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0answers
5 views

Analytical Bond Price under Rendlemen-Bartter?

Assuming the short rate $r_t$ follows the risk-neutral (so $W_t$ is a $Q$-Brownian motion) process $$ dr_t = ar_t dt + \sigma r_t dW_t, $$ does anyone know of an analytical bond price formula? We ...
10
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3answers
2k views

How to estimate real-world probabilities

In the world of finance, Risk-neutral pricing allow us to estimate the fair value of derivatives using the risk free rate as the expected return of the underlyings. However, the behavior of ...
19
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3answers
955 views

How are distributions for tail risk measures estimated in practice?

Let's say you want to calculate a VaR for a portfolio of 1000 stocks. You're really only interested in the left tail, so do you use the whole set of returns to estimate mean, variance, skew, and shape ...
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1answer
77 views

Prove $Y_t$ is a martingale by considering $dZ_t$ and $dL_t$

Suppose we are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard ...
0
votes
1answer
18 views

Scaling of probability mass function

Given a histogram and the probability mass function values for each observation, when plotting the histogram and the curve (this is bell curve since the data is assumed to be normal) on the same ...
0
votes
1answer
21 views

Creating the histogram for the distribution of the portfolio returns

Given log returns for some stocks $A$ and $B$, which are the constituents of our hypothetical portfolio in equal weights, how does one actually come up with a distribution of the log returns of the ...
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0answers
18 views

Altman Z-Score to Probability of default

I have computed the Altman Z Score for approximatly 2500 companies. I was wondering if mathematically I am allowed to use a logistic function ? Such as: ...
7
votes
1answer
247 views

Distribution of hitting time of the integrated CIR process

If an increasing process $X_t$ has a known Laplace transform $\mathbb{E} e^{-s X_t} = m_t(s)$, define its hitting time $\tau$ to some level $B$ to be $$ \tau = \inf\{ u > 0 : X_u \geq B \}. $$ Can ...
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9answers
3k views

Lévy alpha-stable distribution and modelling of stock prices.

Since Mandelbrot, Fama and others have performed seminal work on the topic, it has been suspected that stock price fluctuations can be more appropriately modeled using Lévy alpha-stable distrbutions ...
0
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0answers
65 views

Large deviations theory and extreme value theory

I'll enter into details of both, sooner or later, but for the moment I'm concerned about the differences (and relationships, if any) between these two theories. Can someone give me a brief, but still ...
5
votes
2answers
89 views

Calculating probability of Yuan's slump from options market

http://www.bloomberg.com/news/articles/2016-01-06/if-options-traders-are-right-the-yuan-s-slump-is-far-from-over Contract prices indicate a 79 percent probability that the currency will weaken ...
3
votes
3answers
105 views

How much would one pay for the max of two stocks?

I'm trying to figure out if stochastic calculus is the right approach for this problem... but I only vaguely understand it and I am trying to gauge if I need to spend the time learning measure theory ...
0
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0answers
22 views

Compute stock price probability distribution from option data (IB method & negative probabilities issue)

I'm using a procedure as described in the interactive brokers article here (https://www.interactivebrokers.com/en/index.php?f=5910&ns=T) to compute a probability distribution from option (call) ...
3
votes
2answers
225 views

Pricing when arbitrage is possible through Negative Probabilities or something else

Assume that we have a general one-period market model consisting of d+1 assets and N states. Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with ...
3
votes
2answers
99 views

Asymmetric Random Walk / Prove that $E[T:= \inf\{n: X_n = b\}] < \infty$

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
1
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1answer
61 views

Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale

We are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $\mathbb ...
1
vote
1answer
39 views

Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
5
votes
1answer
83 views

Radon-Nikodym: Changing Distribution vs Changing Random Variable

Let $X \sim \mathcal{N}(\mu,\sigma^2)$ under the probability measure $P$ on the measurable space $(\Omega, \mathcal{F})$. We may define a Radon-Nikodym derivative $Z$, also defined on $(\Omega, ...
1
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1answer
41 views

Negative risk neutral probabilities economic argument

We know of plenty ways to extract risk neutral distirbutions from option prices (for example Breeden Litzberger) but there is no real analysis on how to interpret negative state prices (Haug 2007 for ...
3
votes
1answer
199 views

open problems in mathematical finance

What are open problems in mathematical finance that use fundamental concepts of mathematics (functional analysis, geometry and topology, algebra and number theory etc.) and not data-driven. I have ...
3
votes
2answers
112 views

Proving Derivative Property of Moment-Generating Function

In Shreve II, exercise 1.8, he walks the reader through proving the derivative of a moment-generating function $\phi$ is equal to the expectation $\mathrm{E}[Xe^{tX}]$; i.e., $$ \phi^\prime(t) = ...
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0answers
16 views

Selling two uncorrelated OTM options lowers the over all probability of profit?

I am trying to simulate shorting two uncorrelated put options, I wrote a python program and used monte carlo method to simulate the PnL on expiration: gist It seems the probability of profit is ...
2
votes
1answer
441 views

Empirical copula

I am trying to find the empirical copula linking two random variables $X$ and $Y$. I have some data available but it's limited with respect to the variable $Y$ and I am not convinced it's enough data ...
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0answers
46 views

How to fit a copula to empirical data?

There are numerous types of Copulas one can choose to fit empirical data. My question is wow to select the 'best' copula to fit the data. More specifically, let's assume empirical data ...
1
vote
0answers
17 views

Multiple similar values simulation

Perhaps some of you came across the following task that I am trying to automate for @RISK, VOSE or other simulation software? I have a question as we are trying to use the software to estimate the ...
5
votes
1answer
191 views

KMV-Merton Probabilties of Default vs Moody's EDF

Moody's used to publish probability of default estimates from their Moody's EDF model, but they have temporarily discontinued it. I understand that the Moody's EDF model is closely based on the Merton ...
2
votes
1answer
102 views

Probability of Stock breaching barrier

If a stock has a process: $dS(t) = sigma*dB(t)$, where $B(t)$ is a standard Brownian motion, and current stock price is $S(0)$. There is a barrier $H>S(0)$. What is the probability that the stock ...
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votes
6answers
12k views

How to calculate stock move probability based on option implied volatility and time to expiration? (Monte Carlo simulation)

I am looking for one line formula ideally in Excel to calculate stock move probability based on option implied volatility and time to expiration? I have already found a few complex samples which took ...
0
votes
2answers
110 views

Future spot price versus current forward price

Which are the two conditions necessary to claim that the future spot price will have as many chances to be above or below the current forward price?
3
votes
2answers
79 views

Probability of Closing Stock Price Over a Defined Period

I found this equation when I was reading "Paul Wilmots on Quantitative Finance" which calculates the probability that of a stock price ending/landing on a particular price (S'). So if the stock ...
3
votes
0answers
118 views

Is there a countably infinite Sigma-Algebra? Why?

Assume $\,\mathcal{F}$ be a nonempty collection of subsets of $\Omega$. $\,\mathcal{F}$ is called a $\sigma$-Algebra whenever if $A\in\mathcal{F}$ then $A^c\in\mathcal{F}$, and if ...
5
votes
1answer
136 views

Beta between stock and option

In Black Scholes model I would like to compute $$ \beta_K = \frac{\mathrm{cov}(C_{K,T},S_T)}{\mathrm{cov}(S_T,S_T)} = \frac{\mathrm{cov}((S_T - K)^+,S_T)}{\mathrm{cov}(S_T,S_T)} $$ with respect to say ...
5
votes
2answers
132 views

Do futures follow physical or risk-neutral distributions

I've spent a while looking for an answer to this question and while I feel it is a simple question I have not found an answer. I know prices of option contracts follow an implied, risk-neutral ...
4
votes
2answers
58 views

Joint probability distribution only measures product sets?

According to these notes (top of p 133), "We say that random variables $X_1, X_2, \ldots X_n : \Omega \to \mathbb{R}$ are jointly continuous if there is a joint probability density function $p(x_1, ...
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1answer
89 views

Proving $\mathbb{E}(g(X)) = \int_{\mathbb{R}} g(x) f(x) dx$

Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, P)$ and let $g$ be a Borel-measurable function on $\mathbb{R}$. In Shreve II (p 28) he proves, using the standard machine, ...
6
votes
0answers
96 views

pdf of simple equation, compound Poisson noise

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t dt + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
1
vote
2answers
193 views

Confidence Intervals of Stock Following a Geometric Brownian Motion

In preparation for my Options, Future's and Risk Management examination next week, I have been presented with a series of questions and their answers. Unfortunately, my lecturer, one of the less ...
4
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1answer
166 views

Lipschitz condition in mathematical finance

I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance. To be specific, suppose we want to ...
5
votes
2answers
5k views

t-statistics for the mean return, using Newey-West standard errors

I have seen that in several papers, where the aim was to evaluate the performance of a certain investment strategy, they use t-statistics to test for significance in the results. However, this seems a ...
2
votes
2answers
89 views

Sums of random variables and independence

I'm having troubles with this proof: Let $\{Z_i\}_{i\in\mathbb{Z}}$ be i.i.d. random variables with zero mean and unit standard deviation. For $(a_0, a_1, ..., a_r)$ a sequence of $r$ real numbers ...
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0answers
73 views

How do you calculate the asset drift rate in the Merton model? (used in N(-d2))

Can I replace it with the internal rentability rate? I only have the financial statements and some market data, but I can't find the expected returns anywhere. The goal is to calculate the probability ...
1
vote
3answers
132 views

Direct use of implied volatility

I am not sure to understand exactly the direct use of implied volatility. Let's take an example: if an instrument has a daily volatility of $\sigma$, there is a 68% probability that its value will be ...
2
votes
2answers
180 views

probability question about brownian motion

Assume $W_{t}$ is a standard Brownian Motion, calculate the the probability that $W_{t}*W_{2t}$ is negative, i.e., $P(W_{t}*W_{2t}<0)$. I find it tricky to calculate the probability.Thank you.
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0answers
18 views

Value of sequential mutually-exclusive options (A variant on the Secretary Problem)

How much should you offer a potential hire in a signing bonus? Imagine you are interviewing a list of candidates for a particular job. Each candidate has a "lifetime value", and probability of ...
0
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1answer
52 views

probability that the stock price is below the strike price

How can I prove that under the risk-neutral probability: $\mathbb{P}[S_{t}<K]=-\frac{\partial{C}}{\partial{K}}(K,T)$ where $S_{t}$ is the stock price, K is the strike price, C is the call ...
9
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0answers
235 views

Quantum Mechanics and Economics… What

I was reading this paper: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2002698&download=yes The author has the model presented here: ...
6
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0answers
88 views

Transition densities in the Heson model

Knowing the Characteristic function $\Phi_{T,t} = \mathbb{E} [ e^{i u S_T} | S_t, V_t]$ (or equivalently, the Laplace transform) of an affine process, it's possible to know the distribution of the ...
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8answers
6k views

Probability of touching

For a vanilla option, I know that the probability of the option expiring in the money is simply the delta of the option... but how would I calculate the probability, without doing monte carlo, of the ...
9
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1answer
352 views

Distribution of Geometric Brownian Motion

Please let me know where I have been mistaken! Let the SDE satisfied by the GBM $S(t)$ be $$ \frac{dS(t)}{S(t)} = \mu dt + \sigma dW(t). $$ Then, the underlying BM $X(t)$ will satisfy $$ dX(t) = ...
1
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1answer
105 views

To lump or not to lump

Suppose I have a very simple asset whose price takes only three possible values: $X_t\in \{-1,0,1\}$. I also got some discrete time series $X = (X_t)_{t\geq 0}$ and I would like to come up with a ...