Questions tagged [finance-mathematics]
The finance-mathematics tag has no usage guidance.
402
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Should we use the conditional expectation to write the value of an option?
So, I've just started looking into financial mathematics and the following question keeps bugging me. From what I understood, if the market is arbitrage-free and a given contingent claim of value $h$ ...
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0
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29
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Calculating market risk premium and expected excess return? [closed]
I need some help with this calculation.
Suppose that the CAPM is valid and that the market portfolio consists of 60% stocks and 40% bonds. The standard deviation of the
returns on stocks and bonds ...
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42
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How to measure the difference between the trend of two curves
I'm looking for a way to measure the trend of two curves. For example, the figure below is my forecast for the S&P 500. It can be seen that although I haven't predicted the exact index price, the ...
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1
vote
1
answer
66
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Vasicek interest rate of T-forward measure [closed]
I know dr of risk-neutrual measure is
There is a price of a pure-discount bond can be derived by computing the expectation, I get:
where A and B are:
why dr becomes to:
under T-forward measure?
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2
votes
1
answer
112
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Macaulay Duration - Liability matching
Can someone provide a detailed example to prove the following statement:
"When the investment horizon is equal to the Macaulay duration of the bond, coupon reinvestment risk offsets price risk.&...
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2
votes
0
answers
91
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How did Bachelier characterize the Brownian motion?
The model for a stock price
$$
dS_t=\mu dt + \sigma dB_t
$$
where $B_t$ is a Brownian motion on $(\Omega, \mathcal{F},P)$, is commonly attributed to the work that Bachelier has carried out in his PhD ...
- 221
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26
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What is the Accumulation/Distribution formula doing geometrically?
My understanding of the essence of the Accumulation/Distribution Index is that it tracks the closing price of a security during each period relative to its price range for that period, something like ...
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56
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What could be a real-life example of sectors and instruments in a Financial Market in the context of this Portfolio Optimization Problem?
Recently I've been reading about mathematical models in finances and economics; however, I encountered this book chapter:
Nagurney, A. (1993). Financial Equilibrium. In: Network Economics: A ...
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1
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1
answer
51
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Does a bond pay a coupon at maturity? [closed]
I know a bond pays an annuity cashflow of coupon payments and then at maturity it pays the face value. But, at maturity, does it pay an additional coupon payment on top of the face value or are we ...
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61
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Elliott and Kopp Mathematics of Financial Markets confusion about proof of Lemma 2.2.5
In the book "Mathematics of Financial Markets" by R. J. Elliott and P. E. Kopp, Second Edition there is a claim made in the proof of Lemma 2.2.5., namely that $V_0(\theta)=V_0(\phi)$.
I don'...
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48
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Adapted Roll measure implementation
I'm currently trying to implement the roll measure adapted by Easley et al. (2020, p. 22).
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3345183
The adapted roll measure is given by the eq below....
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21
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Counterparty Credit Risk - Reference book for a mathematical introduction [duplicate]
Which book would you recommend for an introduction to CCR from a mathematical perspective? I would like to complement the Jon Gregory's book "the XVA Challenge" with a more quant oriented ...
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1
vote
1
answer
146
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Game theoretic description of stock market
I read the book "Theory of Games and Economic Behavior" by John von Neumann and Oskar Morgenstern. I think that stock market may be described by game theory. But here are the problems:
...
- 279
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61
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Simultaneous Stochastic Differential Equations
I was thinking about cointegrated time series and came up with the following simultaneous equations model:
$dY_t = \alpha (Y_t - \gamma X_t)dt + \sigma dB_t$
$dX_t = \beta (Y_t - \delta X_t)dt + \tau ...
1
vote
0
answers
46
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Heston model (closed-form moments of returns)
Are there closed form expressions for unconditional moments (first, second) for stock returns under Heston model? Most sources provide the moments for the variance process in Heston model, but not for ...
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votes
1
answer
595
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Where does 1/2 in Fourier Transform method of pricing options come from?
I am reading Jianwe Zhu's Applications of Fourier Transform to Smile Modeling. On page 26, the author is describing how to use the Fourier tranform to price vanilla European call options. If $f_j$ is ...
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73
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Mechanism design in continuous time models
I am interested in mechanism design and information design. Is there any part of the literature in financial mathematics that is associated with them. I mean could we bring somehow these topics under ...
- 163
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answers
64
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Zero coupon price using Vasiceks model under the Real-world P measure model
I'm wondering if there is a way to work out the formula for the price of the zero-coupon bond using the Vasicek's model (P measure). I have tried to find reference on it but could not, I don't know if ...
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50
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Disecting a log diff transformation for time series analysis and prediction
I have been working in a predictive ML model that uses financial time-series as predictor variables. In one of the academic papers I used as reference, and to do feature engineering for building the ...
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0
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41
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Assymptotic behaviors of European options
In some of the numerical works on Black-Scholes generalized models, the boundary conditions on the truncated domain taken from the asymptotic behaviors of European call options, which is given by
$$\...
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2
votes
1
answer
173
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Aggregating greeks to portfolio level
I have been asked to calculate/aggregate certain Greeks (delta, gamma, and vega) up to portfolio level for a portfolio consisting of a range of (long and short) equities, convertible bonds, and ...
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2
votes
1
answer
145
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Interpolation of $\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$
Let's assume that we have SDE
$$dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$$
and we simulate it on a time grid which contains points $t_k$ and $t_{k+1}$. How can we then calculate value of $X$ at time $...
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62
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Correlated Geometric Brownian Motion - Drift rate for different stocks from different countries
I am valuing a structured product where the payout function depends on the paths of two assets. The key in my valuation is to use Monte Carlo simulations of a payout function tied to a geometric ...
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answers
97
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Where is the Quadratic Variation Coming from in this One-Factor Cheyette Model?
I am having difficulty switching from a general interest rate model (the quasi-gaussian or cheyette model) and a specific version of this model. In particular, I assume the following instantaneous ...
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73
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How do you do a geometric average if the numbers are unequally weighted?
If you have an investment of 54 months with the following returns,
year 1 = 15%
year 2 = -12%
year 3 = 14%
year 4 = 21%
last 6 months / year 4.5 = 8%,
what is the ...
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0
answers
43
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Difference between number of stocks and number of bonds: Predictable vs adapted
Let $\nu_k$ and $\eta_k$ denote the number of stocks and number of bonds in the portfolio. According to Schweizer, we need $\nu_k$ to be predictable and $\eta_k$ to be adapted. In the text, the ...
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57
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How to calculate spot rate for maturity which does not have a zero-coupon bond?
How do I calculate zero-coupon yields for a maturity which does not have an equivalent zero-coupon bond?
For instance, let's say we have this spot rate curve:
t0.5=1%
t1=2%
And a bond which has a ...
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43
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Difference between Risk minimization and local risk minimization
According to the survey paper "A Guided Tour through Quadratic Hedging Approaches" by Schweizer the risk function is defined by
$$R_t(\phi)=E[(C_T(\phi)-C_t(\phi))^2|\mathcal{F}_t]$$
When ...
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1
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220
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Game theory and stochastic calculus
Does anybody know any details of game theory literature combined with stochastic calculus in finance? If yes, please recommend some papers of any authors who are doing exceptional work on the filed. ...
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1
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56
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Evaluating principal and interest at different points in time [closed]
Consider simple interest and suppose we have a certain principal and interest at t=7 months, we want to find the value of that amount of money when t=3 months. I would like to do it in two different ...
3
votes
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285
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Deriving Bachelier Greeks
I am working on the Bachelier Model with r not equal to 0 as described in the first and most upvoted answer in following link:
Bachelier model call option pricing formula
This is fairly easy to code ...
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0
answers
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A question in information strucutres and probability measures - How are they connected?
Suppose that $\mathcal{I}=(X,\sigma^{\mathcal{X}},\mu)$ is an information strucutre, which is a probability space, where
$X=X^1\times X^2$ is the cartesian product of the individual finite sets of ...
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votes
1
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261
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Mathematical solution to return decay of daily leveraged products (leveraged ETFs)
Daily leveraged ETFs have an inherent path dependence. An index performing (5%, -5%, 5%) on 3 days has an overall performance of 2.9%. A -1x leveraged ETF would perform -3.1%. At a higher volatility, ...
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2
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417
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How to derive this mathematical equation from the perspective of the mean-variance portfolio optimization?
Question
I found a simplified inequation to decide whether the new asset A should be added to my current portfolio B. If the following inequation is satisfied, the new asset A should be added to my ...
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1
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84
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Difference in pricing of American call and put
In Paul Wilmotts quantitative finance books he says that the the value of an American option satisfies the following
$$
\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2 \frac{\partial^2V}{\partial ...
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3
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0
answers
86
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Independent variable in pricing of strongly path dependent options
I am reading Paul Wilmott on quantatative finance where he discuss the pricing of strongly path dependent options.The payoff at expiry T depends on the path taken by the asset in the sense that it ...
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0
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1
answer
499
views
How to derive the relationship between gamma and theta?
I am trying to derive this formula Θ = –0.5 × Γ × S^2 × σ^2 to see where it comes from.
My thinking is that PnL = delta dS + Vdσ + 0.5Γ(dS)^2 + Θdt.
Assume we delta hedged and vega hedged, first and ...
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3
votes
0
answers
70
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Manual Computation of Python QuantLib's NPV for Pricing of a Forward Rate Agreement
Following the question that I have asked on this link and response obtained, I have managed to price a 3x6 FRA using QuantLib Python, using the following codes:
...
- 191
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votes
0
answers
15
views
net working capital impact on free cash flow
Why does net working capital impact cash (why is it included in free cash flow calculation)? NWC is found by subtracting current assets with current liabilities none of which impact cash.
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2
votes
1
answer
202
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Reflection principle of the Brownian motion
really appreciate some guidance on how to get the following equality:
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1
answer
239
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Is IRR a Nominal or Effective Interest Rate?
The definition of the internal rate of return is the interest rate that causes the net present value to equal zero. However, interest rates can be given in two forms: nominal and effective. So, is the ...
2
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0
answers
42
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Equivalence of expectation condition for contingent claims attainable and contingent claims super replicable
We have the following definitions for set of contingent claims attainable and contingent claims super replicable
I want to prove the following result
How do I show iii $\implies $ ii.I understand ...
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36
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In-sample forecast accuracy of Beta (Kalman filter) CAPM
One can calculate time-varying betas (known from the CAPM) using the Kalman filter.
For example, one can calculate the in-sample forecast accuracy using the MAE.
$MAE = \frac{1}{T}\sum_{t=1}^T|\hat{R}...
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2
votes
1
answer
246
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Proving the discounted stock price is martingale
Let $\mathcal{K}_s$ be
$$ \mathcal{K}_s=\{\tilde{V}_t(\theta):0\leq t<\infty,\,\theta\text{ a simple strategy}\},$$
where $\tilde{V}_t(\theta)$ is the discounted value process of the self financing ...
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votes
1
answer
281
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Explanation for Different Piecewise Yield Term Structures from QuantLib Python
I am new to QuantLib on Python, but as far as I understand, there are different types of piecewise yield term structures which exist on QuantLib which are bootstrapped on a number of interest rate ...
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votes
1
answer
266
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Realised variance under simple rough volatility model
Using the Mandelbrot-Vann Ness representation of fractional Brownian motion in terms of Wiener integrals, increments of the logarithm of realized variance $v = \sigma^{2}$, under the physical measure $...
- 446
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29
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Simulating FX OPTION PRICE for Counterparty credit exposure under SA-CCR
I am looking to generate/simulate the prices for FX option price to calculate the counterparty credit exposure under SA-CCR.
However, I have some doubt since I want to use PDE (Black-Scholes model) ...
2
votes
1
answer
73
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relation between risk averson coefficient and maximum Sharp ratio in Black-Litterman context
BL model compute the implied returns based on the reverse optimization where the objective is:
$${\underbrace U_{{\rm{investor's \ risk \ utility}}} \buildrel \Delta \over = {\bf{w}}_M^T{\bf{\Pi }} - \...
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answers
49
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Mathematical Model in Overlapping games
Does anybody know, where can I find more details about the mathematical model of Karatzas and Shubik A Stochastic Overlapping Generation Economy with Inheritance, July 2000? I need some help and ...
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1
answer
158
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In what cases characteristic function of (log-)price process is known?
Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant ...
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