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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 $f : \mathbb R^n ... 5 For the binary tree model the full replication property of all possible options can be shown using basic algebra and the no-arbitrage argument. It's beautiful how simple it is actually. You can find the complete derivation in Shreve's Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. 5 The general idea is to bootstrap the discount factors in the correct order, based on the data you have given. I'm going to make some assumptions that your bonds are paying annual coupons. The longest maturity is 2.5 years, meaning you need discount factors for 6M, 1.5Y and 2.5Y. The 6M deposit has a rate of 5%, this tells you that you should use the 5% rate ... 4 You see, you added something new to the source formula, i.e. a dependence between weights of different assets:$w_2 = 1 - w_1$. Let's try to forget that they are related to each other and vary them independently: $$\sigma(x)=\sqrt{w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_{1,2}}$$ Now$\frac{dw_2}{dw_1} = 0$and equation becomes: $$\frac{d ... 3 A very good book covering such fundamentals with no or only a minimal amount of maths — highly recommended! Puzzles of Finance: Six Practical Problems and Their Remarkable Solutions by Mark P. Kritzman The topics that are covered here are: Siegel's Paradox Likelihood of Loss Time Diversification Why the Expected Return Is Not To Be Expected Half Stocks ... 3 The essence of discounting is that now is less risky than later. So a contract to deliver £1 in 1 year is more risky than one to deliver £1 tomorrow, (the counterparty could suffer a credit event) so it is worth less. Discount factors multiply; if I know that £1 at 1y is worth £0.98 today, and £1 at 2y is worth £0.98 at 1y (i.e. equal rates for both ... 2 The state price vector are the prices of securities which pay \1 if and only if that state of the world occurs. This is just a question of being able to replicate the payoffs$$ \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 ...