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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 ...


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) = ...


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

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 ...


3

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 $U$ ...


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 ...


2

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

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 ...


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

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 \begin{equation} E(t|\mathcal{F}_{s}^{W}) = t \end{equation} due to the reasoning in the question. Sorry for the noise.


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

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 ...


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

There's more than one way to do this. One common approach among indices is to take an iterative approach. For instance, you might identify the stocks with weights about 5%, then re-weight so that everything adds up to 1. Then you might identify the sectors that break the 10% limit and re-scale them to be less than 10%. Then re-scale everything to add up to ...


1

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 ...



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