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Questions tagged [continuous-time]

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The fundamental topics of continuous time finance?

I just finished Bjork's Continuous Time finance, but it doesn't cover certain things like risk management and stochastic volatility models, and it's treatment of interest rate models is minor. I ...
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Simulation algorithm for discretized continuous-time markov chain?

I need to simulate a discrete time markov chain with a given probability transition matrix P that has 0s on the diagonal (self-transitions are not possible). There is another parameter s, which is the ...
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Why Girsanov's theorem used here?

It is written in Bjork's ArbitrageTheoryInContinuousTime that ... Assume a martingale measure Q exists. This implies (see the Girsanov theorem) that the price processes have zero drift under $Q$ .....
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What topics come after continuous finance a la Bjork?

Ok so I've understood stochastic calculus and continuous finance. Basically, all of Bjork's "Arbitrage Theory in Continuous Time". What books/topics come next? I was thinking of taking a more ...
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What's a good book to learn computational finance topics?

I know continuous finance theory roughly equivalent to what's in Bjork's Arbitrage Theory In Continuous Time (most chapters). I'd like to supplement that knowledge with a more hands-on practical ...
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1answer
54 views

which method is the roubust method to estimate the Hurst parameter?

I know there exist lots of method to estimate the Hurst parameter, such as R/S, V/S, GHE, DFA, DMA, Wavelet Spectral Density, Whittle and so on. Can you tell me which one is the best one. Is anyone ...
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Is the 'constant weight in the risky asset' portfolio-strategy self-financing?

My question concerns a topic in quantitative finance that I feel is often brushed under the table: is a given strategy self-financing. We have two assets, one risky and one riskless, defined by the ...
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110 views

Stochastic Continuously Dividend paying stock

I am a beginner and I got confused on the concept of continuously paying dividend. Let say the process of dividend payment evolve as $$ dD_t = \mu D_tdt + \sigma D_tdZ_t$$ where $Z_t$ is a Wiener ...
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Why can't/doesn't the Fed adjust the federal funds interest rate continuously?

Maybe the question I'm asking doesn't make sense-- but this is something I've wondered about since I learned about the Fed in high school. The media typically talks about Fed interest rate changes as ...
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2answers
713 views

Black Scholes in Practice: Delta Hedging

From the Wikipedia page, we know call option as an example is price through delta hedging. $$\Pi=-V+V_SS$$ and over $[t,t+\triangle t]$ $$\triangle\Pi=-\triangle V+V_S\triangle S$$ My questions ...
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Bjork exercise 7.6: Claim that depends on $T_1$ and $T_0$

See the solution to Exercise 7.6 here. The solution calculates $E^Q (S(T_1)/S(T_0))$ and then just plugs that into the risk neutral valuation formula. But why? The risk neutral valuation formula ...
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Pricing Secured Barrier Call

A European barrier call with barrier $B = 50$, expiration $T = 31$, and strike $K = 33$ costs $12$. The investor is interested in a product that, unlike this barrier call, offers some protection for ...
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487 views

How to construct a continuous price time series out of futures raw data in Excel?

My object of research is corn futures: It is well known that corn futures expire 5 times per year: March, May, July, September and December. Due to their finite life that is limited by their maturity,...
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1answer
247 views

Figure of Stopping and Continuation Region

I am reading Alternative Characterizations of American Put Options by Carr et al. It is stated there that: Consider an American put option on the stock with strike price $K$ and maturity date $T$. ...
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1answer
354 views

How to derive the relationship between log yield and log price?

Usually, people write $y_t^{(n)}=-\frac{p_t^{(n)}}{n}$ where $y, p$ and log yield and log price respectively. My question is how do one derive this expression? Note that $e^{-Y_t^{(n)}\cdot n}=P_t^{(...
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1answer
103 views

What is this ratio: expected returns on stock divided by risk free rate?

So this ratio has come up in some work I'm doing and I can't seem to figure out if it is attested in the literature. Here's the setting: Given a risk free rate $r(t)$ and a stock price which follows ...
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3answers
456 views

Numerical Solution to BS PDE - Digital Option

Here is a relatively simple question about PDE's pricing. Assume that we are within the BS framework and moreover that interest rate is zero. The price $V(t,S_t)$ of the digital is known to be $\Phi(...
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2answers
172 views

Markov Pricing kernel

I'm reading about Markov pricing kernels in the lecture notes of a course I'm following, but I have a big doubt on an application of Ito's lemma. The setting is the following: We define the pricing ...
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1answer
275 views

Arbitrage Strategy Proof in Bjork

In Tomas Bjork's Arbitrage Theory in Continuous Time (or here), $\exists$ this proposition Proposition 2.9 Suppose that a claim X is reachable with replicating portfolio h. Then any price at t=0 of ...
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Why does [dz(t)]^2 converge to dt over infinitesimally short time periods?

I have some trouble understanding a chapter in George Pennacchi textbook "Asset Pricing". Here the author shows that the square of a Wiener Process $[dz(t)]^2$ converges to $dt$ for infinitesimally ...
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1answer
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GNP/GDP and modelling [closed]

Is GNP a continuous, static or a dynamic model ? What about GDP ? What I do know is that it has yearly discrete values. However, when it is modeled, it becomes a continuous graph. So what exactly is ...
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1answer
244 views

Examples of non-increasing variance of a time homogeneous Markovian process

This is an edit to the previous question, on stationary process, which was answered by Richard below. Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from $x_0=0$. ...
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1answer
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From $AR(p)$ to SDE

Let the Vasicek model to be $$\Delta r_{t}=k(\theta - r_{t-1})\Delta t+\sigma\Delta z_{t}$$ Due to the fact that $$\Delta r_{t}=r_{t}-r_{t-1}$$ if you let $\Delta t=1$, it is easy to see by ...
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3answers
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Analyze raw tick data

I'd like to work with raw tick data and naturally this data is unevenly spaced (for example, a couple of quotes are at the same second etc.) For example ...
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4answers
571 views

Position management in presence of continuous forecast

Let's say we have an equity liquidity-providing model that was fitted on 1 minute bar periods. The model forecasts the 1-min next period return given the activity of the previous bars. Now, when we ...
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What are the main differences between discrete and continuous time models when modeling asset price dynamics?

My intuition says that both approaches, discrete time models and continuous time models will be models (i.e. approximations) of reality. Therefore it should be possible to develop useful models in ...