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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Relationship between Beta distribution and its inverse
I am attempting to transform a real world density into risk-neutral density via calibration through the beta distribution.
Calibration in this context is transforming the rw density into the rn ...
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Complete market price and incomplete market price specification
We know that if a liquid market of an asset exists, then the standard derivative pricing theorem implies an equivalent martingale measure exists, not necessarily unique, under which the discounted ...
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Use of Non-Risk-Neutral Measure for Pricing Derivatives
While trying to understand the risk-neutral pricing of derivatives when the underlying is the spot price of a commodity, I encountered the situation that the measure used for pricing derivatives is ...
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Looking for Options Which Pay Exactly When A Random Barrier is Reached
Supper that I fix two barriers $a<b$ and I consider a price process $X_.$ starting in the interval $(a,b)$. Let V be a payoff function and let $\tau:= \inf \{t>0: X_t\not\in (a,b)\}$.
Are there ...
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Functional Form of Radon-Nikonym Derivative
Suppose I have empirical values of dQ/dP in the context of risk-neutral and real-world probabilities of asset returns. Would it be possible to fit a functional form of the derivative? I'm particularly ...
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Call Option, Delta and Expected Payoff
In Dynamic Hedging by N. Taleb, at pag. 283-284, there is an argument about the relationship between risk neutral evaluation of a binary call and the delta about a call.
The author states that:
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Justification of the Risk Neutral Measure in the Schwartz One Factor Commodity Model
I have been trying to understand the form of the risk neutral measure in the Schwartz one factor model for commodities (Model 1 on page 6 here) where the spot price of a commodity follows the process ...
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Arbitrage-Free, Breeden-Litzenberger and Risk-Neutral Measure
Let $\sigma_{\text{BS}}(K, T)$ a given IV slice at $T$, which is implied by a price slice $C(K, T)$.
From this price slice, we can infer the Risk Neutral density of the price distribution at $T$ using ...
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Understanding the Impact of Illiquidity on Equivalent Martingale Measures (EMMs) in a Simple Market
I'm currently studying a simple market model with an asset $S$ whose price follows a geometric Brownian motion ($dS_t=S_t(μdt+σdW_t)$) and a risk-free asset $B$ ($dB_t=B_trdt$) over a finite horizon $...
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Calibration of Heston-Nandi GARCH Model Using Historical Data: Risk-Neutral vs. Physical Measure
I am currently working on calibrating the Heston-Nandi GARCH model using historical asset return data and am faced with a decision on whether to use the risk-neutral or physical measure for this ...
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Why are random coupons not priced using risk-neutral evaluation?
Assume a fixed coupon bond has a coupon which, randomly, is 5 % or 4 %, each occuring with a 50 % probability. The issuer flips a coin on payment date to decide which it should be.
I would value this ...
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Benth: Risk-neutral measure in incomplete markets
I am currently working on Benth and Benth "THE VOLATILITY OF TEMPERATURE AND PRICING OF WEATHER
DERIVATIVES" and i am stuck at following paragraph at page 10, which is about risk-neutral ...
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How to prove that a market is incomplete using the concept of EMMs?
Question
Consider a one-step trinomial tree, where there are two traded assets, a bond with risk-free rate, $r$, a stock with initial price, $S_0$, and terminal price
$$S_T =
\begin{cases}
S_0u,& ...
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Why is my Risk Neutral Density recovery failing?
I'm working on a project to recover a known Risk Neutral Density from option prices, using the Breeden-Litzenberger formula (assuming a continuum of option_price(strike_price), the second derivative ...
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How to simulate a conditional expectation given a filtration
I had a question regarding how to simulate a certain conditional expectation. I am given two processes $X_1(t), X_2(t)$ which both follow their own SDE, but both are of the form
\begin{equation*}
dX_i(...
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Validator for Risk-Neutral Distributions Derived from Option Prices
I've developed a validator for risk-neutral distributions. I did this for the purpose of testing the risk-neutral distributions generated by a Spectral Analysis risk-neutral density recovery method, ...
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Computing Derivative Security with Change of Numeraire
Under Black-Scholes, price a contract worth $S_T^{2}log(S_T)$ at expiration.
This is a question from Joshi's Quant Book (an extension question).
Ok, so I solved this with 3 different methods to make ...
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Why A Derivative With Intrinsic Arbitrage Cannot Be Valued & Hedged With Assets In Risk Neutral?
I'm attempting to concisely show why a derivative that, by nature, introduces arbitrage cannot be valued using risk neutral pricing tools.
Derivative:
Buyer is sold a 'call option', with time 0 value ...
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Validating an option-implied risk-neutral distribution by integrating it twice and comparing the resulting "prices" with the original ones
From Breeden-Litzenberger, we know that the second derivative of a European call option's price with respect to the strike price is equal to the risk-neutral probability density function of the ...
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Verifying my understanding of replicating portfolio, hedging and option pricing
Under risk neutral measure, we use replicating portfolio to mimic the value of derivative (for example European options).
Many literatures use the word "hedging" to describe the replicating ...
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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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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)$$
Is there any risk-neutral measure?
My thoughts are pretty simple: $μ$ is ...
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Dynamics of independent Geometric Brownian Motions under risk-neutral measure Q
Suppose I have two Geometric Brownian motions and a bank account:
$$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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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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