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Questions tagged [finance-mathematics]

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12 views

Understand conditional expectation w.r.t. sigma -algebra [migrated]

When a random variable $X$ is discrete, the definition of conditional expectation of $X$ with respect to a decomposition $\mathscr D$ is $$ E[X|\mathscr D] = \sum_{i = 1}^m x_i \sum_{j = 1}^n P(X|\...
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17 views

Survival rate for mortgage loans

I am trying to search for some papers that calculate the survival rate for mortgage loans based on a number of characteristics like loan-to-value, FICO, etc... The survival rate is essentially a ...
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0answers
21 views

The deflator of reinsurance market is unique

How to prove that the deflator $\phi$ of the reinsurance market is unique when working in the equilibrium model? That is, if we have a pricing function $\pi$, which satisfies: $$\pi(Y)=E(\phi Y)$$ ...
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0answers
18 views

Prove coherency scenario risk measure

The scenario risk measure is defined as follows: $max\{L_i(x) : x \in X\}$, Monotonicity, translation invariance and positive homogeneity follow trivially, but i'm wondering how to prove ...
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1answer
56 views

Why do most interest rate formulas, and indeed finance in general, add 1 to a rate and then subtract afterwards? [closed]

For example, in the formula that shows the relationship between the nominal and effective interest rate shown below, 1 is first added to in/m and then 1 is subtracted from the result. What is the ...
3
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1answer
77 views

Hedging Value-Financial Mathematics

EXERCISE We consider a free from arbitrage financial market $(Ω,F,P,S_0,S_1)$ with $α<S_0^{1}\cdot(1+r)<β$,where $$0<α:=min_{ω \in Ω} S_1^{1}(ω), β:=max_{ω \in Ω}S_1^{1}, α<β$$ Let ...
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1answer
66 views

Equivalent martingale measure exists if and only if $a < S_0^1(1+r)< b$

Exercise : We consider a market of one period $(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$, where the sample space $\Omega$ has a finite number of elements and the $\sigma-$algebra $\mathcal{F} = 2^...
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34 views

How to draw a binomial option tree graph Using matlab/R/C++/JAVA?

I am doing a project and need to create a image/pdf for binomial option price tree. I want to create like this picture : Where I have u(up rate),d(down rate) ,Stock price , Strike price and Time. I ...
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0answers
18 views

Estimating monthly return based on known yearly return and different monthly dataset

I have two data sets: (1) Data set of monthly equity returns (Index 1) and (2) data set of yearly equity returns (Index 2) I want to estimate the monthly returns in Index 2 using the known yearly ...
2
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1answer
89 views

Showing that a market model has arbitrage and describing martingales

This is an exercise which I came upon while studying an introduction to financial mathematics. Exercise : Consider the finite sample space $\Omega = \{\omega_1,\omega_2,\omega_3\}$ and let $\...
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0answers
47 views

Hull White 2 Factor

I have calibrated HW 1 Factor using Caps using closed form solutions like these. Could someone please direct me to a paper where they done something similar for 2 factor which i can code in python......
3
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1answer
133 views

Total Returns From Adjusted Close Prices

I'm trying to understand why the total return (return including dividends) that I get from calculating return using adjusted close price, does not equal the total return calculated in another manner. ...
1
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1answer
41 views

Approximation of Forward Rates in discrete time

The forward rate from time $t$ to $T$ ($f_{t,T}$) can be approximated by: $$ f_{t,T}= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{{T-t}}}-1 \sim \frac{(1+r_T)^T-(1+r_t)^t}{T-t} $$ Why is ...
1
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1answer
76 views

Replicating portfolios [closed]

Prices of a stock are modeled using a two-period binomial tree, with each period being six months. The continuously compounded risk free interest rate is 7 % The stock pays 2 % continuous dividend. ...
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0answers
51 views

Pricing methods in the real world when there is more than one free arbitrage price

Perhaps this question sounds trivial and obvious, but I am starting to study this new field. When we are in a complete market without arbitrage opportunities there is only one risk-neutral martingale ...
3
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0answers
50 views

Price of a stochastic game between an agent and the market

In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as \begin{align} V(x,C) = \dfrac{1}{\...
2
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1answer
74 views

The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market

I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
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1answer
42 views

standard brownian vs brownian motion

We say Xt with paramters (µ,σ) is brownian process if (Xt-s - X t) ~N (µs,σ2 s) AMONG other conditons . Here we don't speak about any particular distribution for X t. We only say it is a brownian ...
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0answers
58 views

Pricing caplet with Bachelier (normal dynamic) using forward measure

I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution. Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing : $C_t(T,T+d)$ ...
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0answers
19 views

How to maximise utility in multi-periode model using martingale method

I am reading chapter 2.2 in Pascucci & Runggaldier (2012), Financial Mathematics - Theory and Problems for Multi-period Models about how to maximise utility in multi-periode model using martingale ...
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4answers
231 views

Determine the right order size with market making strategy

In a market market strategy https://web.stanford.edu/class/msande448/2017/Final/Reports/gr4.pdf, how can we determine the right order size? Assuming I use a market making strategy and on a specific ...
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1answer
87 views

Cash Flow News and Discount Rate News + Return

I will appreciate If someone help me to understand how the final expansion is made. Specifically, how CF & DR are drived. This model is introduced by Chen et. al. (2013).What Drives Stock Price ...
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0answers
29 views

Arbitrage and state price formulation

I must be missing something really obvious due to my temporary obtuseness. Can someone please help me see the obvious? :-P Thank you. I am just browsing Darrell Duffie's Dynamic Asset Pricing Theory. ...
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0answers
28 views

Utility Maximization on a finite Probability Space. Possible mistakes in a paper?

I am currently reading this paper on utility maximization in a financial market model. On page 5 the author starts with the case of a finite probability space and on page 19 he considers the ...
4
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1answer
208 views

Struggling with tau in Black-Litterman

According to the omega formula in B-L tau is used in the Omega estimation to determine the degree of uncertainty given to views of the investor: So, if tau is given a low value then the inverse of ...
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0answers
68 views

Dubious math in Thorp's magnum opus

I started reading Thorp's "Beat the Market" book and stumbled on a formula I can't figure out: https://imgur.com/a/xqfViKt What's the point in adding time to price and the whole probabilites ...
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0answers
27 views

How to examine the impact of the parameters in the Hull White Model on the yield curve

I want to examine the yield curve resulting from the 2 Factor Hull-White model. Is there any way to examine the influence of the parameters on the yields curve without calibrating the model?
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0answers
31 views

Impact analysis of parameters in the 2 Factor Hull White Model

Through the 2-Factor-Hull White Model you can model the yield curve if you have the parameters $a, b, \sigma, \eta$ given. Is there any way to measure the impact of these parameters on the yield ...
10
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1answer
347 views

Market Making Strategies Found by Hamilton-Jacobi-Bellman Equation

Im working my way through the book "Algorithmic and High-Frequency Trading" (AHFT) by Cartea, Jaimungal and Penalva and i'm curious to see how the market making model with an exponential utility ...
0
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1answer
53 views

Develop a pricing formula for an American digital put option

This problem comes from concepts and practice of mathematical finance by Joshi Chapter 8 problem 9. Develop a pricing formula for an American digital put option Joshi's solution - He states that ...
5
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1answer
218 views

Modelling EUR/USD rate with Ornstein-Uhlenbeck model

I have a data set of daily EUR/USD rate for time period 2000-2018. My goal is to model future behaviour of this financial time series using Ornstein-Uhlenbeck model: $$d X_t = \alpha (\theta - X_t) ...
3
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1answer
242 views

Change of numeraire in options with currency exchange features

FV of an EUR denominated option under "COP" risk measure is given by: $$V_t^{COP} = D^{COP} \mathbb{E}_t^{COP} \left[X_T(S_T -K)^+\right]$$ where $X_T$ is the exchange rate COP/EUR. Pricing the ...
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1answer
117 views

Negative VaR equivalent Volatility (VEV) and its meaning?

Can a VaR equivalent Volatility (VEV) as defined by KID/PRIIPS law be negative and what does it mean if it has a negative value?
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0answers
42 views

Return / yield of an ATM-Call Option on a zero coupon bond

The zero-coupon bond with unit face value and maturity S for a call option with maturity T and strike K is given by: The bond prices $P(t,T)$ and $P(t,S)$ $$\begin{aligned} ZBC(t,T,S,K) = & P(t,S) ...
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0answers
34 views

Literature recommendation subordinator models

I'm looking for relevant papers covering subordinator models for stock price modelling. I have alreay read the paper 'A Subordinated Stochastic Process Model with Finite Variance for Speculative ...
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0answers
85 views

are there quantitative tools to work with python pandas?

I have a code written in python and I have all my data in pandas dataframe. Are there quantitative tools to apply to these dataframes? For example, I want to calculate rsi, macd and other financial ...
1
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1answer
101 views

Finding the process of $X/Y$

This comes from Mark Joshi's concepts of mathematical finance exercise 4 chapter 11. If $$dX_t = \alpha X_t dt + \beta X_t dW_t$$ $$dY_t = \alpha Y_t dt + \gamma Y_t d\tilde{W}_t$$ with $W$ ...
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1answer
40 views

Calculate min. win ratio needed for a bet to be profitable [closed]

If a bet 12000 to win 4000 my risk/reward ratio is .33 . How often must I win the bet to be profitable? I know it's 75% but have not found the formula yet.
1
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1answer
79 views

Mathematical definition of a hedge?

For two given portfolios/trading strategies I want to know what criteria need to fulfilled in order to call the one portfolio a hedge to the other. In other words; what is the mathematical definition ...
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2answers
352 views

Stop-loss start-gain paradox: Why is it a 'paradox'?

The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic and Time Value, by Peter P. Carr and Robert A. Jarrow, in The Review of Financial Studies, Volume 3, Issue 3, ...
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0answers
50 views

Computing the Expectation to a Max function

if $X_T$ is log-normally distributed and $k$ is a constant, how do I compute: $$E[\max(X_T-k,0)]$$ I can compute $E[X_T-k]$ and $P(X_T-k>0)$. I was thinking that an approach will be compute to $$E[...
5
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2answers
1k views

Long Gamma vs Vega

What is the difference between being long gamma and being long Vega? I understand that gamma is the vol of delta and that vega is the vol of the underlying. However, I have also found that being long ...
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0answers
36 views

Annuity calculcation

Let $A_k$ denote the annuity paid at the end of year $k.$ If $V_0$ is the amount borrowed during $n$ years with an annual interest rate $r$ and the borrower pays at the end of each period $k$ ...
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3answers
117 views

Does longer time horizon necessarily imply reduced risk?

Is there a mathematical/statistical basis for the commonly-held belief that the longer certain assets (particularly equities) are held, the less risk the investor is exposed to? Alternatively, is ...
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0answers
37 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-...
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1answer
43 views

4-point Trapezium rule for numerical integration

Background: This is in reference to Mark Joshi's concepts of mathematical finance ch.7 problem 11. Question: We have in the Black-Scholes model: $S_0 = 1, T = 1, \sigma = 0.1, r = 0$. A ...
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0answers
41 views

Anti-thetic sampling and second moment matching

Background: This is in reference to ch 7 problem 10 of Mark Joshi's concepts of mathematical finance. Question: A normal random generator produces the following draws: $$0.68, -0.31, -0.49, -0....
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1answer
49 views

Withdrawing monthly from a bank for 40 years [closed]

Consider you have $\$104107.4099$ in the bank with a $.33\%$ monthly effective interest rate. You plan to withdraw a fixed amount X every month for 40 years, such that you make 480 withdrawals in ...
0
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1answer
64 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} ...