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.

Filter by
Sorted by
Tagged with
2
votes
0answers
24 views

EMM, Supremum and Expectation

I asked this question on MSE recently. https://math.stackexchange.com/questions/3922347/supremum-and-expectation I want to prove this when $\mathcal{M}$ is a set of equivalent martingale measure. ...
1
vote
0answers
82 views

Value at risk, risk-neutral vs real-world probability measures

Does anyone know if there is any link between the Value at Risk of risk-neutral distribution and of the real-world distributions of asset rate of returns?
0
votes
0answers
56 views

Risk-neutral pricing for processes other than Geometric Brownian Motion

For pricing derivative with payoff $H$ in BS model we use formula: $$\Pi_t(H)=e^{-r(T-t)}\mathbb{E}_Q[H|\mathcal{F}_t]$$ Now I have market with one risk free asset and one risky asset whose price is ...
1
vote
0answers
46 views

Why does higher volatility for ATM Call Option lead to a lower risk-neutral probability of expiring ITM?

This is a follow-up question on the discussion in the thread here, from which I borrow the graph below depicting $N(d_2)$ (i.e. the risk neutral probability of a Call option expiring in the money) ...
2
votes
1answer
114 views

Discounted price process - martingale

I have a process $S_{t}=S_{0}e^{\left(r-q\right)t+mt+X_{t}}$, where $X_t$ is a Levy process and I want to check for which $m$ the process $e^{-(r-q)t}S_t$ is a martingale. The third condition of a ...
1
vote
0answers
72 views

Derivation of $u=e^{\sigma\sqrt{dt}}$ and $d=e^{-\sigma\sqrt{dt}}$

Anyone could provide me a proof of how, starting from $\frac{dS_T}{S_t}\sim \operatorname{N}(\mu dt,\sigma^2 dt)$ with $p:=\frac{e^{rdt}-d}{u-d}$, we can obtain the parameters $u$ and $d$ as from ...
1
vote
0answers
44 views

Replicating portfolio

I have a doubt about the replicating portfolio methodology. Example - Consider an European Call with $K=21$ and underlying with current price $S_0=20$. We assume that, at the maturity, the underlying ...
1
vote
1answer
108 views

Objective probability of default from CDS spread

I have the risk neutral probability of default extrapolated from the market data of the CDS spreads. How can I empirically estimate the market risk price of the objective probability of default (i.e. ...
1
vote
0answers
28 views

Mismatch of periods with numeraire compared to the forward rates

In Joshi's The Concepts and Practice of Mathematical Finance Page 323--324 I believe that there may be a mismatch of periods with forward rates: Consider time partition $t_{0} < ... < t_{n}$ ...
2
votes
2answers
282 views

Help reconciling incorrect reasoning in options pricing brain teaser

I'm trying to reconcile an interesting brain teaser I was recently posed and I need help understanding the flaw in the reasoning. The problem states there is an asset which after an announcement has ...
2
votes
1answer
78 views

No-arbitrage Pricing

We have a contract whose value is $A(S_t,t) = S_t^3$ at all times, not just at expiration. $S_t$, the underlying stock, follows a Geometric Brownian Motion, $\frac{dS}{S} = \mu dt + \sigma dB$. How ...
3
votes
1answer
63 views

General Dynamics of a Tradable Asset under the Risk Neutral Measure

Is it true that every tradable asset must have a log-normal dynamics under the risk neutral measure where the drift term is the short rate $r$? I.e., is it true that if $X$ is a tradable asset then $$\...
0
votes
0answers
51 views

Discounted stock price under a NON risk-neutral measure

Under a risk-neutral measure $\mathbb{Q}$, the discounted stock price is a $\mathbb{Q}$-martingale. Does it mean that under the actual probability measure $\mathbb{P}$ the discounted stock price is ...
1
vote
1answer
47 views

Risk-Neutrality: Discount factors of the $P$ world according to risk preferences?

I am coming to terms with the connections between the so-called $P$ world and the $Q$ world. In my understanding, the risk-neutral measure $Q$ induces a probability space under which investors are ...
0
votes
1answer
51 views

Log-normal risk-neutral price derivation from binomial trees, not clear about step in derivation process

At page 64 of the book Concepts and practice of mathematical finance, 2nd edition by M. Joshi, paragraph 3.7.2 (Trees and option pricing - A log-normal model - The risk-neutral world behaviour) a ...
6
votes
0answers
60 views

Can the risk-neutral measure depend on the option type?

In an ideal Black-Scholes setting, the Risk-Neutral measure $Q$ is unique and so, obviously, does not depend on what derivative instrument we want to price. Assume some deviation from perfect markets (...
1
vote
1answer
114 views

Under what measure is the SABR stochastic differential equations

The SABR Model is a CEV (constant elasticity of variance) Cox asset process with correlated lognormal stochastic volatility. A forward rate $F(t,T)$ to time $T$, observed at $t$, and the instantaneous ...
0
votes
1answer
47 views

How does $1 + R = q_u · u + q_d · d $ follow from $d ≤ (1 + R) ≤u$ in the Binomial Pricing Model?

I've been reading Tomas Bjork's 'Arbitrage theory' and it says: To say that $d ≤ (1 + R) ≤u$ holds is equivalent to saying that $1 + R$ is a convex combination of u and d, i.e. $1 + R = q_u · u + q_d ...
1
vote
1answer
87 views

Calculating European call option, the Bjork way

We have a 3 period binomial tree with values: ...
0
votes
0answers
54 views

Risk free rate application to option pricing

We have $S_o = 50, u = 1.0606, d = 1/u, K = 54.50,$ risk free rate $r = 0.1$ per week, maturity in 9 weeks, given a binomial tree (3 steps)with the probabilities given by $q = (1+e^{r(T-t)}/u-d)$, no ...
0
votes
0answers
33 views

Equivalent martingale measure for Levy processes

Hey what is the easiest way to find equivalent martingale measure for Levy processes (in Merton model and Kou model for example)? I would like to write the dynamics of the stock price process under ...
1
vote
1answer
66 views

We have a two LIBOR contracts, how to compare their values by change of change of numeraire

We have two LIBOR contracts: contract 1 pays $L\left(T_{1},\:T_{2}\right)-K$ at time $T_{1}$ contract 2 pays $L\left(T_{1},\:T_{2}\right)-K$ at time $T_{2}$. Now, $F_{1}$ is the par strike such that ...
5
votes
1answer
112 views

How to choose the martingale measure in incomplete markets

Hey I know that when market is incomplete, then we have to choose an equivalent martingale measure (I heard about Escher Transform martingale measure, Mean correcting martingale measure, minimal ...
1
vote
0answers
46 views

Change of Measure for Jump Process with Drift and no Brownian motion

If on $(\Omega, \mathcal{F},\mathbb{P})$, $r>0$ is a constant and $Z_t =\sum_{i=1}^{N_t} Y_i$ where $Y_i$ are i.i.d with $E[Y_i]=L$ denotes the size of the jump and can have distributions like ...
2
votes
1answer
90 views

Variance-Covariance Matrix under $\mathbb{P}$ and $\mathbb{Q}$

I'd like to understand why $\Sigma$ is the same under both measures $\mathbb{P}$ and $\mathbb{Q}$. Is it an assumption or a general fact based on theoretical concepts?
2
votes
1answer
79 views

How to derive the expected loss from the credit risk of a bond?

I am trying to work out a formula to derive the expected loss from the credit risk of a bond. My idea is to tie the credit risk to credit valuation adjustment and derive the expected loss from there, ...
2
votes
1answer
122 views

Black-Scholes Formula under $T$-forward measure

The Black-Scholes price of a European call option is given by $$ C_0^{BS}(T, K) = \mathbb{E}_Q[e^{-rT}(S_T - K)_+] = S_0 \Phi(d_1) - Ke^{-rT}\Phi(d_2) ,$$ where $$ d_{1,2} = \frac{\log\big(\frac{S_0}{...
1
vote
1answer
215 views

Power Options & Forwards on Stock Squared

Short story: the process for Stock price squared is not a martingale when discounted by the money-market numeraire under the risk-neutral measure. How can we then compute derivative prices on $S_t^2$ ...
4
votes
1answer
106 views

On Girsanov Theorem to switch from Risk-Neutral to Stock Numeraire

Summary: long-story cut short, the question is asking for what types of functions $f(.)$, the Cameron-Martin-Girsanov theorem can be used as follows: $$ \mathbb{E}^{\mathbb{P}^2}[f(W_t)]=\mathbb{E}^{\...
0
votes
0answers
33 views

missing part of code for calculation of bondarenko positive convolution method for risk neutral density

I am adjusting code provided to Wm. Burknecht (see this post) to calculate risk neutral density using Bondarenko positive convolution method. The SLQSP minimization problem has the form minimize sum ...
3
votes
3answers
373 views

Which measure is used to price a swap?

When we value the floating leg of a standard vanilla swap, we replace the expectation of the future floating rates by the forward rates known today. However my understanding is that the forward rate ...
0
votes
0answers
26 views

What is a succinct description of the difference between neutral and indifference pricing as per Srdjan Stojanovic?

His book (Neutral and Indifference Portfolio Pricing, Hedging and Investing With applications in Equity and FX, by Srdjan Stojanovic, Springer, 2011) is not quickly readable. Wikipedia entry on ...
1
vote
0answers
44 views

Unique risk neutral measure for jumps or incomplete markets for jumps

I wanted to understand why the market is incomplete in jump-diffusion models. whereas if we have a model following geometric Brownian motion then we can get a risk-neutral measure and hence a complete ...
8
votes
3answers
1k views

What is the Risk Neutral Measure?

What is the Risk Neutral Measure? I don't believe this has been answered on the internet well and with all the parts connecting. So: What is the risk neutral measure/pricing? Why do we need it? How ...
3
votes
2answers
248 views

Value (price) of defaultable zero coupon bond with credit risk involved

I'm trying to derivate the Value (price) of defaultable zero coupon bond, but there some steps (math) in between I can't figure out. From the default process modelling, we have: $$P(t ≤ \tau < t+dt ...
2
votes
0answers
30 views

Does equity premium puzzle affect option-implied RWDs using Arrow-Debreu equilibrium?

I am researching and learning about option-implied RNDs (risk neutral densities) and transformation to RWDs (risk world densities) using expected utility theory to compute risk aversion values. This ...
1
vote
0answers
64 views

R: How do i finish the tails in the risk neutral density, obtained from option prices

Im currently working on constructing the risk neutral probability distribution of a stock, based on the option prices. In doing so, i calculate the implied volatilities from the option prices, and ...
1
vote
0answers
43 views

Replicating portfolio of an option and to find inital price

I am very new to financial math so I am not sure how to do with this question. A friend sent me this question to practice but I am unsure how to begin. I read about call option . Can that be used for ...
0
votes
0answers
24 views

question about code posted for calculation of risk neutral density using Bondarenko convolution method

I have questions about the code (found here Estimation of Risk-Neutral Densities Using Positive Convolution Approximation - Python). The synthetic price in Bondarenko paper includes two terms before ...
3
votes
1answer
186 views

Multiple Risk-Neutral measures in incomplete market

This question is in regards to incomplete markets where multiple risk-neutral measures exist. I am a little bit confused by this idea. Say we have an incomplete market with only one stochastic process ...
0
votes
1answer
47 views

Is the differential between risk free rates the drift of an exchange rate only in the risk neutral world?

Take for example this passage from "Monte Carlo Methods in Financial Engineering". Is this a result of the risk neutral world or is this the real world drift as well? I've never seen the explicit ...
0
votes
0answers
77 views

complete python code to calculate risk neutral density from option prices [duplicate]

bAsic python code to implement Litzenberger formula for risk-neutral probabilities implied by option prices. Use S&P 500 option prices whose strike intervals are typically 5 points apart use at ...
1
vote
0answers
66 views

Does simulating price as GBM automatically implies risk neutrality?

I am using a dynamic programming approach to price European options where formulate the pricing as a discrete-time continuous-space Markov Decision process. The MDP is risk-sensitive as in I don’t ...
2
votes
2answers
113 views

Heath–Jarrow–Morton under real-world measure

In HJM model (framework), the drift of the forward is determined by its diffusion coefficient: $$ \mu(t,s) = \sigma(t,s)\int_t^s \sigma(t,v)^Tdv $$ My understanding, is that the change of measure ...
3
votes
2answers
63 views

Estimating risk aversion from option bid-ask spreads

Is it possible to use bid-ask spreads on contracts from a specific tenor to estimate risk aversion and use it to transform risk-neutral density into real-world density?
1
vote
0answers
25 views

In BS model, is there a way to show that the risk-neutral Q is unique without using MRT nor the fact that the market is complete?

In Black-Scholes model, is there a way to show that the risk-neutral probability measure is unique without using the martingale representation theorem nor the fact that the market (in BS model) is ...
3
votes
1answer
66 views

What are the relation between the risk neutral measures in binomial tree and in Black Scholes model?

I appreciate that both are the direct result of constricting a replicate portfolio using stock and bonds. Are there deeper relationship between the two?
0
votes
0answers
35 views

Can you explain the Black-Scholes fair option equation with RND?

I am trying to learn Black-Scholes risk-neutral densities with only prior knowledge of fundamental B-S equations (not the derivation). Sorry if this was asked already or if I sound completely clueless....
2
votes
1answer
33 views

Properties of risk aversion

What are some common properties for risk aversion? I know the basic definition of the risk premium, absolute risk adversion, relative absolute risk adversion. Besides the basic definition, what are ...
1
vote
1answer
54 views

Why are the risk neutral probabilities constant in the Cox Rubinstein model when delta needs to be changed at each time step

Consider the Cox Rubinstein binomial pricing model with N steps, with stock price change given by parameters u and d so that at step $i$ we have $S_{i+1} = uS_{i}$ or $S_{i+1} = dS_{i}$ with $0\leq i \...

1
2 3 4 5
7