# Questions tagged [risk-neutral-measure]

A risk-neutral measure is a probability measure that yields an expected present value (discounted at the risk-free rate) which is equal to the current market price. The risk-neutral measure is also called an equivalent martingale measure.

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### EMM, Supremum and Expectation

I asked this question on MSE recently. https://math.stackexchange.com/questions/3922347/supremum-and-expectation I want to prove this when $\mathcal{M}$ is a set of equivalent martingale measure. ...
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### Value at risk, risk-neutral vs real-world probability measures

Does anyone know if there is any link between the Value at Risk of risk-neutral distribution and of the real-world distributions of asset rate of returns?
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### Risk-neutral pricing for processes other than Geometric Brownian Motion

For pricing derivative with payoff $H$ in BS model we use formula: $$\Pi_t(H)=e^{-r(T-t)}\mathbb{E}_Q[H|\mathcal{F}_t]$$ Now I have market with one risk free asset and one risky asset whose price is ...
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### Why does higher volatility for ATM Call Option lead to a lower risk-neutral probability of expiring ITM?

This is a follow-up question on the discussion in the thread here, from which I borrow the graph below depicting $N(d_2)$ (i.e. the risk neutral probability of a Call option expiring in the money) ...
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### Discounted price process - martingale

I have a process $S_{t}=S_{0}e^{\left(r-q\right)t+mt+X_{t}}$, where $X_t$ is a Levy process and I want to check for which $m$ the process $e^{-(r-q)t}S_t$ is a martingale. The third condition of a ...
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### Derivation of $u=e^{\sigma\sqrt{dt}}$ and $d=e^{-\sigma\sqrt{dt}}$

Anyone could provide me a proof of how, starting from $\frac{dS_T}{S_t}\sim \operatorname{N}(\mu dt,\sigma^2 dt)$ with $p:=\frac{e^{rdt}-d}{u-d}$, we can obtain the parameters $u$ and $d$ as from ...
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### Replicating portfolio

I have a doubt about the replicating portfolio methodology. Example - Consider an European Call with $K=21$ and underlying with current price $S_0=20$. We assume that, at the maturity, the underlying ...
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### Objective probability of default from CDS spread

I have the risk neutral probability of default extrapolated from the market data of the CDS spreads. How can I empirically estimate the market risk price of the objective probability of default (i.e. ...
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### Mismatch of periods with numeraire compared to the forward rates

In Joshi's The Concepts and Practice of Mathematical Finance Page 323--324 I believe that there may be a mismatch of periods with forward rates: Consider time partition $t_{0} < ... < t_{n}$ ...
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### Help reconciling incorrect reasoning in options pricing brain teaser

I'm trying to reconcile an interesting brain teaser I was recently posed and I need help understanding the flaw in the reasoning. The problem states there is an asset which after an announcement has ...
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### No-arbitrage Pricing

We have a contract whose value is $A(S_t,t) = S_t^3$ at all times, not just at expiration. $S_t$, the underlying stock, follows a Geometric Brownian Motion, $\frac{dS}{S} = \mu dt + \sigma dB$. How ...
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### Power Options & Forwards on Stock Squared

Short story: the process for Stock price squared is not a martingale when discounted by the money-market numeraire under the risk-neutral measure. How can we then compute derivative prices on $S_t^2$ ...
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### Does equity premium puzzle affect option-implied RWDs using Arrow-Debreu equilibrium?

I am researching and learning about option-implied RNDs (risk neutral densities) and transformation to RWDs (risk world densities) using expected utility theory to compute risk aversion values. This ...
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### R: How do i finish the tails in the risk neutral density, obtained from option prices

Im currently working on constructing the risk neutral probability distribution of a stock, based on the option prices. In doing so, i calculate the implied volatilities from the option prices, and ...
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### Replicating portfolio of an option and to find inital price

I am very new to financial math so I am not sure how to do with this question. A friend sent me this question to practice but I am unsure how to begin. I read about call option . Can that be used for ...
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### question about code posted for calculation of risk neutral density using Bondarenko convolution method

I have questions about the code (found here Estimation of Risk-Neutral Densities Using Positive Convolution Approximation - Python). The synthetic price in Bondarenko paper includes two terms before ...
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### Multiple Risk-Neutral measures in incomplete market

This question is in regards to incomplete markets where multiple risk-neutral measures exist. I am a little bit confused by this idea. Say we have an incomplete market with only one stochastic process ...
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### Is the differential between risk free rates the drift of an exchange rate only in the risk neutral world?

Take for example this passage from "Monte Carlo Methods in Financial Engineering". Is this a result of the risk neutral world or is this the real world drift as well? I've never seen the explicit ...
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### complete python code to calculate risk neutral density from option prices [duplicate]

bAsic python code to implement Litzenberger formula for risk-neutral probabilities implied by option prices. Use S&P 500 option prices whose strike intervals are typically 5 points apart use at ...
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### Does simulating price as GBM automatically implies risk neutrality?

I am using a dynamic programming approach to price European options where formulate the pricing as a discrete-time continuous-space Markov Decision process. The MDP is risk-sensitive as in I don’t ...
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### Heath–Jarrow–Morton under real-world measure

In HJM model (framework), the drift of the forward is determined by its diffusion coefficient: $$\mu(t,s) = \sigma(t,s)\int_t^s \sigma(t,v)^Tdv$$ My understanding, is that the change of measure ...
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Is it possible to use bid-ask spreads on contracts from a specific tenor to estimate risk aversion and use it to transform risk-neutral density into real-world density?
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### In BS model, is there a way to show that the risk-neutral Q is unique without using MRT nor the fact that the market is complete?

In Black-Scholes model, is there a way to show that the risk-neutral probability measure is unique without using the martingale representation theorem nor the fact that the market (in BS model) is ...
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### What are the relation between the risk neutral measures in binomial tree and in Black Scholes model?

I appreciate that both are the direct result of constricting a replicate portfolio using stock and bonds. Are there deeper relationship between the two?
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### Can you explain the Black-Scholes fair option equation with RND?

I am trying to learn Black-Scholes risk-neutral densities with only prior knowledge of fundamental B-S equations (not the derivation). Sorry if this was asked already or if I sound completely clueless....
Consider the Cox Rubinstein binomial pricing model with N steps, with stock price change given by parameters u and d so that at step $i$ we have $S_{i+1} = uS_{i}$ or $S_{i+1} = dS_{i}$ with \$0\leq i \...