# 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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### Is the initial value of the portfolio replicating a forward zero?

This is from the book Financial Calculus: An Introduction to Derivative Pricing by Martin Baxter. By choosing appropriate weights in a portfolio of a stock and cash bond you can replicate the payoff ...
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### Why is the market price of risk in the one factor Schwartz model different from the usual one?

Assume that the commodity spot price follows the stochastic process (see Schwartz article page 926) $$dS = \kappa(\mu-\log S)Sdt+\sigma SdW,$$ where $\kappa >0$ measures the degree of mean ...
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### Q determined by the market in Binomial Model

I read in a book about change of measure, so that the discounted stock price in a binomial model is equal to the current price. Namely: $$E_{Q}[S_{1}/ \beta |S_{0}]= S_{0}$$ It then says: " Q is ...
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### Poisson process under equivalent martingale measure

I have a stochastic process $N(t)$ which is equal to $n$ with probability $P\{N(t) = n\}=\frac{\left(\lambda t \right)^{n}}{n!}e^{-\lambda t }$ where $t$ represents the time period. In other words, ...
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### How to compute the Present Value of this path-dependent option?

I have an option whose payoff depends on its value at two times $T_1$ and $T_2$ as follows. $$V(t) = \mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B} (S(T_2)-K)^+)],$$ where the stock price follows the GBM ...
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### Lognormal SABR symmetries

Consider the lognormal SABR model ($\beta=1$) for an FX forward process $F$: \begin{align} dF&=aF dW\\ da&=\nu a\left(\rho dW+\sqrt{1-\rho^2}dW^\perp\right) \end{align} where $(W,W^\perp)$ is ...
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### Risk-Neutral probability deduction [closed]

Could anyone show me how to get the second row equation from the first row equation please? For each letter, $p$ is the risk-neutral probability in the risk-neutral world, $u$ is the up factor for the ...
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### Martingale pricing with time-dependent risk-free rate

I want to find the price of a European call-option under the assumption that the risk-free rate $r$ is time-dependent, i.e. $$d\beta = r(t)\beta dt \leftrightarrow \beta(T) = e^{\int_0^T r(u)du}$$ I ...
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### Show that stochastic integral is $F_W(t)-$measurable

In some notes, my professor writes the following for the price function of an geometric asian option: \begin{align} \text{Price}(t)&=\tilde{\mathbb{E}}\left[\left(S(0)\exp\left(\frac{T}{2}\left(r-\...
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### Non attainable claim - Incomplete market

I am wondering whether there is a standard procedure to find a non attainable (i.e. non replicable) asset in an incomplete market. As an example, let us have the following market ($B = (B^1, B^2, B^3)$...
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### What is the interpretation if the real world measure $\mathbb P$ is equal to the martingale measure $\mathbb Q$

Out of interest, is there anything noteworthy about a market when its real world measure $\mathbb P$ is actually also its martingale measure. In other words the real world measure $\mathbb P$ is equal ...
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### Why do stock prices follow a martingale?

I have a quick question: why does the Efficient Market Hypothesis (EMH) assume that stock prices follow a martingale process? I understand that discounted prices under the risk-neutral probability ...
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### State Price Deflators For RW to RN Scenario Generation

I have real world stochastic scenarios that model equity returns for "the market". Growth is calculated by modeling the risk free rate, then applying a risk premium on top of that. For the ...
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### Exercise on Delta-Neutal-Hedging

Suppose you have three positions in the following assets in euros: long on 10.000 calls (maturity T = 3 months, strike= 0.55, Delta (1 call) =0.533), short on 210000 calls (maturity T = 3 months, ...
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### Would it be possible to combine long butterfly with long straddle, achieving profit no matter the outcome?

This has been bugging me for a while, I feel like I'm missing something. Simply put, a long butterfly will make profit if the price at maturity does not change much, as shown below A long straddle is ...
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### Illustrating the change of measure in Black-Scholes-Merton

Say that we have the following environment: \begin{align} dS_t &= \mu S_t dt + \sigma S_t dZ_t \\ dB_t &= r B_t dt \end{align} where $S_t$ is the price of a stock, $B_t$ is the price of ...
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### Risk neutral probability for stock with continuous dividend

Setting: binomial tree with one step over time $\Delta t$. I'm trying to derive the risk neutral probability for a stock which pays a continuous dividend, say $\delta$. i.e. probability $p$ such that ...
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. ...