Questions tagged [math]
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1
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1answer
29 views
Why the variance of a process is $\left( \frac{dS_T^2}{dt}\right)^2$?
Consider an Ito process $dS_t = f(t,S_t) dt + g(t,S_t)dW_t $
What is the reason that we can compute the variance as:
$\sqrt{VaR(S_t)} = \frac{(dS_t)^2}{dt}$
1
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0answers
14 views
How is hypothesis testing work in population sampiling?
I am learning the basics of quant trading from quantconnect's tutorial Confidence Interval and Hypothesis Testing. I understood the first part of the article but I dont understand "Hypothesis Testing"...
3
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1answer
140 views
Finding optimal trading of option on a foward
Assume you have a option on a forward $F$ with a payoff: $\max(F_T - K, 0)$.
Assume also, that you have a bullish view on the forward in such a way that $E_{0}[F_T] > F_0 = E_{0}^{*}[F_T]$ (where ...
1
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1answer
51 views
Finding the extrinsic value of an option with conditions
Background:
Consider a spread option with the payoff $\max (P_{T} - HR\times G_T, 0)$, where $P$, $G$ are underlying prices and $HR$ is a constant.
Let's also assume, that the correlation ...
1
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0answers
39 views
Evaluating contract $D$ where the stock follows the Black Scholes assumption
Ch.7 Mark Joshi Problem 14
A contract, $D$, pays $30\%$ of the increase (if any) of a stock's value in a year. If $S_t$ follows Black-Scholes assumptions, give a formula in terms of the Black-...
0
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1answer
70 views
Properties of Brownian motion and filtration, Exercise 6.22, Joshi Concepts and applications to mathematical finance
Let $W_t$ be a Brownian motion, and let $F_t$ be its filtration then for $t > s$ we are asked to compute
$$\mathbb{E}\left[W_t^2|F_s\right]$$
We have $$W_t = W_s + (W_t - W_s)$$
and
$$W_t^{2} ...
3
votes
1answer
107 views
Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 20,21
Find the Black-Scholes price of an option paying
$$(S_T^{\alpha} - K)_{+}$$ at time $T$.
Solution - The forward price is given by
$$F_T(t) = e^{r(T-t)}S_t$$
So,
$$F_T(0) = e^{rT}S_0$$
and
$...
0
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1answer
150 views
Joshi, Exercise 2.7 Concepts of Mathematical Finance
Let $D(K)$ pay $(S - K)^2$ if $S > K$, zero otherwise. Show that if $D(K)$ is differentiable function of $K$ then the third derivative w.r.t $K$ is non-negative.
From what the hint in the book, we ...
1
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0answers
70 views
Moving average variance [closed]
I have generated a random series of returns drawn from a normal distribution and generated a random price series by compounding these returns (X) so $P_i = P_1(1+X)^i$. I want to show the analytic ...
1
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0answers
161 views
What jobs in Finance are most math intensive? [closed]
I'm a math major and I've always been really interested in Finance; however, I'm starting to enjoy math more and more and would like to know which jobs in Finance use the most/more advanced math. Also,...
1
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1answer
641 views
Given two risky stocks calculate the rate of return, standard deviation, beta, and risk-free rate
Consider a world where there are only two risky stocks, $A$ and $B$, whose details are listed in the table below:
Furthermore, the correlation between the returns of stocks $A$ and $B$ is $\rho_{A ...
0
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1answer
152 views
Using CAPM to derive the following
Background Information:
Say there are $s = 1,\ldots,S$ possible future outcomes (states) with known probabilities $\pi_s > 0$, $\sum_{s=1}^{S}\pi_s = 1$. Define the expected payoff as $\mathbb{E}_\...
-1
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2answers
96 views
Given three stocks what is the fraction of each stock's risk is diversified away
Consider an equally weighted portfolio of three stocks, each of which
is independently distributed of the others but have the same risk. I.e.,
$cov(r_i, r_j) = 0$; $\forall i \neq j$, and $\...
-2
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1answer
222 views
How to solve $dX_t = X_t(\sigma_t dW_t + \mu_t dt)$?
Solve the SDE $$dX_t = X_t(\sigma_t dW_t + \mu_t dt)$$ where $\sigma_t$,$\mu_t$ are deterministic.
Attempted solution
We have $$dX_t = X_t(\sigma_t dW_t + \mu_t dt)$$
Let $f(x) = \log X$, applying ...
0
votes
1answer
36 views
Price of every asset in discrete market model strictly increasing
If the price of every asset in a discrete model is strictly increasing, with probability one, then does the market admit arbitrage?
Thoughts: I believe this is true but I am not sure how to give an ...
1
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1answer
59 views
Implied Expected Stock Return from European Option Prices
We can calculate the expected stock return (under the measure $Q$) from at-the-money ($K=S_t$) option prices as:
$$E\left(\frac{S_T-S_t}{S_t}\right)=\frac{e^{rT}}{S_t}(C_t-P_t)$$
The result is ...
