The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
1answer
13 views

Is probability implied by binary FX options risk neutral or real world?

If we consider binary FX options in the market and estimate the market implied probabilities of certain FX rates occurring, would these resulting probabilities be risk neutral or real world? I hear ...
6
votes
1answer
114 views

What's Risk-Neutral in an Interest Rate Model?

In Shreve II, on p. 265 he states the Hull-White interest rate model as $$ dR(u) = \left( a(u) - b(u)R(u)\right) dt + \sigma(u)d\tilde{W}(u), $$ and then mentions "...$\tilde{W}(u)$ is a Brownian ...
2
votes
3answers
89 views

Unique risk neutral measure for Brownian Motion

For a standard geometric Brownian motion model of stock prices: $$ dS = a S dt + \sigma S dZ$$ we can transform the process to be under risk neutral measure: $$ dS = r S dt + \sigma S d \tilde{Z}$$ ...
2
votes
3answers
137 views

Physical or Real-world Probability Measure

For counterparty credit risk, in particular, for potential future exposure computation, people use the real-world probability measure to evolve the underlying risk factors. My question is that whether ...
8
votes
1answer
336 views

Consistency of economic scenarios in nested stochastics simulation

I am interested in references on research regarding the consistency of economic scenarios in nested stochastics for risk measurement. Background: Pricing by Monte-Carlo: For pricing complex ...
0
votes
1answer
38 views

The State-Price Deflator in a Binomial pricing model

This question comes from a Financial Economics exam and I'm very confused about a state-price deflator which doesn't seem to exist. I've included the whole question for completeness, but my actual ...
1
vote
2answers
136 views

Intuitive Reasoning for Using Risk-Neutral Measure

Although we thoroughly covered risk-neutral pricing in university I never fully understood it in the context of continuous-time processes. But first of all, lets consider a discrete time example: ...
0
votes
1answer
42 views

probability that the stock price is below the strike price

How can I prove that under the risk-neutral probability: $\mathbb{P}[S_{t}<K]=-\frac{\partial{C}}{\partial{K}}(K,T)$ where $S_{t}$ is the stock price, K is the strike price, C is the call ...
0
votes
2answers
118 views

Does a forward price have a drift component in any measure?

Going by intuition, a forward price should already take into account the drift in the underlying price process. Further, assuming interest rates are deterministic, the stochasticity in the forward ...
0
votes
1answer
47 views

Good book about replicating portfolios

I want to know if anybody can suggest me a good textbook which explains in detail and in an understandable way how to create replicating portfolios of financial instruments like options "cash or ...
0
votes
0answers
29 views

Risk Neutral Measure [duplicate]

I have been looking for a good intuitive reasoning for introduction of the risk neutral measure and its uses in quantitative finance, but I have yet to find one. I was wondering if any one could ...
3
votes
5answers
375 views

Why the Black-Scholes formula can be used in the real world?

The BS formula is deduced using the risk neutral measure. Why can it be used in the real world?
0
votes
1answer
67 views

“The drift of stock price becomes the risk-free interest rate” under RNP

Assume that the evolution of a stock price is geometric Brownian Motion $$ dS=\mu Sdt+\sigma SdW(t) $$ where $S$ is the stock price at time $t$ (current time). It says in my book that "under the ...
0
votes
0answers
11 views

Consumer (Borrowers and Lenders) risk free curve

I was thinking about this topic: how to construct a "risk free" curve for the generic consumer. Imagine we want to price a debt security done by a private, lended by another private (say, normal ...
0
votes
1answer
89 views

Bivariate Black-Sholes Model

Let us propose bivariate Black-Sholes Model. Assume, we have an arbitrage-free complete market. $r_{f}$ is risk-free rate. Under real-world measure $P$: $dS_{1} (t)=S_{1} (t) ...
1
vote
1answer
50 views

What is the filtration described?

What is the filtration $(\mathfrak{F}_t)$ encircled below? Is it $(\mathfrak{F}_t) = (\sigma(W_t)) = (\sigma(\tilde{W_t})), t \in [0,T]$? Or is it $(\mathfrak{F}_t) = (\sigma(\hat{W_t})), t \in ...
0
votes
1answer
103 views

Why does expected price of OTM option not equal to BS price?

If I assume that stock returns follow normal distribution with drift = 0% and S.D. = 10%. In the long, if I keep investing in this stock for a year with the same capital every year for a consecutive ...
8
votes
6answers
4k views

Why Drifts are not in the Black Scholes Formula

This question has puzzled me for a while. We all know geometric brownian motions have drifts $\mu$: $dS / S = \mu dt + \sigma dW$ and different stocks have different drifts of $\mu$. Why would ...
-2
votes
1answer
49 views

Magrabe Exchange Option: not equal drifts

I need to calculate the price of exchange option between 2 assets $S_1$ and $S_2$ The formula is given here Wiki: Magrabe formula or here Quant Stack Exchange. In the derivation of the formula it is ...
3
votes
1answer
72 views

When are implied and real world parameters the same?

Suppose $T$ the maturity of a risky bond which defaults with probability $p$ over its lifetime. If it defaults it pays zero. Thus to price this bond in risk neutral terms would give ...
3
votes
1answer
147 views

Data Selection for Empirical Pricing Kernel Estimation (Stochastic Discount Factor)

I want to estimate an empirical pricing kernel for an index. Hence, I need to estimate a physical and risk neutral density. For estimating the physical density, only the index data in an observed time ...
1
vote
1answer
86 views

If the risk neutral probability measure and the real probability measure should coincide

Sorry if this may be a stupid question. I have not had that much mathematical finance, I've only learned about discrete time models. But lets for the argument say that you have a stochastic process ...
5
votes
1answer
496 views

What are the main flaws behind Ross Recovery Theorem?

Stephen Ross’ new paper claims that it is possible to separate risk aversions and historical probabilities if the Stochastic Discount Factor is transition independent using Perron-Frobenius Theorem. ...
1
vote
0answers
164 views

Provide a bond pricing differential equation and invoke Feynman-Kac

Grateful for any assistance. Consider the process: $dZ=r(t)Z\,dt$ , where $r(t)$ is stochastic and $Z=Z(r,t;T)$ is a zero coupon bond. Provide a bond pricing differential equation and invoke ...
8
votes
3answers
1k views

How to estimate real-world probabilities

In the world of finance, Risk-neutral pricing allow us to estimate the fair value of derivatives using the risk free rate as the expected return of the underlyings. However, the behavior of ...
3
votes
2answers
74 views

Risk Neutral Pricing Necessary Condition

Suppose that I have an option on a single stock expiring at time $T$ and I replicate the payoff of this derivative by investing in the stock market and the money market. So this condition reads $$X(T) ...
4
votes
2answers
97 views

How does this follow from the separating hyperplane theorem?

This is from Pliskas book in mathematical finance. I do not know what was best to write the question so I included the pages from the book. He has not written what form of the separating hyperplane ...
13
votes
6answers
2k views

Formal proof for risk-neutral pricing formula

As you know, the key equation of risk neutral pricing is the following: $\exp^{-rt} S_t = E_Q[\exp^{-rT} S_T | \mathcal{F}_t]$ That is, discounted prices are Q-martingales. It makes real-sense for ...
2
votes
2answers
235 views

Exchange rate model and Martingales

In exchange rate model explanation, "...If under the domestic risk neutral measure $Q_d$, the process $X(t)$ satisfies $\displaystyle \frac{dX(t)}{X(t)}=\sigma dZ_d(t)$ Since $Z_d(t)$ is ...
1
vote
1answer
114 views

Prove that the binomial algorithm implies the arbitrage free price at t=0 of a T-claim

In Tomas Bjork's Arbitrage Theory in Continuous Time (or here), $\exists$ these propositions How does the first formula follow from from the algorithm? I get that $\Pi(0;X) = V_0(0)$, but I don't ...
1
vote
0answers
85 views

How to convert HJM model risk-neutral measure $\mathbb{Q}$ to real measure $\mathbb{P}$?

HJM model, $df(t,T) = \mu(t,T) dt + \xi (t, T)dW(t)$, is defined in risk-neutral measure $\mathbb{Q}$, according to Brigo's "Interest Rate Models" book. I wonder, how could I transform it to real ...
8
votes
2answers
1k views

How to transform process to risk-neutral measure for Monte Carlo option pricing?

I am trying to price an option using the Monte Carlo method, and I have the price process simulations as an inputs. The underlying is a forward contract, so at all times the mean of the simulations is ...
3
votes
1answer
236 views

Arbitragefree Pricing: Q vs. P

I read that the Fundamental Theorem of Asset Pricing states, that a market is arbitrage-free if and only if there exists an equivalent martingale measure Q, under which the discounted asset price ...
5
votes
1answer
404 views

Risk-neutral pricing in incomplete markets

I know that in order to use the risk-neutral valuation principle, that is, pricing options as their payoff function under a risk neutral measure, one has to have a complete market. But in the ...
0
votes
3answers
83 views

Why is it enough to know the expected present value of cash flow in risk-neutral framework to price derivatives?

Wilmott book states that its enough to know the expected present value of all cash flow in risk-neutral framework to price derivatives. As I know, to obtain arbitrage-free market we need our ...
3
votes
2answers
451 views

When to use the real world drift and when the risk neutral one for a Monte-Carlo simulation?

Under what conditions should the drift be real world and when risk neutral when simulating Delta Hedging option pricing trading strategy any other? For 2. it should be risk neutral. For 1., it ...
2
votes
3answers
170 views

Difference betweem martingale property and adapted filteration

What is the difference between a random process that is adapted to a filteration and one that had the martingale property. It seems the two notions are quite similar and would be helpful to construct ...
1
vote
1answer
93 views

Simple pricing example confusion

This it taken from "Heard on the Street", Section B. Consider a market with $0$ risk-free rate, no transactions costs etc. The IBM stock costs \$75 and does not pay dividends. Design a security ...
2
votes
2answers
175 views

Pricing forward contract on a stock

Please tell me where I've gone wrong (if I did in fact make a mistake). I'm pricing a long forward on a stock. The usual setup applies: This has payoff $S(T) - K$ at time $T$. We are at $t$ now. ...
3
votes
1answer
165 views

Numéraire — couldn't understand the wiki explanation

I'm trying to understand Numéraire concept so am reading the wiki page: I couldn't understand the last formula's 2nd equation: $$ ...
6
votes
3answers
330 views

How to choose a risk-neutral measure when the market is incomplete?

I am more of a probabilist than a financial mathematician. I am currently working on the features of American put options under a particular stochastic volatility model. Like most stochastic ...
4
votes
1answer
173 views

Does risk-neutral measure have anything to deal with risk-neutrality in utility theory?

Or simply: why do we call equivalent martingale measures as risk-neutral measures? In the utility or game theory, when we consider a person's preferences to certain outcomes, we often deal with the ...
5
votes
2answers
231 views

Risk neutral Esscher transform of exponential Levy processes

Let $X_t$ be a Levy Process and $e^{X_t}$ the corresponding exponential Levy process. Using the Esscher transform for a change of measure for which the Radon-Nykodym derivative is ...
3
votes
2answers
221 views

How can I show that $u=e^{\sigma\sqrt{\Delta t}}$ in the binomial option pricing model

Given that $e^{r\Delta t}(u+d)-ud-e^{2r\Delta t} = \sigma^2\Delta t$ I would like to show that $u=e^{\sigma\sqrt{\Delta t}}$ I know I must somehow use Taylor's approximation $e^x = 1 + x + ...
0
votes
1answer
58 views

Under an EMM, does there necessarily exist a replicating portfolio?

In general, under an EMM, does there necessarily exist a replicating portfolio for every derivative? I believe the answer to this is false. A simple example is a discrete time, trinomial model. ...
1
vote
3answers
291 views

Risk Neutral Evaluation - Exchange/Spread Options

I have two assets, $S_1$ and $S_2$, which follow geometric Brownian motion processes. This implies that both $S_1$ and $S_2$ have a lognormal distribution. I'm trying to get the exchange option price ...
4
votes
2answers
1k views

Is Vasicek risk neutral?

I am a bit new to this, and am trying to understand the concepts of the risk neutrality in interest-rate models. What I can't seem to understand is why the Vasicek model is risk-neutral? Following ...
2
votes
1answer
188 views

SDE simulation: P or Q?

Let's take a GBM under $P$: $dS=\mu dt+\sigma dW_{t}^{P}$ and then under $Q$ $dS=r dt+\sigma dW_{t}^{Q}$, where $dW_{t}^{Q} = dW_{t}^{P} + (\mu - r)/\sigma dt $ Now, let's say that I have ...
1
vote
0answers
95 views

What are the industry standard models for monte carlo simulation of basket options?

I would like to simulate an equity index, a risk free cash account and the yield curve for the purposes of valuing a guarantee on an insurance product that is being backed by both equities and cash. ...
4
votes
0answers
89 views

Risk neutral measure in exponential levy model

Is there a method of finding a risk-neutral measure for assets driven by the levy process? I understand there is the esscher transform but I think it tends to transform the processes into ...