# Questions tagged [finance-mathematics]

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### Replicating a portfolio with a certain payoff function

Assume there are two stocks $S_1$ with price $p_1(t)$ and $S_2$ with price $p_2(t)$ where $t$ indicates time. Assume, there is a hypothetical derivative $D$, which is such that, price of $D$ at a time ...
173 views

### reference for portfolio / margin calculations in backtesting tool

I have been tasked with writing a backtesting tool from scratch. I understand a lot of trading operations, but I am primarily a researcher. I need to support futures and equities trading. I need to ...
134 views

### How to consider open interest & volume change in option pricing?

Is there publically available option pricing model or theory that considers open interest/volume % change? I believe that laws of supply and demand effect options like any tradable good. However, I ...
458 views

### CAPM Calculations

Im trying to calculate Alpha using CAPM & I have data on everything necessary. $$R_t-R_f={\alpha}+{\beta}\times(R_m-R_f)$$ i.e. $${\alpha}=R_t-R_t-{\beta}\times(R_m-R_f)$$ In more detail, I ...
303 views

### Likelihood Ratio Method - Delta

I was checking Glasserman(2004) - Monte Carlo for Financial Engineering and got to the likelihood ratio method. I am also looking in my textbook (M. Cerrato: The Mathematics of derivatives securities ...
586 views

### Shall I use the Longstaff and Schwartz method or the forward Monte Carlo method to price an American call?

I am comparing two methods: Least squares by Longstaff and Schwartz and A Forward Monte Carlo method. I am not sure what price I should consider as the "true value" to compare these two approaches. ...
4k views

Corporation $A$ has an excellent credit rating and can borrow at a fixed rate of $5\%$ or a floating rate of LIBOR + $1\%$. Corporation $B$ has a somewhat less excellent credit rating and can borrow ...
926 views

### Pathwise Derivative To Estimate Delta

I am trying to estimate delta using the pathwise derivative method (Broadie and Glasserman (1996)) and I stuck on this part: Here is the other notation defined: Here is my C++ code I have written so ...
555 views

Let's write $S(T) = S_T$ and $S(0) = S_0$. We want to compute $\frac{d}{dS_0}\mathbb{E}[f(S_T)]$. From a previous discussion this is equal to $$\mathbb{E}_{S_0}\left[f(S_T)\frac{g'_{S_0}(S_T)}{g_{S_0}(... 1answer 268 views ### Extending risk neutral measure to insurance/mortality filtration In insurance mathematics, one often models the underlying of an insurance policy with a Black Scholes model on a filtered probability space (\Omega,\mathbb{Q},\mathcal{F},\mathbb{F}=(\mathcal{F}_{t}))... 0answers 157 views ### Portfolio with several risky assets and one risk-free Suppose we have N risky assets r_1, r_2, ... , r_N with a covariance matrix C. If we want to build a portfolio \omega = (\omega_1, \ldots, \omega_N)^t (I loosely denote the portfolio with ... 2answers 117 views ### Force of interest third degree polynomial Just struggling with a question here, any help would be appreciated. For its deposits, a bank offers a force of interest per annum over a given year, which equals 4.99% at the beginning of the year, ... 1answer 252 views ### Using crude Monte Carlo Background Information: The crude Monte Carlo algorithm for the arithmetic Asian call option is$$Y = e^{-rT}(\overline{S}_A - K)^{+}$$and the control is$$C e^{-rT}(\overline{S}_G - K)^{+}$$The ... 1answer 482 views ### Dividend yield under Black 1976 formula for futures options? I have a question regarding the BS 1976 formula for futures options. https://www.glynholton.com/notes/black_1976/ How do I deal with dividends under this model, assuming that the dividend yield is ... 1answer 68 views ### Valuation functional Consider an economy with J = 2 assets and S = 3 states. The J\times S payoff matrix for the two assets is$$X = \begin{pmatrix} 0 & 3 & 3\\ 1 & 1 & 0\\ \end{pmatrix}$$and the ... 1answer 215 views ### A forward Monte Carlo method for American Options Pricing I am trying to implement the forward Monte Carlo algorithm from the paper "A Forward Monte Carlo Method for American Options Pricing" by Daniel Wei-Chung Miao and Yung-Hsin Lee. I am a little bit ... 0answers 198 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,... 1answer 714 views ### Which book would you recommend for beginners in Quantitative Finance? [closed] I have a normal high school level of mathematics including some statistics and find the world of Quantitative Finance both alien and invigorating at the same time so, would like to learn more. What ... 1answer 249 views ### Monthly returns annualized vs annual returns [closed] Lets say that I have a stock with annual returns, a_i  for year i\in \left\{1,...n\right\} and monthly returns m_{i,j} for month j\in \left\{1,...12\right\}. Lets define monthly returns to be ... 2answers 500 views ### Why are some utility functions widely used? There are some von-Neumann utility functions that I come across quite often in different articles / books like:  U(x)=\ln(x), U(x)= \frac {1}{\gamma}x^\gamma with \gamma <1 and U(x)=\frac {1-... 1answer 100 views ### Is there a quick way to see why this claim C(S, t) on S does not satisfy the Black-Scholes PDE? I'm self-studying for an actuarial exam on financial economics and encountered the below practice exam problem. An exam problem should typically take 5-6 minutes to complete, so I'm wondering if ... 1answer 91 views ### Transform raw forecasts into orthogonal forecasts I am trying to combine multiple forecasts on each of N assets in line with Grinold and Kahn's methodology, taken from Active Portfolio Management, 2nd ed. On p.311, they suggest transforming the raw ... 1answer 1k 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 ... 1answer 222 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}_\... 1answer 221 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 ... 0answers 98 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! 2answers 6k 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, ... 2answers 628 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 ... 0answers 376 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 "... 1answer 594 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 (... 0answers 198 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 = ... 0answers 57 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 ... 1answer 474 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) - ... 1answer 532 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 ... 1answer 367 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 Black-... 1answer 456 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 contract ... 1answer 54 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 ... 1answer 122 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 that ... 1answer 186 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? 1answer 159 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 ... 1answer 635 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 ... 1answer 100 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(S_t,... 1answer 511 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 ... 1answer 68 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 ... 1answer 341 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 ... 1answer 264 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)$$Let f(x) = \log X, applying ... 1answer 88 views ### Do we have a Brownian motion Background Information: The process W = (W_t:t\geq 0) is a \mathbb{P}-Brownian motion if and only if i) W_t is continuous, and W_0 = 0 ii) the value of W_t is distributed, under \mathbb{... 1answer 796 views ### Problem on Characteristic function in Heston model I know the Heston model .In this model, we have$$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,...
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 ...
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}...