approach: Risk and Asset Allocation, by Meucci; for microstructure: Market Microstructure in Practice, by Lehalle and Laruelle.

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