Quantitative Finance Stack Exchange
Quantitative Finance Stack Exchange

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.

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 and differs from the physical measure (aka "real world probability") by only a mean shift.

The risk-neutral measure may be recovered from option prices using the technique of Breeden and Litzenberger (1978). While this is appealing, efforts to then find the necessary mean shift have been unsuccessful. Carr and Yu (2012), later extended by Ross (2015) can be promising for some instruments but does not work for any instrument in zero-net-supply (e.g. futures, forwards, swaps, options). Furthermore, Borovička, Hansen, and Scheinkman (2016) and Jackwerth and Menner (2017) have cast doubt on whether these techniques are even successful for any instruments.

445 questions
13
votes
3 answers

Version of Girsanov theorem with changing volatility

Is there a version of Girsanov theorem when the volatility is changing? For example Girsanov theorem states that Radon Nikodym (RN) derivative for a stochastic equation is used to transform the expectation where the sampling is done in one mesaure…
adam
  • 539
  • 4
  • 12
13
votes
6 answers

Risk-neutral vs. physical measures: Real-world example

Taken from a mid-July Wall Street Journal news story: Surging optimism in financial markets hasn’t translated into a big pickup in economic growth. Stocks hit records Friday and big U.S. banks reported stronger-than-expected earnings. But new…
arni
  • 581
  • 1
  • 4
  • 11
6
votes
2 answers

How is Radon-Nikodym derivative different from the likelihood ratio?

I see that the Radon-Nikodym derivative is the ratio of probability measures, $dP/dQ$. How is this different, in general, from a likelihood ratio of two continuous distributions? I understand the RN-definition broadly applies for…
Bravo
  • 671
  • 3
  • 15
6
votes
2 answers

Why is future price process defined to be a martingale under the risk neutral measure?

In Shreve's book, future process is defined to be a stochastic process that satisfies the following two conditions: (1) $Fut_s(T,T) = S(T)$ where $Fut_s(T,T)$ is the future price at expiration and S(T) is the price of the underlying. (2) At any…
user1559897
  • 515
  • 4
  • 11
5
votes
2 answers

"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 risk-neutral probability measure, the drift of stock…
Kian
  • 830
  • 2
  • 8
  • 23
3
votes
1 answer

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 derivative value based on those funds is meaningful. I…
Preston Lui
  • 341
  • 2
  • 8
3
votes
1 answer

Risk Neutral Probability and invariant measure

Is a risk-neutral probability a special case of an invariant measure?
Jeff
  • 33
  • 2
3
votes
1 answer

When are implied and real world parameters the same?

Suppose $T$ the maturity of a risky bond which defaults with probability $p$ over its lifetime. If it defaults it pays zero. Thus to price this bond in risk neutral terms would give $$P=\mathbb{E}^{\mathbb{Q}}\left[e^{-r(T-t)}(1-p)\right].$$ If such…
Nathan Meibergen
  • 41
  • 3
3
votes
2 answers

Risk Neutral Pricing Necessary Condition

Suppose that I have an option on a single stock expiring at time $T$ and I replicate the payoff of this derivative by investing in the stock market and the money market. So this condition reads $$X(T) = V(T) \quad \text{almost surely}$$ where $X(T)$…
Calculon
  • 595
  • 4
  • 14
2
votes
1 answer

Is "risk-neutral probability" a misnomer?

Aside from not being a probability in the common sense (i. e. not concerning the odds of events), as far as I understood it, the "market's attitudes towards risk" are actually factored into / built in the "risk-neutral probability". For pricing we…
Amaterasu
  • 151
  • 2
2
votes
0 answers

Why a currency is not considerend as a numéraire for a risk neutral measure

We often say that "A risk neutral measure is associated with the money market account, not the currency. Currency pays a dividend because it can be invested in the money market." How is a currency paying a dividend ? if like it is advocated,…
user30614
  • 187
  • 7
2
votes
3 answers

Risk neutral measure doubt

For a derivative in a complete market, we can say that: $h_0 = E(h_t)$ assuming 0 risk free rate. Is the above relation also valid for a stock/ non derivative i.e. $s_0 = E(s_t)$ under the same risk neutral measure?
dayum
  • 343
  • 1
  • 12
2
votes
1 answer

How to express the conditional expected variance under the risk neutral measure?

The conditional expection of variance under risk neutral measure is $$\mathbb{E}^Q[V_T |S_T=K]$$ where $S_T$ and $K$ represent the spot price at maturity and strike price, respectively. Assume I know the risk neutral density $q(V,S,t)$, I want to…
Smirk
  • 53
  • 5
2
votes
1 answer

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 can describe risk-neutral distribution = subjective…
Alkibiades
  • 41
  • 3
1
vote
2 answers

Estimating the Market Price of Risk (Hull's Section 36.3)

I'm currently trying to understand risk-neutral valuation and transforming real-world stochastic processes to their risk-neutral version. If I understood it correctly, the main point of risk-neutral valuation is to not have to deal with real-world…
Merwin
  • 21
  • 3
1
2