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### Given Brownian motion $B_t,B_s$ and $t>s$, how to calculate $P(B_t>0,B_s<0)$?

As stated, this is an interview question. Given Brownian motion $B_t,B_s$ and $t>s$, how to calculate $P(B_t>0,B_s<0)$?
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### Can the concept of negative probabilities be used to price a call option?

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 ...
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### $\mathbb{P}$ and $\mathbb{Q}$ probability measure/distribution interpretations

I'm trying to understand probability distributions implied from market prices and was reading through this reference explaining the interpretation of $N(d_1)$ and $N(d_2)$ in the log-normal vol Black-...
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The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \... 2answers 296 views ### How can we have negative probabilities in finance? Can we have negative payments in bonds? If not, how else can we have negative probabilities? In Half of a Coin: Negative Probabilities, the author mentions bond duration. Suppose we have payments at times$t = 1,2,...,n$denoted respectively by$R_1, R_2, ..., R_n$and the discount factor is ... 0answers 31 views ### logistic regression multivariable fractional ploynomials stata vs. R I a going through Hosmer, Lemenshow and Sturdivant's (HLS) Applied Logistic Regression (2013) and trying to interpret the difference between what STATA is doing and what R is doing. Concerning the fit ... 0answers 27 views ### Negative probabilities - what are the two ordinary pgfs that correspond to the gf of a half-coin? In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable,$X = 1_H$: $$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{... 2answers 254 views ### Brexit implied probability It is possible to bet on the Brexit e.g. on this page: https://sports.ladbrokes.com/en-gb/betting/politics/british/eu-referendum/uk-european-referendum/220800266/ The quotes are 8/15 for remain, and ... 1answer 57 views ### Paper on the use of probability theory in finance? I have taken probability theory course in college and want to see how it is used practically in finance. What papers should I read? I want it to be not too difficult (undergraduate probability theory ... 0answers 23 views ### Modeling the distrubution of future swap rates I'm interested in better understanding the unwind cost/value of a swap at various points in the future. Suppose that we have entered a 7Y swap (paying fixed) and want to understand the unwind cost/... 1answer 61 views ### CIR model - nth moment generation E^*[r_T^n] I am analyzing the nth moment generation process for r_t with dynamics defined by CIR model r_t has following dynamics$$dr_t=a(b-r_t)dt+\sigma \sqrt{r_t} dW_t^* \quad \quad (1)$$for some ... 1answer 48 views ### Determining confidence level of directional signals With regards to technical analysis, are there ways of determining the confidence level of a directional signal? Taking a relative strength index (RSI) as an example, can the extent to which an asset ... 1answer 65 views ### Option delta - Conditional probability definition? Can someone help me interpret this definition of delta? Delta is a conditional probability of terminal value (St) being greater than the Strike (X) given that St > X for a call option. Is the ... 2answers 235 views ### Understanding the solution of this integral The following integral represents an expected value of a geometric brownian motion for S_T>K (i.e. part of the Black-Scholes call option price):$$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\... 1answer 81 views ### What are the answers to these questions on card deck and option pricing? here are 3 questions I have some trouble dealing with. Your help will be greatly appreciated! 1 - We have a deck card: 26 red, 26 black. we play a game: you draw a card from the deck without putting ... 0answers 16 views ### Exercise probabilities in Black Scholes [duplicate] In the Black Scholes Formula, why are the probability of an Asset or Nothing Call and Cash or Nothing Call being exercised different. The probabilities are N(d1) and N(d2) respectively. 1answer 74 views ### BSM Model - Actual probability Actual probability of exercise of put option under BSM model is: PD = N(-d2(u)) (using expected return of stock, u) Risk-neutral equivalent is ... 0answers 43 views ### On the construction of a Brownian motion from a Gaussian process Let$X\$ a Gaussian process defined by $$X_t=\int_{0}^{t}\left(\frac{1}{\sigma}\left(r_s-\frac{\sigma^2}{2}\right)-\rho\sigma_P(s,T)\right)\mathrm{d}s+\sqrt{1-\rho^2}Z_2(t)+\rho Z_1(t);\;\;t\in[0,T] ... 0answers 57 views ### Modeling Interest-only Mortgages First post on this forum - happy to be here. Please give feedback if this is off-topic so I can more meaningfully contribute moving forward. Can we infer a range of future all-in costs for I/O ARMs ... 0answers 15 views ### Affect of choosing different combinations of variables for multivariate regression [closed] If I have variables x1,x2,x3,and x4 that have correlation coefficients âˆ’0.9,âˆ’0.5,0.5, and 0.9 to another variable y, what is the effect of choosing different combinations of them in a multivariate ... 0answers 60 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 ... 0answers 23 views ### Determining Monthly Premium with Credit default swap I hold a 10 year, 100 million bond. In order to minimize risk, I enter into a credit default swap in which I am paid every time (monthly) the bond rating drops to a new low. I have the probabilities ... 0answers 61 views ### Stock price distribution from options marks I am reading the following link: on "recovering probability distributions from option prices" - how to subtract influence of stochastic volatility? At the end of the derivation it seems ... 1answer 97 views ### Density plot of the skew-t distribution I am using the sgt package in R to recreate the plot from Hansen's paper ( available here http://www.ssc.wisc.edu/~bhansen/papers/ier_94.pdf on page 8) using random ... 1answer 75 views ### Calculating probability of options with normal/lognormal distribution: does time make a difference? I'm trying to calculate the probability of a calendar spread resulting in a profit at expiration, when estimating it is modeled as a lognormal distribution, by getting: ... 1answer 36 views ### How were the probabilities of recession over the next four quarters calculated in this table? http://www.bloomberg.com/news/articles/2016-02-08/goldman-sachs-says-defy-mr-market-as-recession-risk-still-low The probability of a slump in the U.S. is just 18 percent and 23 percent over ... 1answer 38 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 ... 1answer 43 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 ... 1answer 41 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 ... 0answers 129 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: ... 0answers 78 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 ... 2answers 105 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 ... 3answers 114 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 ... 0answers 38 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) ... 1answer 69 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 ... 1answer 86 views ### Prove uniqueness, and 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 ... 2answers 249 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 ... 1answer 41 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 \... 2answers 104 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 \... 1answer 113 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, \... 1answer 46 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 ... 1answer 532 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 ... 0answers 23 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 ... 0answers 56 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 f(x_1,y_1)f(... 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 ... 1answer 402 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 ... 1answer 111 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 ... 2answers 120 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) = \...
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 ...