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Swap Just to be clear, (3.4c) leads to (3.5a) when we assume lognormal $R(\tau)$. Lognormal $R(\tau)$ means we can write $$R(\tau) = R_0 e^{-\frac{1}{2}\sigma^2 \tau + \sigma \sqrt{\tau} Z}$$ with $Z$ normal, and I'm assuming a zero mean -- which I think is required. Then for (3.4c) we have for the expectation value: E\left[(R(\tau) - R_0)^2 \right] = ... 4 All the topics you've mentioned are wonderful and shouldn't be eschewed by reading some finance-oriented review book. I recommend these instead. Linear algebra: Hoffman and Kunze and Halmos Set theory: Halmos Measure theory: Rudin and Tao 3 The general effect of quantitative analysis of the markets is to enforce randomness. Suppose a strategic quant finds a predictable pattern where a stock always rises on Tuesdays. His institution will commence buying the stock every Monday, and selling on Tuesday. The trading itself pushes the stock price up on Monday and down on Tuesday (in general), so if ... 3 I would recommend the books from Steven Shreve. Here is a link to some one of his older online pdf's (1997 but nevertheless true) so you can check if that fits the bill. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.137.6951&rep=rep1&type=pdf 3 1) Gatheral expresses everything in forward terms: forward value of the spot and of the call. Consider an asset A. You need to hold A at time T but since you don't need it now you don't want to buy it now. Instead you enter a forward contract with someone that says that at time T you will pay the amount K and get the asset in exchange. What ... 3 We use Derman and Kani's notations. Arrow-Debreu prices The Arrow-Debreu price \lambda_i is the price of the security \Lambda_i paying \1 in node (n, i), and \0 in all other states (n, j), for j \neq i. Let \mathbb{P}_{n,j} be the risk-neutral probability of getting to state (n,j), from state (1,1). The price of \Lambda_i is the ... 2 How to solve this, you can generate random portfolios based on constraints see method="random" in optimize.portfolio in PortfolioAnalytics in R See (1) as those would solve the above, however you do not have an objective function so ANY solution that meets your constraints would be accepted, see below for examples of objective functions as they would give ... 2 Sorry, but despite being used as a popular example in machine learning, no one has ever achieved a stock market prediction. It does not work for several reasons (check random walk by Fama and quite a bit of others, rational decision making fallacy, wrong assumptions ...), but the most compelling one is that if it would work, someone would be able to become ... 2 For a martingale \{M_t \mid t\geq 0\} and the stochastic integral \begin{align*} I_t = \int_0^tZ_s dM_s, \end{align*} we have that \begin{align*} E((I_t)^2) = E\bigg( \int_0^tZ_s^2 d\langle M\rangle_s\bigg), \end{align*} where \langle M\rangle is the quadratic variation. That is, the ito's isometry holds for a martingale integrator only. However, in ... 2 The claim payoff you describe, g(M), looks to me like a tight butterfly spread that pays off only in one state of the world. Can't you just replicate that by short two calls with strike K_0 and long two calls, with strikes one either side at K_0\pm 1? Then the price of your option would be C(K_0+1)+C(K_0-1)-2\cdot C(K_0). This is effectively the ... 2 The above question was a typo due to the author -- the expression should be evaluated as $$E(t|\mathcal{F}_{s}^{W}) = t$$ due to the reasoning in the question. Sorry for the noise. 1 Let \begin{align*} w_t = \frac{1}{\sqrt{\sigma_1^2+\sigma_2^2 -2\sigma_1\sigma_2 \rho}}(\sigma_1\tilde{w}_t^1-\sigma_2\tilde{w}_t^2). \end{align*} Then, using Levy's characterization, we can show that \{w_t \mid t \geq 0\} is a standard Brownian motion. 1 A convex function is when the line between two points on the graph always lies above the graph. And this does hold for the put, its also sometimes called a sublinear function. Also see http://en.wikipedia.org/wiki/Convex_function So the author is correct in saying that (K-s)^+ is convex. 1 t τ----T A FRA from \tau to T pays the difference between the fixed rate and the actual fixing (Libor), discounted from T back to \tau at the Libor rate. This is from when that was a good measure of the risk free rate, with the idea that you would receive this and invest at Libor from \tau to T. Thus the cash flow at \tau is: ... 1 Both the quantity \frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+ and the quantity (A-p(T_{i-1},T_i))^+ are known at time T_{i-1}. Then the payment \frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+ at time T_i discount back to time T_{i-1} is the equivalent payment. That is \begin{align*} \bigg[\frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+\bigg] \times ... 1 Could you please be more specific with your question and post the text here? This will be more helpful for other people visiting the site. Now as far as to where the 1/2 went, usually people put 1/2 in front of the second order term because this will simplify to 1 after the derivation: \frac{\partial x^2}{\partial x} = 2x $$vs$$ \frac{1}{2} \cdot ...
You might want to give us the exact statement of the author. Let the Wiener process $W_{s}$ be a r.v. from $\left(\mathcal{F}_{s},\Omega\right)\to\left(\mathcal{B}\left(\mathbb{R}\right),\mathbb{R}\right)$. The Borel-$\sigma$-algebra $\mathcal{B}\left(\mathbb{R}\right)$ contains all intervals of the form $\left[x,y\right]$ for $x\neq y\in\mathbb{R}$, ...