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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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### What is the arbitrage opportunity in this simple one-period market?

I have a single period market, and three states, and I have 3 risky assets. I assume no interest. So I have three states $\Omega=\{\omega_1,\omega_2,\omega_3\}$. All assets start with the value 1, ...
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### $E[F_T] = F_0 \ \rightarrow \ \text{or} \ \leftarrow \ p = \frac{1-d}{u-d}$?

From Ch 12 in Hull's OFOD, we compute the risk-neutral probabilities for a futures contract: Later in Ch 17, futures options are valued, and we have the same result: In relation to ...
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### Estimating Parameters - Vasicek

The Vasicek model for the short rate $r_t$ is given by the SDE $$dr_t = \alpha(\beta - r_t)dt + \sigma dW_t,$$ where $W_t$ is a Brownian motion under the physical measure. I'd like to compute bond ...
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I am trying to computing the price of an option at time $t$, with payoff $X = \frac{S_{T_2}}{S_{T_1}}$, at time $T_2$, where $t < T_1 < T_2$. Here how I compute it: Using the forward measure $... 1answer 118 views ### Relationship between risk-neutral probability and subjective probability I recently came across a Paper by a paper of Rubinstein and Jackwerth (1997): http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.5214&rep=rep1&type=pdf where they assume that you ... 1answer 440 views ### Consistency of economic scenarios in nested stochastics simulation I am interested in references on research regarding the consistency of economic scenarios in nested stochastics for risk measurement. Background: Pricing by Monte-Carlo: For pricing complex ... 9answers 7k views ### Why Drifts are not in the Black Scholes Formula This question has puzzled me for a while. We all know geometric brownian motions have drifts$\mu$:$dS / S = \mu dt + \sigma dW$and different stocks have different drifts of$\mu$. Why would ... 1answer 236 views ### Risk neutral drift vs real world I was of the understanding that risk neutral drift was always the risk free rate. A section from Gregory's book on Credit Value Adjustment seems to say risk neutral drifts are typically estimated from ... 2answers 78 views ### does there need to be risk-neutral agents in the market to enforce risk-neutral pricing? I'm trying to understand a fundamental link between mathematical finance and economics. I understand that risk-neutral pricing is free of arbitrage with replicating portfolio. Does risk-neutral ... 2answers 160 views ### Option with payoff$K^2/S^2$Given the dynamics of the risky asset ( with dividend$q$), $$\frac{dS_t}{S_t}=(\mu-q)dt + \sigma dW_t^P$$ Consider a european option with payoff, $$P_0(S) = \begin{cases} 1, & \text{... 0answers 69 views ### how best to equalize individual pair risk in a portfolio of stock pairs? I am building a portfolio of stock pairs in which each pair is individually hedged via beta/hedge ratio adjustment. I am looking for a method to ensure that I am taking the same risk in each pair that ... 1answer 132 views ### Bond price in Ho-Lee Model I know Ho-Lee model and want to extract the price at t, of a European call option with strike price K and exercise date T, on an underlying S-bond, but I don't know what way should I choose: ... 2answers 196 views ### Real world monte-carlo (P-measure) Consider the 2 following approaches to pricing a security: Monte-carlo (\mathbb{Q}-measure) $$C = \frac{1}{N} \sum_{i=1}^{n} e^{-rT} max(S_i(t) - K, 0)$$ Monte-carlo ... 1answer 397 views ### Provide a bond pricing differential equation and invoke Feynman-Kac Theorem Grateful for any assistance. Consider the process: dZ=r(t)Z\,dt , where r(t) is stochastic interest rate and Z=Z(r,t;T) is a zero coupon bond Price. Provide a bond pricing partial ... 2answers 135 views ### How to discretize a GBM under P- and Q-measures? Under the P-measure, a geometric Brownian motion can be specified using the following SDE:$$dS_t=\mu S_tdt+\sigma S_tdW_t^P$$and its Euler discretization is$$S_{t+\Delta t}=S_t + \mu S_t \Delta ... 1answer 85 views ### Is probability implied by binary FX options risk neutral or real world? If we consider binary FX options in the market and estimate the market implied probabilities of certain FX rates occurring, would these resulting probabilities be risk neutral or real world? I hear ... 1answer 248 views ### What's Risk-Neutral in an Interest Rate Model? In Shreve II, on p. 265 he states the Hull-White interest rate model as $$dR(u) = \left( a(u) - b(u)R(u)\right) dt + \sigma(u)d\tilde{W}(u),$$ and then mentions "...$\tilde{W}(u)$is a Brownian ... 3answers 110 views ### Unique risk neutral measure for Brownian Motion For a standard geometric Brownian motion model of stock prices: $$dS = a S dt + \sigma S dZ$$ we can transform the process to be under risk neutral measure: $$dS = r S dt + \sigma S d \tilde{Z}$$ ... 3answers 330 views ### Physical or Real-world Probability Measure For counterparty credit risk, in particular, for potential future exposure computation, people use the real-world probability measure to evolve the underlying risk factors. My question is that whether ... 2answers 162 views ### Intuitive Reasoning for Using Risk-Neutral Measure Although we thoroughly covered risk-neutral pricing in university I never fully understood it in the context of continuous-time processes. But first of all, lets consider a discrete time example: ... 1answer 55 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 ... 2answers 267 views ### Does a forward price have a drift component in any measure? Going by intuition, a forward price should already take into account the drift in the underlying price process. Further, assuming interest rates are deterministic, the stochasticity in the forward ... 1answer 69 views ### Good book about replicating portfolios I want to know if anybody can suggest me a good textbook which explains in detail and in an understandable way how to create replicating portfolios of financial instruments like options "cash or ... 0answers 35 views ### Risk Neutral Measure [duplicate] I have been looking for a good intuitive reasoning for introduction of the risk neutral measure and its uses in quantitative finance, but I have yet to find one. I was wondering if any one could ... 5answers 735 views ### Why the Black-Scholes formula can be used in the real world? The BS formula is deduced using the risk neutral measure. Why can it be used in the real world? 1answer 227 views ### “The drift of stock price becomes the risk-free interest rate” under RNP Assume that the evolution of a stock price is geometric Brownian Motion $$dS=\mu Sdt+\sigma SdW(t)$$ where$S$is the stock price at time$t$(current time). It says in my book that "under the ... 1answer 104 views ### Bivariate Black-Sholes Model Let us propose bivariate Black-Sholes Model. Assume, we have an arbitrage-free complete market.$r_{f}$is risk-free rate. Under real-world measure$P$:$dS_{1} (t)=S_{1} (t) [\mu_{1}dt+\sigma_{1}...
What is the filtration $(\mathfrak{F}_t)$ encircled below? Is it $(\mathfrak{F}_t) = (\sigma(W_t)) = (\sigma(\tilde{W_t})), t \in [0,T]$? Or is it \$(\mathfrak{F}_t) = (\sigma(\hat{W_t})), t \in [0,T]...