Questions tagged [risk-neutral-measure]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
1answer
61 views

Why use the risk-free rate for discounting in a risk neutral world?

I am reading Options, Futures, and other derivatives by John C. Hull. In the chapter on Binomial trees, he remarks: A risk-neutral world has two features that simplify the pricing of derivatives: ...
2
votes
0answers
25 views

State price deflator in the Vasicek model

I am trying to implement a simple bond pricing model using state price deflators in a Vasicek model. I am simulating paths of the processes $$\mathrm{d}r^{P} =\kappa^{P}(\theta^P - r^P(t))\mathrm{d}t ...
2
votes
0answers
92 views

Model-Free Option Pricing

From Breeden and Litzenberger (1978) and subsequent work, we may find the risk-neutral density $q_{S_T}$ of $S_T$ from European option prices - assuming there are enough traded options (e.g. SPX) via ...
2
votes
1answer
38 views

Valuation of Cash-Or-Nothing option

Studying options pricing, I'm stuck with the following problem: The price of a stock is described by the dynamic: $$dS_t = \mu\, dt + \sigma\,dW_t$$ Compute the fair price of a Cash or Nothing ...
0
votes
0answers
38 views

Risk neutral measure & change in numeraire

There are two questions about risk neutral and change in numeraire I am not so sure if my answer is correct. Question 01: Risk neutral Let says I have 2 risky asset A and B. Each has stochastics ...
0
votes
0answers
41 views

What discount rate should I use in domestic/foreign context?

I am trying to price a quanto option by monte carlo simulation via quanto adjustment. SDE: $dS_t^f=S_t^f(r_f - \rho \sigma_s \sigma_{d/f})dt + S_t^f\sigma_s dW_t^d$, where $S_t^f$ is the underlying ...
1
vote
1answer
63 views

How can the forward risk neutral measure be used to derive Black's model?

In the Hull textbook's derivation of Black's model (Section 27.6), they apply equation (27.20), which is $f_0 = P(0,T)E_T(f_T)$, where $P(0,T)$ is the value of a zero coupon bond at time $0$ expiring ...
4
votes
1answer
78 views

Equivalence of Put Pricing Formulas

I have to show that: \begin{equation} P_{t,T}(K)=e^{-r(T-t)} \int_0^{\infty}\left(K-S\right)^+ q_T^S(S)dS \end{equation} is equivalent to: \begin{equation} P_{t,T}(K)=e^{-r(T-t)}\int_{-\infty}^{...
1
vote
2answers
119 views

Risk-neutral pricing and statistical arbitrages

I'm studying the martingale approach to asset pricing. Dealing with the concept of risk-neutral probability, I came up with a question about the possibility of "arbitrages in expectation". I'll be ...
3
votes
0answers
51 views

American Perpetual Put Option

I want to compute the expected payoff of a (classical) perpetual American put option in the Black-Scholes-Merton (BSM) framework with an optimal strategy of exercising the option at time $\tau=\inf\{t:...
0
votes
0answers
27 views

Risk neutral interest rate calibration

I've only worked with RW model before but not RN interest rate models, so I'm looking for some practical insights on how RN calibration is done for interest rate models. Let's say I want to start ...
0
votes
0answers
29 views

Expected Payoff of stock using risk neutral Valuation

I recently saw a calculation where the expected 10-state-payoff-diagram of a stock with mean 6% and variance 20% was calculated through the risk neutral measure. The method was as follow: ...
1
vote
0answers
43 views

Unconditional Expectation vs. Conditional Expectation at time $0$

In most mathematical finance books I have read (all of them actually), the expectation, with respect to the sigma algebra at time $0$, $\mathcal F_0$, is considered the same as the unconditional ...
1
vote
1answer
57 views

Deriving the risk neutral probability with the arrow debreu Price vector

today I had an oral exam about Stochastic Finance. With one of the questions I was pretty helpless. We were talking option pricing in a scenario where we have Portfolio with n-assets and k-states. But ...
2
votes
2answers
77 views

Are all changes of measures for continuous diffusion processes given by the change of drift?

In elementary discussions on change of measure for geometric Brownian motion, one often find statements like "change of measure = change of drift". Given a general continuous diffusion process of the ...
2
votes
0answers
52 views

Risk-Neutral Pricing with Regime Switching

As the title suggests, I am currently trying to implement a dual regime-switching options pricing model. In its simplest form, I am fitting a risk-neutral GARCH(1,1) to a crash and normal regime. ...
4
votes
1answer
129 views

Estimation of Radon–Nikodym derivative from hisotrical returns and option price data

Say we have an estimate of empirical density function $f^{\mathbb{P}}_S(s)$ of historical log-returns on a stock $S$ over a 30-day period under the real-world objective measure $\mathbb{P}$. We also ...
3
votes
1answer
75 views

What happens in the binomial model if the real-world probability is $0$

Consider a binomial model. Suppose we know that the price of a stock will become a certain value at the next timestep. That is, one of the two outcomes has $0$ real-world probability. Then it should ...
1
vote
0answers
73 views

Change of measure from physical to risk-neutral under Radon-Nikodym and Girsanov Theorem

Given a stochastic process, how do we prove and generate the change-of-measure? I have been trying to prove the change-of-measure as under the Radon-Nikodym theorem and Girsanov Theorem, but ...
3
votes
1answer
102 views

Introduction of a stochastic discount factor in martingale pricing

The example below is taken from Björk (2009). Let Radon-Nikodym derivative be $$L=\frac{dP}{dQ} \;\; \text{on} \; \mathcal F$$ or written analogously $$P(A) = \int_AL(\omega)dQ(\omega) \;\; \text{for ...
1
vote
1answer
106 views

Barrier Option Valuation

Good day, A reverse knock-out barrier call option expires worthless if the asset price ever goes above a given barrier level. Calculate the value of this barrier option struck at $K = 3$ with ...
1
vote
1answer
135 views

Hedging strategy for American Option

Good day, I was asked to devise a hedging strategy for an American Option given the following claims. Note, $r=0$ and the underlying stock pays a dividend of $1$ at time $t=1.5$ \begin{...
4
votes
0answers
56 views

Why does risk-neutral price processes do not, in general, compose all arbitrage-free price processes?

I was reading reviewing my mathematical finance notes and I came across a remark I cant understand fully Remark :Contrary to discrete time models, the risk-neutral price processes do not, in general, ...
1
vote
1answer
148 views

Measure of a Brownian motion = normal distribution?

Consider some model where the process increments are normally distributed, e.g. Vasicek: $$dr(t) = \left(\theta - ar(t)\right)dt + \sigma dW(t).$$ We usually say that $W(t)$ is a Brownian motion ...
2
votes
1answer
149 views

Risk neutral modelling of a stock

Suppose a stock $S$ follows $$dS(t) = \alpha(t)S(t)dt + \sigma(t)S(t)dW(t),$$ where $W(t)$ is a Brownian motion under $P$. Also suppose there is a short rate process $r(t)$. My question would be is ...
1
vote
1answer
76 views

Why can only non-dividend paying assets serve as numeraire?

In Kerry Back, A Course in Derivative Securities, Sect. 1.4 (page 29), the author stated the FTAP in the following form (in boldface): If there are no arbitrage opportunities, then for each (non-...
5
votes
2answers
197 views

Estimation of Risk-Neutral Densities Using Positive Convolution Approximation - Python

I'm trying to estimate the risk-neutral density through positive convolution approximation (introduced by Bondarenko 2002: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=375781). I'm currently ...
3
votes
1answer
131 views

Do *all* non-dividend paying assets have the risk-free instantaneous return rate under the risk-neutral measure?

For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). ...
4
votes
1answer
194 views

How to determine the risk-neutral measure in a Heston model?

To clarify, I'm quite familiar with the risk-neutral pricing framework, and I know one can efficiently Monte-Carlo a Heston model via the non-central $\chi^2$ distribution approach. But so far we're ...
0
votes
0answers
48 views

Change of measure price put option

I hope you can help me out. I'm really stuck understanding this. In my lecture notes we calculated the price of a put option (maturity m,with strike price $(1+i)^m$, where i is some interest rate) as ...
2
votes
2answers
99 views

Can the historical probability be the same as the risk neutral probability measure?

In particular lets consider a zero-beta asset $i$ (in the CAPM sense). Let $R_f$ be the risk free rate $R_i$ the return on the asset $i$ $R_m$ the return on the market portfolio $\beta=\frac{Cov(R_i,...
4
votes
1answer
94 views

The choice of portfolio in the proof of the Black-Scholes formula

Consider a stock whose price $S$ satisfies $$dS_t=\mu S_tdt+\sigma S_tdW_t$$ for constants $\mu,\sigma$ and where $W$ is a $\mathbb{P}$-Brownian motion. Further assume that the stock pays out ...
3
votes
1answer
87 views

Fair value of a binary cash-or-nothing option with a barrier

I want to find the fair value of a European cash-or-nothing option that pays \$1 if $S_t>K$ and $S$ breached the level $M<0<K$, where $S$ is the risk-neutral process $dS_t=\sigma dW_t$. My ...
4
votes
1answer
148 views

Uniqueness of Risk-neutral measure: Probabilistic view

Suppose we are working on the Black and Scholes Framework. There are only two assets, the risk-less bank account and a stock. The discounted process is a GBM under the physical measure with drift term ...
0
votes
1answer
67 views

Risk neutrality coherence with risk aversion

I haven't been able to find an understandable explanation why the risk neutrality is coherent with the risk aversion implication of the expected utility hypothesis. I can see that when using the risk ...
0
votes
1answer
87 views

Why can derivatives be viewed as a portfolio of the underlying and the riskless asset?

I am struggling with the statement: "Every derivative of the underlying can be viewed as a portfolio of the underlying asset and the riskless asset." Is this based on the put-call parity? Also I ...
3
votes
1answer
213 views

Equivalent martingale measure exists if and only if $a < S_0^1(1+r)< b$

Exercise : We consider a market of one period $(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$, where the sample space $\Omega$ has a finite number of elements and the $\sigma-$algebra $\mathcal{F} = 2^...
1
vote
1answer
159 views

Risk neutral valuation formula

I am totally new to Finance and Arbitrage theory and I have started reading Björk (2018) Arbitrage theory in continuous time. Can anyone please explain to me what is the risk-neutral valuation formula ...
0
votes
1answer
170 views

Suggestions of papers for computing market implied probability distribution function

I need suggestions of papers that propose simple and fast methods (not heavily dependent on simulations, nut can depend on simulation) to derive the market implicit probability distribution function ...
3
votes
1answer
264 views

condition of risk neutral pricing

The theorem says if $U$ is a numeraire and let $\mathbb{Q}^U$ be the corresponding measure. Then for every tradable asset $S$, the relative price $S_t/U_t$ is a martingale under $\mathbb{Q}^U$. But I ...
4
votes
1answer
269 views

Change of numeraire between T-forward and Bank Account

I follow a course, and get to the point that one bond price discounted by another one is a martingale: $$ \frac{P(t,T_0)}{P(t,T_1)} - \text{ is a } \mathbb{Q}^{T_1} \text{ martingale } $$ I can not ...
2
votes
0answers
123 views

Dividend paying asset, why can't be taken as numéraire?

Why when considering numéraires, one cannot use a dividend paying asset to define a risk neutral measure? Here's where I got my question : (Shreve - Stochastic Calculus For Finance II)
3
votes
1answer
174 views

Libor Market Model (LMM) under risk neutral measure

I would like to establish the equations of forward libors under risk neutral measure. Here is how I do it, and what I get : Under the $P_{T_j} $ measure, forward Libor $L_j$ is martingale. Thus: $$ ...
1
vote
0answers
86 views

Checking arbitrage for the SABR model - analytical vs numerical approach

I wish to check if the fitted volatility smile/surface from the SABR model for a fixed time period is arbitrage free. Through my research, I've learnt the following need to be checked: The RND (risk ...
2
votes
0answers
69 views

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 ...
1
vote
1answer
217 views

Confusion regarding the risk neutral and physical measures

I may be confused. I am looking at the risk neutral vs. physical measures. We know that knowing the short interest rate stochastic process $r$, a bond maturing at time $T$ can be considered as a ...
1
vote
1answer
223 views

Vasicek short rate: Risk-neutral measure into real-world measure

I consider the Vasicek model under the risk-neutral measure $\mathbb{Q}$: $$ dr_t=\kappa(\theta−r_t) dt+\sigma dW^{\mathbb{Q}}_t.$$ I have already determined $$\mathbb{E}^{\mathbb{Q}}\left[e^{−\int\...
3
votes
0answers
98 views

Equivalent martingale measure in time changed Levy models

I am investigating time changed Levy models. As far as I have seen, these models are usually directly described under the risk neutral measure $\mathbb{Q}$. However, I'm interested in first modelling ...
2
votes
1answer
37 views

calibrating two (or X) equity diffusion trees

I have two equities S1 and S2. Each one follows the following tree evolution : $$S_1 \rightarrow \left \{ \begin{matrix} S_1 (1+u_1) & \text{with probability } p_1 \\ S_1 (1-d_1)...
1
vote
0answers
149 views

What is the risk neutral density and how is it estimated? [closed]

I don't understand the words "risk-neutral density". Please explain what it is, and how it can be estimated in practice. My guess would be that we have an underlying probability space $(\Omega, P)$. ...