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### Risk-Neutral Probabilities, Trinomial Model

My professor has many grammatical mistakes and errors in his questions, so apologies ahead of time. I am just trying to understand what he wants for this question, In trinomial model, let $S_0 = 1$, ...
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### Binomial Model, Number of nodes from $t = 0$ to $t = n$

How many paths are there in a binomial model from time $t = 0$ to time $t = n$? How many nodes (states) are there? Intutively it seems that there are $2^n$ paths and $2n - 1$ nodes. But I am not sure ...
Let $\tilde{W}_t := (1+R)^{-t}W_t$ and $\tilde{S}_t := (1+R)^{-t}S_t$ be respectively discounted wealth process and discounted asset price. Then, show that $$\tilde{W}_t = w_0 + ... 0answers 35 views ### Risk-Neutral Probabilities Consider one period Arrow-Debreu model with N = 2 and M = 4 show in Figure: Find all the possible risk neutral probability \pi. What I am confused about is how D_1 and D_2 have an up and ... 0answers 29 views ### Arbitrage free price I could not find much information on this question: what is an arbitrage free price in trinomial model? If there is, is there any way of replicating a call option under this model? Although I have ... 1answer 37 views ### How to rightfully balance the share of the organization between departments after variable changes? This is an abstracted version of the problem I'm facing and I have to tell you first, my question might not be precise and or even correct, so I hope you understand and in that case can improve the ... 7answers 688 views ### Proof that no trading system always wins I am pondering on the existence/impossibility of a trading system (or algorithm) that ALWAYS ends up winning money, no matter how the price of a futures moves. In a context where one can go long or ... 0answers 29 views ### Return, STD and CAPM based on Continuously compound return on daily prices Mission: For some ETF, Get 1, 3, 5 years: Return STD CAPM parameters (alpha, beta) Reference if I calculated correctly: Yahoo finance performance & risk data Raw data: Daily adj. close ... 0answers 15 views ### Question in the proof of “Optimization of conditional value-at-risk” I'm reading the paper "Optimization of conditional value-at-risk" by Rockafellar and Uryasev. The state two theorems within the paper which are proven in the appendix. Let me introduce some notation ... 1answer 22 views ### How to calculate 5 years return & STD for ETF? I want to calculate by-myself 5 year return & STD for SPY ETF. What I did: Downloaded to Excel from yahoo finance historical data for the ETF (daily Adj. Close) from ... 1answer 96 views ### Clarify a derivation in Pat Hagan's Convexity Conundrums I am looking for help in understanding the algebraic derivation to go in between some of the lines in Pat Hagan's famous Convexity Conundrums paper e.g. how he goes from 3.4a to 3.5a. 1answer 80 views ### Derivation of Magrabe formula I'm going through the following note by Davis, link. In chapter 3 he derives the Magrabe formula. I got stuck at equation (3.16). We have two assets: ... 5answers 6k views ### What is a martingale? What is a martingale and how it compares with a random walk in the context of the Efficient Market Hypothesis? 3answers 331 views ### How is stock data objectively different to this random walk? I have a random walk that is generated as so using python, numpy, and matplotlib ... 1answer 161 views ### Value of European Call equals Value of American Call, Question on Explanation/Proof I am reading S. Shreve, Stochastic Calculus for Finance, Vol. I. There he proves that American Call Options have the same value as European Call Options. In the proof he uses that for a Call option ... 10answers 71k views ### How can I go about applying machine learning algorithms to stock markets? I am not very sure, if this question fits in here. I have recently begun, reading and learning about machine learning. Can someone throw some light onto how to go about it or rather can anyone share ... 1answer 91 views ### Stochastic calculus: what am I doing wrong? it is just the computation of a second moment but however is creating debate !!... Can someone spot the error? 2answers 371 views ### Why is that a risk averse consumer buys the optimum insurance when there is actuarially fair insurance? I think I understand the fact that when marginal utilities of the same function are equal (a consequence of the actuarially fair insurance), the independent variables in it must be equal -- right? But ... 0answers 38 views ### Show that in an arbitrage-free and non-redundant market a certain set is compact Some notation: We consider a financial market with d+1 assets, the 0-th asset is considered the risk-free asset, the others are the risky ones. The vector \overline \pi \in \mathbb R^{d+1} ... 1answer 81 views ### Pricing digital options in discrete time I am stuck in this exercise from my textbook: Consider a one-period market model with N+1 assets: a bond, a stock and N-1 call options. The prices of the bond are B_0=1 and B_1 = 1+r, ... 2answers 88 views ### Weighting with restrictions, but no clear objective function? I have 40 shares in an index and I want to weight them based on their market value, define the known value as x_i In the traditional way, the weight of each share is calculated as: w_i = x_i / ... 2answers 311 views ### Book recommendation: math toolkit for quantitative finance and statistics I am looking for a book which teaches mathematical topics which are relevant to master quantitative finance and statistics. Please note, I do not mean a book which would explain how math is applied ... 1answer 49 views ### Desperate for help with simple derivative Can someone help explain how differentiating the following with respect to x:$$ \frac{1}{2} \alpha \mathbf{x}^T \Sigma \mathbf{x} + (\mathbf{\mu} - R\mathbf{1})\mathbf{x} $$Yields the following: ... 2answers 67 views ### Conditional expectation of a non stochastic process In an example I was working through it was shown that W_{t}^{2} - t was a martingale with respect to the Brownian motion filtration \mathcal{F}_{s}^{W} with t>s. Everything was fine except a ... 1answer 102 views ### How can we write swap as a chain of FRA's For the rest of my question I use the notation from Brigo. The discounted payoff of a receiver interest rate swap (RFS) at t<T_{\alpha}, where T_{\alpha} is the first resetting date, is given ... 1answer 54 views ### Is it possible that some types of financial systems can resonate? Financial systems can certainly be modeled using the same tools physicists use to model dynamic physical systems. The validity of such is evidenced by models such as that developed by Black and ... 1answer 69 views ### \frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+ at time T_i is equivalent to a payment of (A-p(T_{i-1},T_i))^+ at time T_{i-1} How can I show that payment of \frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+ at time T_i is equivalent to a payment of (A-p(T_{i-1},T_i))^+ at time T_{i-1} ? Where A is a deterministic ... 1answer 132 views ### derivation of heston pde in gatheral Following Gather (the volatility surface, chapter 2) we assume the following process:$$ dS_t = S_t(\mu_t dt+\sqrt{\nu_t}dZ^1_t) d\nu_t= -\lambda(\nu_t-\bar{\nu})dt+\eta\sqrt{\nu_t}dZ^2_t$$... 1answer 160 views ### Arrow-Debreu Price in “The Volatility Smile and its implied Tree” I was reading the old, but still interesting paper "The volatility smile and its implied tree" by Derman and Kani. I have a two questions about the derivation of the 2n+1 equations, both of them ... 2answers 233 views ### Is linear programming important for quant? I'm thinking about taking a course on Linear and Convex Programming, but I don't know how useful it is in the real world finance. Which areas in finance is mathematical programming used? 1answer 103 views ### Distribution of minimum of hazard functions Suppose I have two random variables, X_1 and X_2, that are independent (but not identically distributed) and assume both have hazard functions \lambda_1(s) and \lambda_2(s), for s > 0. ... 0answers 71 views ### Understanding Price Elasticities in Discrete Choice Models (Derivative) I'am in the midst of a paper on mutual fund product differentiation by Li and Qiu. Here, the authors model the utility an investor derives from investing in a mutual fund using a Discrete Choice Model ... 1answer 89 views ### Math basics of Equally-weighted Risk contributions i'm writing my BA Thesis about "Equally-weighted Risk contributions". Can anyone recommend math books for further understanding of Risk contributions? 2answers 1k views ### Why Ito calculus? Coming from physics, I am used to the fact that the Ito interpretation of most natural stochastic equations is wrong, and one should be using Stratonovich calculus instead (of course they are ... 9answers 7k views ### Recommendations for books to understand the math in quantitative finance papers? Can anyone recommend books that explain the math used in quantitative finance academic papers? 3answers 236 views ### Budget Constraint in Sharpe Ratio Optimization I am a math student and I am trying to understand the budget constraint in Sharpe Ratio optimization for portfolio design. Recall the budget constraint requires that the sum of the portfolio weights ... 1answer 256 views ### PDE pricing of barrier options in BS Path-dependent options in BS framework is intuitive to price with monte-carlo under risk-neutral measure, however it appears that several kinds can be priced with PDEs. I understand how does the story ... 1answer 105 views ### backward Kolmogorov equations - Markov properties I'm a physicist who's research has lead him into the theory of stochastic differential equations. If this question is not appropriate for this forum, please feel free to delete it. So I've been ... 5answers 3k views ### Random matrix theory (RMT) in finance The new kid on the block in finance seems to be random matrix theory. Although RMT as a theory is not so new (about 50 years) and was first used in quantum mechanics it being used in finance is a ... 0answers 75 views ### Sampling and/or asymptotic distribution of a function Assume we have the following function:$$f(p) = \frac{1}{(1-p)d}\ln\left(\frac{1}{T}\sum_{t=1}^{T}\left[\frac{1+X_t}{1+Y_t} \right]^{1-p} \right)$$where d is a constant T is a constant X_t ... 2answers 586 views ### Is it possible to understand financial theory without mathematics? I am trying to develop a short course on financial theory, covering the fundamentals of forward and options pricing, and 'efficient market' theory. I want to reduce the amount of mathematics to a ... 0answers 58 views ### FTAP in the model independent case, paper by Schachermayer I have a question about the following paper by Beatrice Acciaio, Mathias Beiglböck, Friedrich Penkner, Walter Schachermayer. At the very beginning of the paper, on page 3, there are two definitions ... 1answer 81 views ### Get discount factors with limited knowledge? I am facing the problem of just having this information: 6% coupon bond with 2.5 years to maturity, traded at a 100% clean price 4% coupon bond with 1.5 years to maturity, traded at a 98% clean price ... 0answers 123 views ### What is the right group of durations? It seems that the group of durations commonly used in quantitative analyse is \mathbf{R} but it seems to me that \mathbf{R_+^*} could also be an interesting choice. While I am not aware of ... 1answer 57 views ### Why is the discount function non increasing if pure cash holdings are feasible? I am struggeling with the question, for example lets take a swap with rate of 3.2 for one year and 3.6 for 2 years and Discount Factor 0.96899 for the first year and 0.93158 for the second year. ... 0answers 194 views ### Proving the asymptotic distribution of Manipulation-Proof Performance Measure (MPPM) (Paper by Goetzmann et al.) In Goetzmann et al.'s (2007) paper, the authors derive a "Manipulation-Proof Performance Measure" (MPPM), which is a performance measure that is impervious to performance manipulation by fund ... 1answer 742 views ### Determine state price vectors? I have 3 states with two assets, stocks and bonds. The bond has a payoff of 1 in every state of the world. And the stock has a current price of S_0 = 100 and ... 0answers 112 views ### Does it make sense to apply complicated mathematics to calculate with precision when the margin of error is +/-10%? [closed] This is more of a philosophical question than general question. Quantitative finance applies highly complicated mathematics and has attracted very smart people to this field lately given the high pay ... 0answers 93 views ### How to show that the risk contribution function is or is not injective? Assume a portoflio w \in \mathbb{R}^n, you can get the total risk contribution \psi_i of asset i by doing:$$\psi_i = w_i \frac{\partial \sigma(w)}{\partial w_i}= \frac{1}{\sigma(w)} \left[ ...
I have often seen the following statement in different paper: As $\sigma$ is homogeneous and of degree 1, we use Euler decomposition and write \$\sigma(x)=\sum_{i=1}^n x_i \frac{\partial ...