# 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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### What will be the payoff equation of a GBPUSD European Exotic option/FX forward with Notional in USD [duplicate]

Given the currency pair , GBPUSD with spot price as $S_t$ at time $t$, Strike price as $K$, $I$ is an indicator function indicating if GBPUSD is below the "Knock-in-Rate" at expiry, $L$ ...
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### Why is it said that Girsanov’s theorem destroys the tractability of the process which is undesirable for quantitative finance applications？

I am reading the paper "Risk-neutral pricing techniques and examples" by Robert A. Jarrow et al., and it is said that Girsanov’s theorem destroys the tractability of the process which is ...
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### Price formula for a claim with payoff $\bf{1}_{W_T \geq K}$

Let $dW^{\mathbb{P}}(t)$ be a standard BM under $\mathbb{P}$ and $dW^{\mathbb{Q}}(t)$ a standard BM under $\mathbb{Q}$. We know that $$dW^{\mathbb{P}}(t)=dW^{\mathbb{Q}}(t) - adt$$ where $a$ is a ...
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### Could a phoenix autocall be priced by a snowball option with zero coupon plus expectation of coupons received in knock out observation dates?

I know that coupons in the phoenix autocall can be received in each observation date if the underlying price in that date does not touch down the knock-in barrier and receiving periodic coupons is ...
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### Monte Carlo methods: Choosing the best measure

When pricing derivatives using Monte Carlo methods, we take outset in the risk neutral pricing formula which states that we need to calculate the expected value of the discounted cashflows. To do this,...
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### Complete market without risk-neutral measure

Let $\mathcal{M}$ be a one-period model with $\Omega=\{\omega_1,\omega_2\}$ and $S_t^0=1$ for $t=0,1$. Find a $D$ such that $S^d$, $d=1,...,D$ yields a complete market without a risk-neutral measure. ...
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### Forward Black Implied Volatility For Within Risk Neutral European Option Pricing

Going to preface this question with an acknowledgement with how silly the ask is, but alas that is the working world; if anyone can share any ideas I'm all ears. We're pricing an exotic option in risk ...
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### Beta Weighting Deltas: What happens to the non-correlation part?

At various informational websites about option trading, it is often mentioned that in order to compare different underlyings in an apples-to-apples comparison, it is useful to beta-weight the deltas. ...
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### How to find a risk-neutral measure for funds with management fee

There are many funds (index funds or actively managed funds) that charge management fees, which inherently makes it underperform the asset it holds. There are some applications where finding the ...
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### Martingale under risk neutral probability

I have a question to prove martingale under risk-neutral measure: Question Consider a discrete time process $S$, which at time $n \in \mathbb{N}$ has value $S_n = S_{0}\prod_{j=1}^{n}Z_j$, $S_0>0$ ...
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### Shreve multiperiod binomial model

In Section 1.2 in Shreve's Stochastic Calculus for Finance I, he introduces the Multiperiod Binomial Model. There is something about it that I don't quite understand. He assumes that coins are tossed ...
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### Can the risk neutral pdf derived from Breeden-Litzenberger Method be used to calculate vega and theta?

I have been researching volatility smoothing techniques and risk-neutral pdf. I noticed one interesting post in Does the risk neutral pdf that is derived using Litzenberger-Breeden Method correspond ...
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### Risk-neutral density versus put-call skew and open interest

I've been experimenting with the Breeden-Litzenberger formula in Python based on some code obtained here: https://github.com/robertmartin8/pValuation/blob/master/ProbabilisticValuation/...
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### Why do we simply assume the risk neutral probabilities to be "0.5"?

I am aware that there was a question similar to this but my question is a little different. Firstly, in context of binomial short rate, why do we simply assume the risk neutral probabilities p=1-p=0.5?...
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### Valuation via decomposition or via simulation of the underlying?

My question might be very straight forward but I have seen both approaches being followed in practice so I am curious to see if there are arguments in favor or against each one. I am explaining my ...
1 vote
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### Is there a risk-neutral measure if there are two stocks with different drift terms?

There are two stocks: $S_t$ and $P_t$ $$dS_t = S_t(\mu dt + \sigma dB_t)$$ $$dP_t = P_t((\mu + \varepsilon) dt + \sigma dB_t)$$ Is there any risk-neutral measure? My thoughts are pretty simple: $μ$ is ...
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### Risk Neutral Pricing - Why the Risk Free rate for Risky security (Intuition) [duplicate]

I am struggling with this concept of risk neutral probabilities. My understanding of how a risk neutral pricing framework works is as follows: (discrete, binomial lattice for simplicity) I do not know ...
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### question regarding relation between expectations on different measures

I am a beginner to the theory of stochastic calculus and measure change. I have derived an equation related to expectations on different measures. I wanted some expert opinion on whether this is true ...
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### ATM Implied Volatility and Expected Variance

This answer claims that $$\sigma^2_{ATM}\approx E^Q\left(\frac{1}{T}\int_0^T\sigma^2_t dt\right)$$ ie implied ATM vol = risk-neutral expectation of integrated variance. Is there some proof available? ...
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### Estimating market price of interest rate risk under CIR model

My goal is to find the market price of risk associated with the interest rate under the CIR model whose stochastic differential equation under the physical measure is given: \begin{eqnarray}\label{...
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### Implication of unique risk neutral measure

I'm reading Shreve Stochastic Calculus II, theorem 5.4.9 (Second fundamental theorem of asset pricing), This is the part that confuses me : suppose there is only one risk-neutral measure. This ...
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### BKM risk neutral moments in python

I am trying to compute the BKM implied moments (Bakshi, Kapadia and Madan 2003) in python by following this paper: Neumann, Skiadopoulos: Predictable Dynamics in Higher Order Risk-Neutral Moments: ...
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### Dividend Dynamics under Q Measure / Using Girsanov Theorem with Covariance

I want to find the value of a dividend stream. I can do it under the P-measure, but now I would also like to do it under the Q-measure but cant figure out how to derive the dynamics of the dividend ...
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### Is the market price of an asset always lower than the expected discounted value under the REAL WORLD measure?

The risk neutral measure is often said to reflect the risk aversion of investors. So intuitively, I would think that an asset's expected discounted value should be lower under the risk neutral measure ...
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### GARCH option pricing

I have been trying to implement GARCH(1,1) model for pricing call options. Suppose I have calibrated Garch(1,1) model for modelling the conditional volatility using the historical data of an equity ...
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### Risk premium of insurance risk

I recently came across an equation in a paper. In short, suppose that $I(t)$ denotes a longevity index at time $t$. An informative indicator that is useful in the absence of any information about the ...