Questions tagged [finance-mathematics]

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1answer
163 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}_\...
3
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1answer
174 views

Payoff of European Call Option with Transactioncosts

I was wondering about the following scenario: assume that you have a underlying which trades under a positive bid-ask spread $S^B \leq S^A$ and that there is also a European Call-Option on this ...
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0answers
90 views

Can someone suggest some good reads on OAS and Spread Duration?

I have been through the CITI Yield book paper and the OAS by Barclays. Is there is anything else that tackles this topic? Any help would be much appreciated. Cheers!
3
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2answers
5k views

What is the difference between pull to par and roll down in both mathematics and conceptual?

I don't really understand the difference. Shouldn't roll down and pull to par be the same technically? If a bond is trading as a discount it "increases" in value because everyday gets closer to par, ...
3
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2answers
504 views

Is mathematical finance relevant in asset managament?

I was hoping to consult on the relevance on the relevance of mathematical finance in the asset management business. Traditionally, mathematical finance focuses more on topics related to stochastic ...
1
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0answers
300 views

Different definitions of arbitrage

Consider the following setup: Let $S=\left(S_1,\ldots,S_n\right)$ be a $n$-dimensional price process and denote by $V$ its value process defined by $V_t=\phi_t\dot\ S_t$ for $t=0,\ldots,T$. In "...
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1answer
361 views

How to price 0 floors in csa agreements for negative ois rates?

A CSA (Credit Support Annex) agreement specifies the interest rate to be earned on the collateral provided to back a derivatives transaction. For cash collateral this rate is generally the OIS (...
2
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0answers
178 views

How to understand the integral in the Girsanov theorem?

Let $W^P$ be a $d$-dimesional $P$-wiener procss. Define $L_t = > e^{\int_0^t \phi_s^T dW_s^P - \frac{1}{2} \int_0^t \| \phi_s\|^2 > ds}$.Assuming $E^PL_T = 1$, then the measure given by $dQ = ...
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0answers
53 views

Concatenation property of a set of semimartingales

Consider as in (1, Definition 2.1) a convex subset $\mathcal{X}_1$ of the set of semimartingales $\mathbb{S}$ satisfying the following properties: $X_0=0$ $X_t\geq -1$ for all $t\geq 0$ for all ...
0
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1answer
407 views

How to price a quanto basket option?

EDIT: Maybe there is no way to get explicit solutions for basket options (maybe the Black-Scholes differential equation can't be solved directly ??). Q3: How do you price and hedge ( S1(T) + S2(T) - ...
3
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1answer
418 views

Step by Step Guide to Learn Quantitative Finance [closed]

Can some one help in creating step by step guide to learn Quantitative Finance? The suggestions should be in the lines of 1- Which Maths topics needs to be learn 1st 2- Which Maths Books or ...
2
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1answer
306 views

Is Black-Scholes complete?

If we have a Black-Scholes model $B_t = \exp{(rt)}$ and $S_t = S_0\exp{(\sigma W_t + \mu t)}$ then is it complete? What if $W_1$ and $W_2$ are independent Brownian motions. Then the two-stage ...
2
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1answer
327 views

Black-Scholes evaluating the squared of the stock price

Consider a Black-Scholes model $S_t = 5\exp{(\sigma W_t + \mu t)}$, $B_t = \exp{(rt)}$, where $W_t$ is Brownian motion with respect to a given measure $\mathbb{P}$. Suppose you hold a forward ...
0
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1answer
51 views

For discrete models, the existence of strong arbitrage is equivalent to a particular self-financing strategy

Background Information: This question is from Lectures on Financial Mathematics: Discrete Asset Pricing. Question: Prove that for discrete models, the existence of a strong arbitrage is also ...
1
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1answer
111 views

All martingale measures price the attainable claim equally

Background Information: This question is from Lectures on Financial Mathematics: Discrete Asset Pricing. Theorem 3.2 First Fundamental Theorem of Asset Pricing - Suppose $\nu$ is any measure such ...
2
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1answer
173 views

Where can I find ideas for strategies? [closed]

Every book I read refers me to many other books, there is practically no way I can read all this text in my life time. Once and for all, where is the best place to fish for ideas?
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1answer
153 views

Show that there exists a fully invested portfolio such that the covariance between their returns is zero

Background Information: I came across this question in chapter 2 of Active portfolio Management by Grinold and Kahn. It pertains to the efficient frontier which is displayed below: Question: If $...
3
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1answer
578 views

What is the difference between state prices and stochastic discount factor?

I was reading a paper on arbitrage and it was mentioned that a positive SDF implies no arbitrage and later on it said that positive state prices imply no arbitrage. I am new to this topic and i am ...
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1answer
97 views

What is the value this “special” forward contract at maturity?

Background Information: I am not sure this is relevant: Terminal value pricing: If the derivative $X$ equals $f(S_T)$, for some $f$ then in the value of the derivative at time $t$ is equal to $V_t(...
1
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1answer
439 views

Calculation of Weighted Interest Rate based on Outstanding Debt

I would like to know what's the way on how to calculate the weighted average interest rate for a loan portfolio properly, especially when looking at periods shorter than a year. The basic definition ...
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1answer
65 views

If there is an inconsistent pricing strategy then by defintion we have strong arbitrage

Background Information: An Inconsistent pricing strategy is a self financing strategy $\phi$ with $V_T(\phi)= 0$ and $V_0(\phi) \neq 0$ A strong arbitrage is a self-financing strategy $\phi$ with $...
3
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1answer
257 views

How to prove we have a $\mathbb{Q}$-Brownian motion?

Background Information: This question comes from the book Financial Calculus by Baxter and Rennie. WE start with looking at the marginal of $W_T$ under $\mathbb{Q}$. We need to find the likelihood ...
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1answer
229 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 ...
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1answer
75 views

Do we have a Brownian motion

Background Information: The process $W = (W_t:t\geq 0)$ is a $\mathbb{P}$-Brownian motion if and only if i) $W_t$ is continuous, and $W_0 = 0$ ii) the value of $W_t$ is distributed, under $\mathbb{...
3
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1answer
666 views

Problem on Characteristic function in Heston model

I know the Heston model .In this model, we have $$f(\Phi,x_t,v_t)=\exp(C_j(\tau,\Phi)+D_j(\tau,\Phi)+i * \Phi * x_t)$$ How can we extract the Characteristic function as follows $$f(\Phi_1,\Phi_2,...
4
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1answer
803 views

What's the intuition behind the transformation of Black-Scholes into Heat equation?

A sequence of transformations can be used to turn the Black-Scholes PDE into the heat equation. Let $C(S, t)$ be the price of a vanilla European option at time $t$, maturing at time $T$, where the ...
0
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1answer
204 views

Showing the discounted stock is a martingale

Background Information: This question follows from here It is tempting to write $$V_0(X) = \beta\left[\left(\frac{\beta^{-1}S_0 - S_1(d)}{S_1(u) - S_1(d)}\right)X(u) + \left(\frac{S_1(u) - \beta^{-1}...
2
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1answer
164 views

Law of One price and the Inconcistent pricing strategy

Background Information: A market satisfies the Law of One Price if every two self-financing strategies that replicate the same claim have the same initial value. An inconsistent pricing strategy is ...
0
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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 ...
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1answer
135 views

Zero-coupon Loan Investment [closed]

Zero-coupon default-free interest rates maturing over the next five years are listed below (in percent per annum, continuously-compounded): Maturity Years -- Yield 1 --------------------1.9 2 ------...
1
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1answer
120 views

Formula for conditional expectation. Related to the Fundamental Theorems of Asset Pricing

Let $\lambda$ be a probability measure on $\Omega$ (finite), with filtration $\{\mathcal{F}_t\}$. Define $\nu(X) = \lambda\left(X\frac{d\nu}{d\lambda}\right)$, where $\frac{d\nu}{d\lambda}$ is a ...
2
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1answer
2k views

Computing Buy-and-hold abnormal returns (BHARs) $= \prod_{t=\tau_1}^{\tau_2}(1+R_{i,t}) - \prod_{t=\tau_1}^{\tau_2}(1+R_{m,t})$

I am doing an event study and wanted to know if was going about this correctly$$ \text{BHAR}_{i(\tau_1,\tau_2)}\quad=\quad\prod_{t=\tau_1}^{\tau_2}(1+R_{i,t})~-~\prod_{t=\tau_1}^{\tau_2}(1+R_{m,t}) $$ ...
0
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1answer
48 views

Verifying value of claim as an expectation

Background: We have so far taken the bond B to be deterministic for simplicity, but some reflection shows that this is not in any way necessary. Everything works out the same way with a stochastic ...
0
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1answer
57 views

Do we have arbitrage if the probability measures are less than zero

Background Information: This question follows from here It is tempting to write $$V_0(X) = \beta\left[\left(\frac{\beta^{-1}S_0 - S_1(d)}{S_1(u) - S_1(d)}\right)X(u) + \left(\frac{S_1(u) - \beta^{-1}...
2
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1answer
98 views

Value of a perfect hedge

Background Information: The price of a portfolio at time $t$ ($t = 0 ,1$) is $$V_t(\pi) = \phi S_t + \psi B_t$$ The portfolio $\pi$ is a perfect hedge for the claim $X$ if $V_1(\pi) = X$ a.s. as ...
0
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1answer
123 views

Utility function for avoiding investment

An investor has initial wealth $30000$ and utility function $\ln{x}$. He is planning to invest in a project where he has $60%$ chance of gaining $\alpha%$ and $40%$ chance of losing $\beta%$. Express ...
1
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3answers
122 views

Suppose i want to track S&P500 index using 15 stocks, how do i adjust their weights?

I am given 15 stocks (which is listed in NYSE), and want to track/replicate the S&P500 index. So i am currently have the information about the stock price, and given some capital to invest in (all ...
7
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2answers
693 views

Does financial math benefit society?

This is an open ended question but just want to hear some of everyone's thoughts on this. How does financial mathematics benefit the economy, the stock market, and the individual investor? I know ...
5
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4answers
183 views

What is the industry standard for annualizing returns over non-contiguous time periods?

Suppose I am invested in the same fund for the first 200 days in 2013, some combination of 150 days in 2014, and the last 100 days in 2015. Further suppose that geometrically linking the daily returns ...
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2answers
363 views

Close form solution for Geometric Brownian Motion

I have a very fundamental problem, please help me out. I am little confused with the derivation for the close form solution for the Geometric Brownian Motion, from the very fundamental stock model: $$\...
1
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1answer
126 views

Can someone check this boundary condition for me?

At the moment I'm comparing plots between the implicit numerical Black-Scholes PDE and the Monte-Carlo Method for the Black-Scholes equation. However, for the particular boundary condition I'm using I'...
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0answers
116 views

Can someone try this Boundary Condition for the Black-Scholes PDE out for me?

I have a bit of a favor to ask and if anyone could help me out with this I'd really appreciate it. At the moment I'm trying to use the triangle wave formula as the payoff for the Black-Scholes PDE i.e....
1
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1answer
211 views

Initial/Boundary Conditions for a Butterfly Option?

What are the initial and boundary conditions for a Butterfly Option? I want to write up a PDE program for it and I have a rough idea of what the payoff should be (is it just a call and a put at the ...
1
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1answer
62 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 ...
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0answers
127 views

How can we observe volatility smile from the market. Drawbacks of Heston Stochastic Volatility Model

Here are two questions related to implied volatilities. a) The set up here is for an European option. We can get its implied volatility smile from calibration, the question is why could we also ...
0
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1answer
112 views

Calculate historical duration based on current duration & historical prices

Suppose I have today current duration of a bond and it's historical daily prices. How from that I can calculate the historical duration? e.g. the value of duration I would saw if yesterday, week ago, ...
1
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1answer
208 views

Time-Value of money exercise problem. Any advice on how to solve?

Problem An investor will receive $365 at the end of each year for thirteen years. The first payment will be received four years from now. Given that the interest rate is 3%, the present value of this ...
4
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1answer
295 views

Black-Scholes Model for portfolios

Given Black and Scholes model, consider the portfolio $a_t$ = 1/2, $b_t$ = $1/2$$S_t$ $exp(-rt)$. Show that this portfolio replicates one share of stock. Show if it is self-financing. Find another ...
5
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2answers
1k views

Machine learning techniques for quantitative finance?

I am a mathematician who wants to learn about quantitative finance, in particular how machine learning can be applied to it. I assume some machine learning techniques are more applicable than others ...
2
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0answers
196 views

Arrow-Debreu Equilibrium Pricing

I have this problem in asset pricing that I don't know how to solve. Here it is: Consider an economy with a complete set of Securities and $N$ states of the world Tomorrow. Assume that there are two ...