Questions tagged [risk-neutral-measure]

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5
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1answer
116 views

What is the connection between the risk neutral implied density and the real world density?

I understand that we can use option prices to imply volatilities and ultimately to imply a risk neutral density. I also understand that this implied density is not the same as the "real world density"....
2
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1answer
90 views

How To Understand the Drift of ln(S) if S Follows Geometric Brownian Motion

As we know, if an asset S follows geometric Brownian motion, under risk neutral measure, it can be expressed as $\frac{dS}{S}=rdt+\sigma dW$, by applying Ito's lemma, $d(lnS)=(r-0.5*σ^2)dt+σdW(t)$, ...
2
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0answers
83 views

Stochastic Volatility Models Real World Calibration

I am trying to find some research pertaining to the historical (or real world) calibration of stochastic volatility models. For example, in applications such as counterparty credit risk (IMM) or ...
2
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1answer
71 views

Risk Neutral Pricing and the Drift

For risk neutral pricing, why do we want to compute expectation of a martingale? why is this so important? Why do we dislike the drift so much? Avoid math heavy answers please.
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2answers
100 views

How to Understand Lognormal Distribution in the Following Case

I got a question and corresponding solution, but have some difficulties in understand the lognormal distribution part of it, so I really appreciate your advice: Question: assume zero interest rate ...
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0answers
28 views

Confirm If Risk-Neutral Measure is Unique in My Following Case

I'm reading a book that discusses about derivatives pricing and have some doubts about a particular problem and really appreciate your advice: Question: Assume a non-dividend paying stock follows a ...
2
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1answer
250 views

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 ...
-1
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1answer
19 views

What's the relationship between the risk-neutral probability in HJM and the risk-neural probability under domestic money market?

In shreve's book, we model the stock price dynamics as: $$S_i(t) = \alpha(t)S_i(t)dt +S_i(t)\sum ^d_{j=1}\sigma _{ij}(t)dW_j(t)$$ and the forward rate can be written as : $$df(t,T) = \gamma(t,T)dt + \...
2
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2answers
162 views

Assumptions in using risk-neutral pricing formula

The well-known risk-neutral pricing formula goes as follows (extracted from Shreve's Volume 2, section $5.2.4$ (Pricing Under the Risk-Neutral Measure)): Given any $T>0$ and any $t\in[0,T],$ if $V(...
2
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2answers
476 views

Risk Neutral and Real World Valuations using Monte Carlo

Assume I'm an investor that wants to sell exotic put options. No one else is selling my kind of put option, so I need to determine my own "Market Price" through Monte Carlo simulation. I know that by ...
0
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1answer
71 views

T-Forward measure and tenors

As far as I understand, a T-forward measure is associated with a situation when a zero-coupon bond with the same maturity, i.e. $P(t,t+T)$, is used as a numeraire. However, given that the yield curves,...
2
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1answer
113 views

Pricing of compounded swaps

As far as I understand, a compounded swap rolls up individual payments into one final payment which becomes: $$ V(t_n) = N \prod_{i = 0}^{n-1}(1 + d_i L_i)-N $$ where $d_i$ is the day fraction for ...
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1answer
48 views

Call price in case of AOA

I have this exercice, and for the last question, i tried to say that with lower bound, $C > S_0 - Ke^{-rT}$ which is $-8$ something but it doesn't make sense so i don't know what to do. Could we ...
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0answers
63 views

Proving an Expectation

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends. Consider the perpetual American put option with payoff $(K-S_\tau)^+$ when ...
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2answers
142 views

Proving a process is martingale under the Risk Neutral Measure

Show that for any $\lambda \in \Re$, the process $Y_{\lambda,t}$ defined as: $$Y_{\lambda,t} = (S_t/S_0)^\lambda e^{-(r\lambda-\lambda(1-\lambda)\sigma^2/2)t}$$ is a martingale under the risk ...
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0answers
73 views

Introducting a new probability measure

I'm trying to understand what means : $$ \frac {d \mathbb {\tilde{P}} }{d \mathbb P } \bigg\rvert_{\mathcal F_t }$$where $\mathcal F_t $ is a filtration I guess (not explicitely mentionned). they ...
2
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1answer
154 views

Floating Strike Lookback Call Option

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate $r$, stock drift $\mu$ and volatility $\sigma$). If $r=\...
1
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2answers
67 views

Equivalent Martingale (/Risk Neutral) Measure Conditions

I am trying to understand EMM's and wanted to understand why we use EMM's over just martingale measures. The way we define EMM's is (for a simple one period model): Given a probability measure $\...
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0answers
44 views

Strictly increasing asset price under a risk-neutral probability measure?

I am reading a paper on option pricing under jump processes in continuous time. There is a section labeled examples where the authors work under a risk neutral probability measure and derive option ...
1
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1answer
79 views

Why Joshi defined option value to be discounted payoff using risk neutral expectation?

Currently I am reading Mark Joshi's The Concepts and Practice of Mathematical Finance. At page $59,$ the author mentioned the following. Instead of requiring that every portfolio should have ...
2
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0answers
75 views

Does an option need to be tradable for Black Scholes pricing formula to hold?

Given the classic Black-Scholes model, e.g. $dS(t)/S(t)=rdt+\sigma dW^{\mathbb{Q}}(t)$ with $S(0)=S_0$ and $dB(t)=rB(t)dt$ with $B(0)=1$, whereby $r$ and $\sigma$ are constants and $\mathbb{Q}$ ...
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1answer
76 views

Calculating the value of Beta - Martingales

Assume a risk free bond $B_t$and the stock St follow the dynamics of the Black & Scholes model. (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Find $\beta$ such that the ...
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0answers
44 views

Deriving the risk-neutral pricing formula for the 2-state credit risk model

I am reading Interest rate models by Cairns—specifically the chapter on credit risk. Cairns introduces first the simple 2-state continuous time Markov model for credit risk—with the two states being "...
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0answers
37 views

Risk neutral measure in the binomial approximation of geometric Brownian motion

Suppose an asset is described by geometric Brownian motion with a drift, i.e. $$dS_t = S_t\mu dt + S_t \sigma dW_t$$ for a Wiener process $W_t$ and $S_0=1$. By some consequence of Girsanov's theorem (...
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0answers
37 views

Risk neutral valuation [closed]

In a world with three possible states (1, 2, 3) and three assets (A, B, C), the payoff matrix looks like this: $r_A;_1,_2,_3 = 110, 110, 110$ $p_A = 100$ $r_B;_1,_2,_3 = 100, 50, 40$ $p_B = 70$ $...
3
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1answer
72 views

Proof that we can price any derivative as the discounted value of its expected return under the risk neutral measure

I am reading a paper which tries to convey the intuition behind the Black-Scholes pricing formula. In that paper, the author states the following two things without proof, and I would like to know why ...
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0answers
17 views

single period security market with two assets

Consider a single period security market with two assets. Assume the current prices are There are two states at time one and the payoff matrix is 1.Suppose the investor believes that each state has ...
2
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1answer
75 views

Hedging an option on a non-traded asset in BS world

I have given the following task given. Suppose you are in a Black-Scholes World where you have the standard assets $$ dS_t = \mu S_t dt + \sigma S_t dW_t $$ $$ dB_t = r B_t dt $$ and now you also ...
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2answers
115 views

In BS option pricing, why is the drift rate of GBM equal to risk free rate for all stocks in risk neutral?

Can the drift rate μ depend on specific stock ? If not what is the rationale for the discounted Stock price to be a martingale ? \begin{align} & dS_t/S_t = \mu dt + \sigma dW_t \end{align} ...
1
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1answer
69 views

stochastic interest rate in binomial pricing model and in continuous models

Is the interest rate allowed to be truly stochastic in the binomial pricing model and in continuous models so that we are still able to switch to the risk-neutral measure? Shreve mentions multiple ...
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0answers
52 views

Switching to forward measure

I am trying to follow the wiki article on forward measure (https://en.wikipedia.org/wiki/Forward_measure), and can not understand how the change of measure was performed here: As I understand the ...
1
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1answer
110 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
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0answers
38 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 ...
4
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149 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
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1answer
74 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 ...
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0answers
48 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 ...
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0answers
51 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
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1answer
69 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
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1answer
89 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}^{...
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2answers
150 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
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0answers
64 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:...
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0answers
42 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 ...
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0answers
33 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: ...
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0answers
81 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
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1answer
126 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
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2answers
129 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
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0answers
55 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
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1answer
225 views

Estimation of Radon–Nikodym derivative from historical 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
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1answer
105 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 ...
2
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0answers
199 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 ...