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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39 views

Is the initial value of the portfolio replicating a forward zero?

This is from the book Financial Calculus: An Introduction to Derivative Pricing by Martin Baxter. By choosing appropriate weights in a portfolio of a stock and cash bond you can replicate the payoff ...
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Why is the market price of risk in the one factor Schwartz model different from the usual one?

Assume that the commodity spot price follows the stochastic process (see Schwartz article page 926) $$ dS = \kappa(\mu-\log S)Sdt+\sigma SdW, $$ where $\kappa >0$ measures the degree of mean ...
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66 views

FX Asian Option Moment-matching in Harmonic case

I need to price a "foreign-paying" fixed-strike Asian (i.e., average) option. Thus, the payoff is: $$\left(\frac{A_T - K}{A_T}\right)^{+} = \left(1 - \frac{K}{A_T}\right)^{+} = K \left(\frac{...
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Fitting parameters given an inverse function. (Orosi, 2015)

In trying to replicate Orosi's (2015) 5-parameter implied volatility model, but I can't wrap my head around the parameter fitting procedure Orosi proposes. My main goal is to calibrate the model to my ...
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83 views

Equivalence of Call Option on $S_T$ and Put Option on $\frac{1}{S_T}$ in FX Markets

Part 1: I am trying to price an option in the FX world. It naturally pays in the domestic currency, but in this case the payout currency must be the foreign currency. For example, consider the payoff: ...
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How to use Girsanov theorem for complicated RN derivatives?

Let $W_t$ be a Brownian motion under probability measure $\mathbb{P}$. Let $X_t$ be defined as follows. $$\mathrm{d}X_t = a \mathrm{d}t + 2\sqrt{ X_t} \mathrm{d}W_t.$$ Also define: $$L_t = \exp\left(-\...
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Q determined by the market in Binomial Model

I read in a book about change of measure, so that the discounted stock price in a binomial model is equal to the current price. Namely: $$E_{Q}[S_{1}/ \beta |S_{0}]= S_{0} $$ It then says: " Q is ...
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Poisson process under equivalent martingale measure

I have a stochastic process $N(t)$ which is equal to $n$ with probability $P\{N(t) = n\}=\frac{\left(\lambda t \right)^{n}}{n!}e^{-\lambda t }$ where $t$ represents the time period. In other words, ...
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83 views

How to compute the Present Value of this path-dependent option?

I have an option whose payoff depends on its value at two times $T_1$ and $T_2$ as follows. $$V(t) = \mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B} (S(T_2)-K)^+)],$$ where the stock price follows the GBM ...
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Lognormal SABR symmetries

Consider the lognormal SABR model ($\beta=1$) for an FX forward process $F$: \begin{align} dF&=aF dW\\ da&=\nu a\left(\rho dW+\sqrt{1-\rho^2}dW^\perp\right) \end{align} where $(W,W^\perp)$ is ...
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Additional requirement for the asset price and payoff to ensure the market is arbitrage-free

Suppose we have two risky assets and one risk-free asset in the market. The market is incomplete in that there are three assets and four states. The price vector at $t_0$ is: $\boldsymbol{p_0}=[p^s_{1}...
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101 views

replicating self-financing portfolio for risk neutral measure

Let the price process $S_{t}, 0 \leq t \leq T$, be a diffusion, and savings account be $\beta_{t}$ such that the Equivalent Martingale Measure $Q$ exists. Let $C_{T}=g\left(X_{T}\right)$ be the claim ...
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How to estimate lambda from NAGARCH submodel in R

I am trying to estimate the model="fGARCH", submodel="NAGARCH" from the rugarch package in R. However, when I am estimating the parameters, only omega, alpha, beta and gamma are ...
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339 views

Risk Neutral Valuation, Drifts and Calibration

Lets consider a pricing model like Vasicek. Apparently, if you calibrate a derivatives pricing model to market prices this gives you risk neutral parameters. Its not clear to me as to WHY this will ...
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119 views

option pricing formula for $S_{t}=S_{0}+\mu t+\sigma B_{t}$ where r = 0

I have been on this for hours and it's not getting me anywhere. Any help is so highly and deeply appreciated. A call option with strike $K$ and expiration $T$ pays $C_{T}=\left(S_{T}-K\right)^{+}$ at ...
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Risk-neutral Probability, Risk-Adjusted Returns & Risk Aversion

When we employ the Fundamental Theorem of Asset Pricing and the existence of an equivalent probability measure, say $Q$ with respect to the historical probability $P$, we often say the expectation ...
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145 views

Instantaneous Forward LIBOR rate formula under the real-world measure: A fundamental question

We know how the formula of an instantaneous forward LIBOR rate looks like: \begin{eqnarray} L(t, t, T) = \frac{1}{\Delta}\left(\frac{1}{P(t, T)} -1\right) \end{eqnarray} where $P(t, T)$ stands for the ...
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109 views

Calculating futures price

Consider a world as follows: $$\frac{dB}{B} = r_tdt$$ $$\frac{dS}{S} = r_tdt - 0.05dW_1 + 0.5dW_2$$ $$dr_t = 0.2 dW_1$$ where $r_0=0$. The Wiener processes $W_1$ and $W_2$ are independent. The price ...
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If arbitrage can happen exactly at one moment, is it really arbitrage?

There are many "interpretations" of what no-arbitrage means in mathematical finance, the most well known is no free lunch with vanishing risk: If $S=\left(S_{t}\right)_{t=0}^{T}$ is a ...
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Martingale measure and replicating portfolio in Risk Neutral Pricing of Defaultable Zero-Coupon Bonds

When pricing a defaultable zero-coupon bond the risk-neutral price is given as the expected value of the discounted payoff of the bond under a risk-neutral measure. My first question is how do we ...
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62 views

How to prove that the following is still a Brownian motion [closed]

Given a Brownian motion $B_t$ on a filtered probability space, how can I prove that $W_t=B_t+\alpha t$ is still a Brownian motion, with $\alpha \in \mathbb{R}$? Is it always true? Do I need necessarly ...
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57 views

Future price in continous time

I am in the following continuous time market: $S_t^0 = rS_t^0dt$ $S_t^1 = (\mu - \delta) S_t^1dt + \sigma S_t^1 dB_t$ where $r, \mu, \delta$ and $\sigma$ are constant values in $\mathbb{R}$. $\delta$...
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251 views

Trading strategy for a misspecified density

I am trying to implement a strategy that exploits potential misspecifications in density predictions (e.g.: long states with too-low probability; short states with too-high probability). In particular,...
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114 views

Real world probabilities from option implied risk neutral density?

The work of Breeden and Litzenberger-formula (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2642349) gives us a risk neutral probability distribution of a stock price, depending on the option ...
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72 views

What is practical meaning of T-forward Measure vs Risk-neutral Measure?

What I understand is that risk-neutral measure use Risk-free product as numeraire and T-Forward measure use Bond Price as numeraire reading material says, T-forward measure make the pricing behavior ...
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44 views

Use Discrete ARMA(1,q) Process to Model Short Rate for Term Structure Fitting

I'm new to this field but I'm reading related literature lately and quite obsessed with the topic. I come to know that people like to model short rate under risk-neutral measure $Q$, because under $Q$ ...
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67 views

If there is a $T$-forward measure and a risk neutral measure, then markets are not complete?

I am trying to understand the connection between market completeness and risk neutral measures. A market is complete if and only if the equivalent martingale measure is unique. But if I change to the $...
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Risk-Neutral probability deduction [closed]

Could anyone show me how to get the second row equation from the first row equation please? For each letter, $p$ is the risk-neutral probability in the risk-neutral world, $u$ is the up factor for the ...
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101 views

Martingale pricing with time-dependent risk-free rate

I want to find the price of a European call-option under the assumption that the risk-free rate $r$ is time-dependent, i.e. $$ d\beta = r(t)\beta dt \leftrightarrow \beta(T) = e^{\int_0^T r(u)du} $$ I ...
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Show that stochastic integral is $F_W(t)-$measurable

In some notes, my professor writes the following for the price function of an geometric asian option: \begin{align} \text{Price}(t)&=\tilde{\mathbb{E}}\left[\left(S(0)\exp\left(\frac{T}{2}\left(r-\...
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Non attainable claim - Incomplete market

I am wondering whether there is a standard procedure to find a non attainable (i.e. non replicable) asset in an incomplete market. As an example, let us have the following market ($B = (B^1, B^2, B^3)$...
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124 views

What is the interpretation if the real world measure $\mathbb P$ is equal to the martingale measure $\mathbb Q$

Out of interest, is there anything noteworthy about a market when its real world measure $\mathbb P$ is actually also its martingale measure. In other words the real world measure $\mathbb P$ is equal ...
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106 views

Why do stock prices follow a martingale?

I have a quick question: why does the Efficient Market Hypothesis (EMH) assume that stock prices follow a martingale process? I understand that discounted prices under the risk-neutral probability ...
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State Price Deflators For RW to RN Scenario Generation

I have real world stochastic scenarios that model equity returns for "the market". Growth is calculated by modeling the risk free rate, then applying a risk premium on top of that. For the ...
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59 views

Exercise on Delta-Neutal-Hedging

Suppose you have three positions in the following assets in euros: long on 10.000 calls (maturity T = 3 months, strike= 0.55, Delta (1 call) =0.533), short on 210000 calls (maturity T = 3 months, ...
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95 views

What's the price of a lookback call option in the arbitrage-free CRR-model?

If we consider the CRR-model in two periods, i.e. T=2. Let $S^1$ be the risky asset with $S_0^1=100$ and $S^0$ the bond with $S_0^0=1$. Furthermore, we assume the model is arbitrage-free with $y_b=-0....
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52 views

Real Option Valuation using simulation: real world vs risk neutral measure

I am trying to value a real option in the form of a software investment using a simulation. The software investment yields to daily revenues $R_t$ and costs $C_t$. Here are the formulas for these: $$...
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Risk neutral probabilities in binomial option pricing with discrete dividends — whose argument is correct?

In trying to discover more about pricing American options with dividend payouts, I found the the post linked here. I notice two disagreeing answers when it comes to determining the replicating ...
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147 views

Digital call under Ornstein-Uhlenbeck dynamics

I am trying to price a digital option with payoff $\mathbb{I}_{S_T>K}$, where $S_t$ follows the Ornstein-Uhlenbeck dynamics $\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma\mathrm{d}W^{\mathbb{Q}}_t$ in the ...
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254 views

Change of numéraire for two risky assets without bank account (Margrabe’s formula?)

I am considering two risky assets following the usual correlated GBM given by $$\frac{\mathrm{d}S^{(i)}_t}{S^{(i)}_t}=\mu_i\mathrm{d}t+\sigma_i\mathrm{d}W^{(i)}_t,\quad i\in\{1,2\}$$ with $$\mathrm{d}...
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Bond price under the risk-neutral measure

Could you point out where I am making mistake in the process below? It follows from the term structure equation and the Feynman-Kac theorem that the bond price is given by $ p(t,T) = E_t^Q\left[ \exp\...
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143 views

Pricing of Asian-like option

I am considering an option which has payoff function $\max\{S_T-\frac1\tau\int_0^\tau S_t\mathrm{d}t,0\}$ for a fixed $\tau$ in the risk-neutral measure $\mathrm{d}S_t/S_t=r_t\mathrm{d}t+\sigma_t\...
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Example of one-period model that satisfies law of one price but is not free of arbitrage

We know that by the law of one price: in a one-period model $(\overline{\pi},\overline{S})$ for an arbitrage-free market model it follows that for two strategies $\overline{\rho}$ and $\overline{\xi}\...
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108 views

For one-period model, construct a risk-neutral measure $\mathbb P^{*}$ such that the density is constant on $\{S^{1} (<,>,=)c\}$

Consider a one-period arbitrage-free model, it has one risky asset $(\pi^{1},S^{1})$ such that $\pi^{1}>0$, with interest rate on the risk-free asset $(\pi^{0},S^{0})$ at $r > -1$.Furthermore $...
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53 views

Given the density function of $S^{1}$ in one-period model, find the risk-neutral measure

Consider the one period market model $\left(\overline{\pi},\overline{S}\right)$ consisting of a risk-free asset $\left(\pi^{0},S^{0}\right)=(1,1+r)$ and a risky $\left(\pi^{1},S^{1}\right)$ Let $ r &...
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884 views

Would it be possible to combine long butterfly with long straddle, achieving profit no matter the outcome?

This has been bugging me for a while, I feel like I'm missing something. Simply put, a long butterfly will make profit if the price at maturity does not change much, as shown below A long straddle is ...
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120 views

Illustrating the change of measure in Black-Scholes-Merton

Say that we have the following environment: \begin{align} dS_t &= \mu S_t dt + \sigma S_t dZ_t \\ dB_t &= r B_t dt \end{align} where $S_t$ is the price of a stock, $B_t$ is the price of ...
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91 views

Risk neutral probability for stock with continuous dividend

Setting: binomial tree with one step over time $\Delta t$. I'm trying to derive the risk neutral probability for a stock which pays a continuous dividend, say $\delta$. i.e. probability $p$ such that ...
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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. ...
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126 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?

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