Questions tagged [finance-mathematics]

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11
votes
1answer
651 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 ...
9
votes
3answers
390 views

Why is it useless to model stochastic volatility when pricing Vanilla style derivatives?

With respect to the answer by user AFK in Ideas about Stochastic volatility models. I am specifically interested in interest rate options (IR Caps/Floors and Swaptions).
8
votes
3answers
6k views

What mathematical theory is required for high frequency trading?

I am an applied math postdoc and I have been presented with the option of leaving academia to work in high frequency trading. I wanted to get a feel for the field and the theory underlying it so I ...
8
votes
1answer
290 views

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 ...
7
votes
2answers
503 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, ...
7
votes
2answers
691 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 ...
6
votes
2answers
3k 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 ...
6
votes
1answer
356 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) ...
5
votes
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 ...
5
votes
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 ...
4
votes
1answer
588 views

Finding arbitrage opportunity

Find an arbitrage opportunity in this market. Can anyone explain how to mathematically solve this exercise with for example solving a system of linear equations?
4
votes
1answer
85 views

Barrier Option from binomial tree

What is the smallest information structure that is required for using the binomial tree to calculate the price of a barrier (up-and-in) option? My gut feeling is any node below the node that reaches ...
4
votes
1answer
411 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 ...
4
votes
1answer
802 views

How to price and find a replicating portfolio for a call spreads using a two-period binomial model?

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.03$ and $l = 0.98$. a.) If the interest rate for both periods is $R = .01$, find the ...
4
votes
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 ...
4
votes
1answer
58 views

How to prove that the expected squared error associated with the optimal combination weight is smaller than the minimum of 2 forecast variances?

I am looking at linear combination of two forecasts (Bates and Granger, 1969). I would like to understand how to prove that the expected squared error associated with the optimal combination weight is ...
4
votes
1answer
802 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 ...
4
votes
0answers
57 views

Why does risk-neutral price processes do not, in general, compose all arbitrage-free price processes?

I was reading reviewing my mathematical finance notes and I came across a remark I cant understand fully Remark :Contrary to discrete time models, the risk-neutral price processes do not, in general, ...
4
votes
0answers
117 views

How to find a probability of VIX moving from one price to another

I asked a similar question on here with a bounty. I decided to modify the question to simplify what I am trying to do. Is there a package on MATLAB or some other tool where I can find the probability ...
4
votes
0answers
115 views

How do I calculate the present value of a credit default swap?

I am paid 20 million every time a bond drops to a new low over a 120 month period. I need to know how to find the present value of such an arrangement if there is a continuously compound interest of 5 ...
3
votes
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
votes
1answer
248 views

Mathematical equation relating $\frac{dV}{dS}$ to $\frac{dV}{dK}$

Please help me figure out what is the mathematical relationship between $\frac{dV}{dS}$ (Delta) and $\frac{dV}{dK}$ ($K$=strike), taking into account vol skew. I ask this because I want to figure out ...
3
votes
4answers
331 views

Mark Joshi Quantitative finance numerical techiniques, writting an algorithm that produces a random variable

Background: I am preparing for interviews and I was told to try and answer as many problems in the Mark Joshi book as possible. Question: Suppose an asset takes values from a discrete set $v_j$ ...
3
votes
2answers
786 views

Understanding the solution of this integral

The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price): $$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\...
3
votes
2answers
774 views

Basic question on Portfolio Theory

I was revising my stuff about portfolio theory and I noticed that every single time, expected return and corresponding variance or covariance are given! (not calculating ourselves). So I'm just ...
3
votes
4answers
353 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 ...
3
votes
3answers
196 views

Getting sets of random correlated variables

For the training of a machine learning model I need to add additional features (macro variables), and these features are correlated. I need to run the model N times, and for each time I have to add ...
3
votes
2answers
296 views

Application of Ito's lemma

Let $X_t$ be some stochastic process driven by wiener process ($W_t)$ so it can be expressed as: $$dX_t=(...)dt+(...)dW_t$$ Let $f(t,x)$ be some $C^2$ function. Define the process $Z_s=f(t-s,X_s)$ ...
3
votes
2answers
411 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-...
3
votes
2answers
502 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 ...
3
votes
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 ...
3
votes
1answer
70 views

How to comprehend this notation?

I learned mathematical finance from Bjork's Arbitrage Theory in Continous Time, and never once did I encounter the "quadratic variation"-thingy with the angle brackets. So now that I am reading ...
3
votes
1answer
234 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^...
3
votes
1answer
474 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. ...
3
votes
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,...
3
votes
3answers
201 views

existence of implied volatility

I read a book where it was written : 1/ "implied volatility is the market's consensus on the volatility of the asset between now and the maturity of the option". -> Could someone explain me this ...
3
votes
2answers
108 views

How to Mathematically Prove Markets are Price-Discovering?

We all know that the Efficient Market Hypothesis is true if you're willing to make enough simplifying assumptions about the market participants. But where can I find a mathematical proof of this in ...
3
votes
1answer
148 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 ...
3
votes
2answers
136 views

Verifying two properties of the Clayton Copula

So I'm trying to verify the first two properties of a copula for the Clayton model. The first two properties being: $C(u_1,…,u_d)$ is non-decreasing in each component, $u_i$ The $i^{th}$ marginal ...
3
votes
1answer
84 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 ...
3
votes
1answer
110 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 $...
3
votes
2answers
246 views

Equivalent martingale measure price dynamics

Assume $S_0(t)=\exp(\int_0^t r(s) ds)$. Then $\mathbb{Q}\sim \mathbb P$ is a martingale measure $\iff$ every asset price process $S_i$ has price dynamics under $\mathbb Q$ of the form $dS_i(t)...
3
votes
1answer
470 views

European call delta derivation

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}(...
3
votes
1answer
173 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 ...
3
votes
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 ...
3
votes
1answer
576 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 ...
3
votes
1answer
256 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 ...
3
votes
2answers
846 views

Mathematical definitioln of Potential Future Exposure

I have come across a risk measure called "Potential Future Exposure" and I have not really understood the meaning of it. Knowing that this has to do with counterparty credit risk, I read different ...
3
votes
3answers
288 views

How to understand nonrandom/random process in Shreve book? [closed]

I have been reading Chapter 4 of Shreve's Stochastic Calculus for Finance II. It is easy to understand the simple process, $\Delta(t)$, defined on Page 126, which is just a constant inside a given ...
3
votes
3answers
2k views

Black Scholes Constant Implied Volatility

I hope someone can clarify my ideas about the constant implied volatility in the classical Black Scholes framework. As well known, market practitioners quote the prices of vanilla call and put ...