Questions tagged [finance-mathematics]
The finance-mathematics tag has no usage guidance.
0
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0answers
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
$$
E[X|\mathscr D] = \sum_{i = 1}^m x_i \sum_{j = 1}^n P(X|\...
0
votes
0answers
17 views
Survival rate for mortgage loans
I am trying to search for some papers that calculate the survival rate for mortgage loans based on a number of characteristics like loan-to-value, FICO, etc... The survival rate is essentially a ...
1
vote
0answers
21 views
The deflator of reinsurance market is unique
How to prove that the deflator $\phi$ of the reinsurance market is unique when working in the equilibrium model?
That is, if we have a pricing function $\pi$, which satisfies:
$$\pi(Y)=E(\phi Y)$$
...
0
votes
0answers
18 views
Prove coherency scenario risk measure
The scenario risk measure is defined as follows:
$max\{L_i(x) : x \in X\}$,
Monotonicity, translation invariance and positive homogeneity follow trivially, but i'm wondering how to prove ...
0
votes
1answer
56 views
Why do most interest rate formulas, and indeed finance in general, add 1 to a rate and then subtract afterwards? [closed]
For example, in the formula that shows the relationship between the nominal and effective interest rate shown below, 1 is first added to in/m and then 1 is subtracted from the result. What is the ...
3
votes
1answer
77 views
Hedging Value-Financial Mathematics
EXERCISE
We consider a free from arbitrage financial market $(Ω,F,P,S_0,S_1)$ with $α<S_0^{1}\cdot(1+r)<β$,where $$0<α:=min_{ω \in Ω} S_1^{1}(ω), β:=max_{ω \in Ω}S_1^{1}, α<β$$
Let ...
1
vote
1answer
66 views
Equivalent martingale measure exists if and only if $a < S_0^1(1+r)< b$
Exercise :
We consider a market of one period $(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$, where the sample space $\Omega$ has a finite number of elements and the $\sigma-$algebra $\mathcal{F} = 2^...
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0answers
34 views
How to draw a binomial option tree graph Using matlab/R/C++/JAVA?
I am doing a project and need to create a image/pdf for binomial option price tree.
I want to create like this picture :
Where I have u(up rate),d(down rate) ,Stock price , Strike price and Time.
I ...
0
votes
0answers
18 views
Estimating monthly return based on known yearly return and different monthly dataset
I have two data sets: (1) Data set of monthly equity returns (Index 1) and (2) data set of yearly equity returns (Index 2)
I want to estimate the monthly returns in Index 2 using the known yearly ...
2
votes
1answer
89 views
Showing that a market model has arbitrage and describing martingales
This is an exercise which I came upon while studying an introduction to financial mathematics.
Exercise :
Consider the finite sample space $\Omega = \{\omega_1,\omega_2,\omega_3\}$ and let $\...
0
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0answers
47 views
Hull White 2 Factor
I have calibrated HW 1 Factor using Caps using closed form solutions like these.
Could someone please direct me to a paper where they done something similar for 2 factor which i can code in python......
3
votes
1answer
133 views
Total Returns From Adjusted Close Prices
I'm trying to understand why the total return (return including dividends) that I get from calculating return using adjusted close price, does not equal the total return calculated in another manner.
...
1
vote
1answer
41 views
Approximation of Forward Rates in discrete time
The forward rate from time $t$ to $T$ ($f_{t,T}$) can be approximated by:
$$
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
vote
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.
...
1
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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
votes
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}{\...
1
vote
1answer
76 views
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 ...
0
votes
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 ...
1
vote
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)$ ...
0
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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 ...
2
votes
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 ...
1
vote
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 ...
0
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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. ...
2
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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?
0
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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 $P(t,S)$
$$\begin{aligned} ZBC(t,T,S,K) = & P(t,S) ...
1
vote
0answers
34 views
Literature recommendation subordinator models
I'm looking for relevant papers covering subordinator models for stock price modelling. I have alreay read the paper 'A Subordinated Stochastic Process Model with Finite Variance for Speculative ...
0
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0answers
85 views
are there quantitative tools to work with python pandas?
I have a code written in python and I have all my data in pandas dataframe. Are there quantitative tools to apply to these dataframes? For example, I want to calculate rsi, macd and other financial ...
1
vote
1answer
101 views
Finding the process of $X/Y$
This comes from Mark Joshi's concepts of mathematical finance exercise 4 chapter 11.
If
$$dX_t = \alpha X_t dt + \beta X_t dW_t$$
$$dY_t = \alpha Y_t dt + \gamma Y_t d\tilde{W}_t$$
with $W$ ...
-2
votes
1answer
40 views
Calculate min. win ratio needed for a bet to be profitable [closed]
If a bet 12000 to win 4000 my risk/reward ratio is .33 .
How often must I win the bet to be profitable?
I know it's 75% but have not found the formula yet.
1
vote
1answer
79 views
Mathematical definition of a hedge?
For two given portfolios/trading strategies I want to know what criteria need to fulfilled in order to call the one portfolio a hedge to the other. In other words; what is the mathematical definition ...
7
votes
2answers
352 views
Stop-loss start-gain paradox: Why is it a 'paradox'?
The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic and Time Value, by Peter P. Carr and Robert A. Jarrow, in The Review of Financial Studies, Volume 3, Issue 3, ...
0
votes
0answers
50 views
Computing the Expectation to a Max function
if $X_T$ is log-normally distributed and $k$ is a constant, how do I compute:
$$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
votes
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 ...
0
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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 $k$ ...
2
votes
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 ...
1
vote
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-...
1
vote
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 ...
0
votes
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:
$$0.68, -0.31, -0.49, -0....
-2
votes
1answer
49 views
Withdrawing monthly from a bank for 40 years [closed]
Consider you have $\$104107.4099$ in the bank with a $.33\%$ monthly effective interest rate. You plan to withdraw a fixed amount X every month for 40 years, such that you make 480 withdrawals in ...
0
votes
1answer
64 views
Properties of Brownian motion and filtration, Exercise 6.22, Joshi Concepts and applications to mathematical finance
Let $W_t$ be a Brownian motion, and let $F_t$ be its filtration then for $t > s$ we are asked to compute
$$\mathbb{E}\left[W_t^2|F_s\right]$$
We have $$W_t = W_s + (W_t - W_s)$$
and
$$W_t^{2} ...
