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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 financial ... 8 As Sanjay said, you can apply Itô's Lemma to$f(t,x)=x^2and obtain \begin{align*} \mathrm{d} S^2_t=\left(2\mu S_t^2+\sigma^2S_t^2\right)\mathrm{d}t+\left(2\sigma S_t^2\right)\mathrm{d}W_t. \end{align*} Thus,(S_t^2)$is again a geometric Brownian motion and hence, for each time point$t$log-normally distributed with drift$2\mu+\sigma^2$and volatility$...

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At the first glance, what you are asking for is a model admitting arbitrage, so there is a zero chance of losing money and positive chance of yielding profits. Well, many equilibrium models start with assuming arbitrage is not possible (otherwise it would be trivial wouldn't it). But, in my opinion, what you actually seek is the Efficient Markets Hypothesis....

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Let $\tau = T-t$. Then \begin{align*} S_T = S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, Z}, \end{align*} where $Z$ is a standard normal random variable, independent of $\mathcal{F}_t$. Moreover, \begin{align*} E\left(S_T 1_{\{S_T >K\}}\mid \mathcal{F}_t \right) &= E\left(S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, ...

6

To price financial instruments such as options, bonds and stocks must be priced so as to be "arbitrage free". The concept of arbitrage can be made precise by one of the fundamental ideas of quantitative finance, the so called Arbitrage Theorem. Put differently the Arbitrage Theorem provides a very elegant and general method for pricing derivative ...

6

Under the risk-neutral measure the discounted (under some numéraire) price process is a martingale. If we have a bank account with dynamics $dB_t = r B_t dt$ then the discounted asset $X_t = \frac{S_t}{B_t}$ will have the dynamics dX_t = \frac{dS_t}{B_t}- \frac{S_t dB_t}{B_t^2} = (\mu - r S_t) \frac{1}{B_t} dt + \frac{\sigma}{B_t} dW_t \end{...

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 ...

5

No, a sum of two GARCH processes is generally not a GARCH process. (I am not even sure whether there exists a nontrivial special case where the opposite holds.) By GARCH I mean the classic definition of GARCH due to Bollerslev (1986), not an arbitrary variation like EGARCH, IGARCH, FIGARCH or whatever else. Let me provide an example. Take two ...

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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 ...

5

Optimization is definitely important in Quantitative Finance, especially for portfolio optimization where we maximize utility of the return of a portfolio as linear weighted vector of asset returns subject to a desired risk level: $$\max_{w\in[0,1]^n} U(\mu_p(w),\sigma_p(w))\quad s.t. \sum_{i=1}^n w_i=1$$ where $w$ being the portfolio weights, and $U$ ...

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I think the main difference even in this little example is the gain-loss asymmetry which is a known stylized fact: When you look at the big bump both time series posses your artificial one is perfectly symmetric whereas the real one takes longer for going up and then crashes in a relatively shorter time frame. This is a known phenomenon in real financial ...

5

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 ...

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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

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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: $$E\left[(R(\tau) - R_0)^2 \right] = ... 5 Here's how i'd have at it; * I happen to know these are okay guesses. ** Let's assume it's just the potential energy, and that as the point of the "5000 years" part of the question. The moon came from the earth - likely it's crust since the idea is that it was formed from an impact. The crust of the earth is less dense than the core, so the moon ... 5 Let define \mathbb{Q} and \mathbb{P} two equivalent probabilities on a filtered space (\Omega,(\mathcal{F}_t)_{t\geq 0}) Let define Z_T=\frac{d\mathbb{Q}}{d\mathbb{P}} restricted to \mathcal{F}_T measurable events. It means that for X_T being \mathcal{F}_T measurable we have:$$\mathbb{E}^{\mathbb{Q}}[X_T] = \mathbb{E}^{\mathbb{P}}\left[...

5

$\require{cancel}$ $$\text{PnL} = -[P(t+\delta t,S+\delta S)-P(t,S)] + rP(t,S)\delta t + \Delta(\delta S - rS \delta t + q S\delta t)$$ Assuming a pure diffusion, at the order 1 as $\delta t \to 0$ P(t+\delta,S+\delta S) = P(t,S) + \frac{\partial P}{\partial t}\delta t + \frac{\partial P}{\partial S}\delta S + \frac{1}{2}\frac{\partial^2P}{\partial S^2}(\... 5 Aside from the independence requirement for the increments, that is, the independence of X_{s+t}-X_s and \mathcal{F}_s, you can check whether the increment X_{s+t}-X_s has the distribution of N(0, t). In fact, note that \begin{align*} X_{s+t}-X_s &= (\sqrt{s+t}-\sqrt{s}) Z\\ &\sim N\left(0,\, (\sqrt{s+t}-\sqrt{s})^2\right), \end{align*} which ... 5 It is actually rather simple. Lets start with the fixed rate market. A can borrow at 5% while B can borrow at 7%. Simply said, A has a comparative advantage of 2% in the fixed rate market. In the floating rate market, A borrows at LIBOR + 1% while B borrows at LIBOR + 2.5%. From here, I'm guessing you already know that A has the comparative advantage as ... 5 You are right about the dropped \sim, it's probably just a typo. Furthermore, remember that in stochastic calculus, you have to take into account second order derivatives, i.e.d\left(\frac{1}{Y_t}\right) = -\frac{1}{Y_t^2}dY_t + \frac{1}{2}\frac{2}{Y_t^3}dY_t^2$$which is the Taylor expansion up to second order. Then you substitute dY_t in the right ... 5 As you mentioned, we have$$\sum_{l=0}^{k}{p}=\frac{k(k+1)}{2}$$You want to know$$\sum_{k=0}^{n}{\sum_{l=0}^{k}{p}}=\sum_{k=0}^{n}{\frac{k(k+1)}{2}}=\frac{1}{2}\sum_{k=0}^{n}{k^2}+\frac{1}{2}\sum_{k=0}^{n}{k}$$you know that$$\sum_{k=0}^{n}{k^2}=\frac{n(n+1)(2n+1)}{6}$$Therefore$$\sum_{k=0}^{n}{\sum_{l=0}^{k}{p}}=\frac{n(n+1)(2n+1)}{12}+\frac{n(n+...

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All books recommended in previous posts are splendid :-) I would like to add one more book for continuous time financial mathematics: Arbitrage Theory in Continuous Time by Tomas Bjork.

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Paul Wilmott on Quantitative Finance.

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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: $$\frac{d \... 4 Note that$$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) = \...

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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

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Trinomial trees give incomplete markets so there is a range of possible risk neutral prices. So you have to find the possible probabilities that make the tree risk-neutral and see what prices you get. You have the correct expressions. Now just have to parametrize the set of solutions. It is one-dimensional and all the probabilities are positive so you need ...

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