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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Understanding completeness in this simple one-period exercise
Let's consider a one period model (t=0, 1) with one risk-free asset that yields r, and one risky asset. $S_t^j$ will be the value of the asset j=0,1 at time t=0,1, where j=0 is the risk-free asset and ...
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How do I find the risk neutral probability? [closed]
I am trying to solve for risk neutral probability, however I do not have an idea how I should calculate it. Is anybody able to help me out on this one? Thanks!
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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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Pricing barrier options under risk neutral measure
I think this must be a really stupid question but I cannot see what I am missing.
Let's assume we're pricing some barrier option under Black -- Scholes model. Under risk neutral measure, the drift $\...
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Straddle Approximation - Directly from Integral
The ATMF straddle approximation formula, given by
$V_\text{Str}(S, T) \approx \sqrt{\frac{2}{\pi}} S_0 \sigma \sqrt{T}$
where $S_0$ is the current underlying spot price, $T$ is the time remaining ...
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How are VIX options priced in a mean-reverting framework?
If a trader assumes that the VIX follows a mean-reverting process like the Orstein-Uhlenbeck process, how would they price this non-martingale asset? My intuition tells me a trader would use doob-...
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Derivative pricing under $\mathbb{P}$
I recently learnt about the Girsanov-Cameron-Martin theorem, which basically says, that if $(\tilde{B}(t),t\in[0,T])$ is some Brownian motion with a (possibly stochastic) drift $\theta(t)$ defined on $...
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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 ...
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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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No arbitrage argument for the price process of a forward contract
I was reading the book Stochastic Calculus for Finance II by Shreve and I read the proof that the forward price for the underlying $S$ at time $t$ with maturity $T$ is given by
$$
For_S(t,T) = \frac{S(...
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Dynamics of independent Geometric Brownian Motions under risk-neutral measure Q
Suppose I have two Geometric Brownian motions and a bank account:
$$dB_t=rB_tdt$$
$$
dS=S(\alpha dt + \sigma dW_t)
$$
$$
dY = Y(\beta dt + \delta dV_t)
$$
Where $dW_t$ and $dV_t$ are independent ...
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Discounted price of an option
If the discounted price of any asset is a martingale under risk neutral measure, why is $E^Q[e^{-rT} (S_T-K)_+ | F_t]$, not merely $e^{-rt} (S_t-K)_+$?
This is something I wanted to clarify, since ...
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Minimal entropy martingale measure and Bayes estimated under Kullback-Laibller divergence loss function
We know that no unique equivalent measure exists in an incomplete market. Therefore, we need to choose a pricing measure equivalent to the physical measure based on a criterion. One typical approach ...
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Money account discounted libor rate is it a martingale under risk neutral measure?
I see that Libor $L(t,S,T)$ is a martingale under $T-$forward measure. Where we used argument that zero-coupon bonds are martingales under $T$-forward measure, as zero-coupon bond is a traded security....
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Floating Strike Geometric Averaged Asian Option Pricing
How can I use the risk neutral evaluation to price an asian option with floating strike using the continuous geometric average? I have tried searching for ways to do it but have found almost nothing.
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What is the risk neutral expectiation of an option price given a move in spot?
Lets say we have a volatility surface for the SPX at time t with spot S. We consequently know the price of some call option at maturity T with strike K. What is the risk neutral expectation of the ...
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Problem matching prices of Black-Scholes vs. GARCH(1,1) in Duan (1995)
In the paper of Duan (1995) the author compare European call option prices using Black-Scholes model vs. GARCH(1,1)-M model (GARCH-in-mean). To be brief, the author fits the following GARCH(1,1)-M ...
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Why fitting $\mathbb{Q}$ vs $\mathbb{P}$ measure Heston model if both fit to market
If both models fit their closed form formulas to market prices, why should I prefer a more complex model? ($\mathbb{Q}$ version has one extra parameter $\lambda$)
Do valuation with dynamics work ...
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Questions about the replicating portfolio in the binomial model
I'm starting to teach myself quantitative finance and I've got several questions (marked in bold) regarding the replicating portfolio of a security in the binomial model. I'm following, among others, ...
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One Period Risk Neutral Probability for Caplet
I am studying some financial modeling put together by the Society of Actuaries in the USA. In it, the following practice problem was given:
Find the Risk Neutral price of an at-the-money interest ...
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What is the dynamic of the forward price process under $\mathbf{Q}$?
Let me define the Spot price process of an underlying as follows: $$dS_{t}=\mu_{S}S_{t}dt+\sigma_{S}S_{t}dW_{t},$$
where $\left(W_{t}\right)_{t\geq0}$ is an appropriate Wiener-process, so $\left(S_{t}\...
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The difference between Credit Curve and CDS Curve
What's the difference between the credit curve and the CDS curve? Can I read the CDS curve from the Bloomberg terminal? for both single name and index?
Also, can someone please explain the difference ...
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Move from risk-neutral probability to historical probability
I am working on a density forecasting project using options. Using the Breeden-Litzenberger formula it is possible to find the implied density at maturity under the risk neutral probability of an ...
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Does every process need to be a martingale under martingale measure?
From the pricing theory, processes need to be martingales when divided by the numeraire asset.
A classical example is a stock option:
Consider a money market $B$ being the numeraire asset. When we ...
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Pricing of Zero Coupon bond under Risk-neutral pricing measure
Pg 242 Topic 5.6.2: Futures contract
Risk-neutral pricing of a zero-coupon bond is given by the below formulae:
$$ B(t,T) \, = \,\frac{1}{D(t)}. \tilde E~[D(T)\mid F(t)], 0\,\leq \,t\,\leq\,T\,\leq\,\...
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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 ...
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Option implied risk neutral distribution vs BKM risk neutral moments
I am doing some research on the option implied risk neutral distribution and methods calculate it, and so far have come across two ways to do so.
The first way is through the Breeden-Litzenberger ...
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Market price of risk ($\lambda$) - Brigo and Mercurio
In page 52 of Interest Rate Models by Brigo and Mercurio the following is stated:
Precisely, let us assume that the instantaneous spot rate evolves under the real-world measure $Q_0$ according to $dr(...
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