Questions tagged [continuous-time]

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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 ...
snth's user avatar
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9 votes
4 answers
780 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 ...
Robert Kubrick's user avatar
8 votes
1 answer
589 views

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 ...
Lisa Ann's user avatar
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7 votes
0 answers
144 views

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 ($B = (B^1, B^2, B^3)$...
notSoSure's user avatar
5 votes
2 answers
2k 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 ...
ZHU's user avatar
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5 votes
1 answer
809 views

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 ...
Deli's user avatar
  • 51
3 votes
1 answer
261 views

Value of contingent claim at a given time

Consider a contingent claim whose value at maturity T is given by $\min(S_{T_0}, S_T)$ where $T_0$ is some intermediate time before maturity, $T_0 < T$, and $S_T$ and $S_{T_0}$ are the asset price ...
aquant6435's user avatar
3 votes
3 answers
2k views

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 ...
user1025852's user avatar
3 votes
1 answer
179 views

Which references would be useful as an introduction to econometrics as it pertains to CONTINUOUS TIME models?

It seems like the problem of trying to estimate model parameters for continuous time models is not commonly covered in standard econometric textbooks, even those focusing on time series. I certainly ...
Stéphane's user avatar
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3 votes
3 answers
790 views

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 ...
Roberto Liebscher's user avatar
3 votes
0 answers
396 views

Black-Cox yield spreads

From Lando (2004)* I am trying to replicate the following figure (Section 2.6 Default Barriers: The Black-Cox Setup): The spreads are computed as follows: $$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$ ...
Sandu Ursu's user avatar
2 votes
1 answer
118 views

Definition of continuously compounded yield for perpetual defaultable coupon bond

In continuous-time asset pricing, the price of a defaultable perpetual coupon bond is given by $$P(V) = \frac{c}{r}\left[ 1- \left(\frac{V}{V_b}\right)^{-\gamma}\right] + (1-\alpha)V_b \left(\frac{V}{...
Luca Gi's user avatar
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2 votes
1 answer
275 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 $x_0=0$. ...
Hans's user avatar
  • 2,806
2 votes
3 answers
860 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(...
Alex Apas's user avatar
2 votes
2 answers
243 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 ...
Abramo's user avatar
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2 votes
0 answers
92 views

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 ...
Maeu's user avatar
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2 votes
0 answers
1k 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,...
LuLuLu's user avatar
  • 21
1 vote
1 answer
780 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 $T$. ...
Monica Sendi Afa's user avatar
1 vote
1 answer
628 views

Is Ornstein–Uhlenbeck process the continuous-time correspondence of AR(1) process?

I see the AR(1) process (with $|\alpha| < 1$) can be written in the following way: $$x_{t+1} = \alpha x_t + \epsilon_t$$ $$\Delta x_t = - (1 - \alpha) x_t + \epsilon_t$$ which looks quite like the ...
DiveIntoML's user avatar
1 vote
1 answer
2k 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^{(...
Kun's user avatar
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1 vote
1 answer
117 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 ...
pdevar's user avatar
  • 111
1 vote
1 answer
44 views

What is the consumption constraint in writing the continuous version of Asset Pricing Model?

In the first chapter of John Cochrane's Asset Pricing textbook, in order to calculate the price in discrete time, we solve the maximization problem of $Max\space E(\Sigma\beta^j U(c_{t+j}))$ when our $...
MarcusAerlius's user avatar
1 vote
1 answer
85 views

Which are the practical implications that the continuously compounded rate of return can be smaller than the expected rate of return?

I'm reading Hull's Options, Futures and other Derivatives and it intrigues me that the distribution of the continuously compounded rate of return x is: $x \sim \phi(\mu - \frac{\sigma^2}{2}, \frac{\...
Lay González's user avatar
1 vote
1 answer
219 views

Can anyone explain to how Hull get from stock returns to continuously compounded stock returns?

I'm reading Chapter 13 of Hull's book and am stuck on how he got from stock returns to continuously compounded stock returns. As a recap, he built the generalized Wiener Process, which describes a ...
confused's user avatar
  • 717
1 vote
1 answer
279 views

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 $Q$ .....
Juliso's user avatar
  • 11
1 vote
1 answer
97 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 ...
steven's user avatar
  • 29
1 vote
1 answer
429 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 ...
BCLC's user avatar
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1 vote
0 answers
54 views

The continuous-time limit of asset price processes where there is more than one asset

I've read Merton's article "On the Mathematics and Economics Assumptions of Continuous-Time Models" (Reprinted in Continuous-time Finance, Chapter 3), where Merton proved that the price of ...
Steve Norkus's user avatar
1 vote
1 answer
84 views

How to construct continuous futures contracts with multiple maturities

I am trying to replicate the Schwartz-Smith (2000) model and having an issue understanding what the data is and how to generate it. Specifically, the authors use a table of continuous futures with ...
user86422's user avatar
1 vote
0 answers
38 views

Comparative statics on $c/r$ using fundamental asset pricing equation

Consider the fundamental asset pricing equation for a perpetual coupon bond: $$rP = c + \mu P' + \sigma^2/2 P''$$ with standard boundary conditions $P(\bar x) = \bar x$ and $\underset{x\rightarrow \...
Luca Gi's user avatar
  • 327
1 vote
0 answers
61 views

Is there a closed form solution to the following system of SDEs?

Suppose we have the system \begin{align} dr_t=\alpha_r(x_t-r_t)dt+\sigma_rdW_t^r\\ dx_t=\alpha_x(\bar{x}-x_t)dt+\sigma_xdW_t^x\\ \end{align} As this system is affine, I believe there should be an easy ...
Carl's user avatar
  • 123
1 vote
0 answers
73 views

Derive Q-dynamics of $\ln S_t$ having multiplicative error structure

From Kwon, T. Y. (2012). Three essays on credit risk models and their bayesian estimation (Doctoral dissertation): Assume the following log equity price model: $$\ln S_t = g_S(V_t,t,\Theta_V) + Z_T$$...
Sandu Ursu's user avatar
1 vote
0 answers
73 views

A hitting time of an open set for a càdlàg process is a stopping time

In Protter Stochastic Integration and Differential Equations, Springer (2003), the following definition is given: Definition. Let $X$ be a stochastic process and let $\Delta$ be a Borel set in $\...
Edouard Berthe's user avatar
1 vote
0 answers
90 views

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 ...
Attila Víg's user avatar
0 votes
2 answers
116 views

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 ...
Marinab's user avatar
  • 11
0 votes
1 answer
136 views

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 ...
Max Wallace's user avatar
0 votes
1 answer
171 views

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 ...
QFi's user avatar
  • 215
0 votes
1 answer
125 views

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 ...
cad's user avatar
  • 1
0 votes
1 answer
181 views

Risk-neutral pricing to determine no-arbitrage price

We are asked to consider a derivative with payoff $C_t = S_{T}^{1/3}$ at maturity $T > 0$ and to use risk neutral pricing to derve the no-arbitrage price process $C_{t}$. Some context: Let $W$ be a ...
George's user avatar
  • 15
0 votes
1 answer
57 views

Compounding Equivalence

I am trying to understand under what circumstances or transformations would $[1+(E_2-E_1)*\frac{d}{360}]$ equal to $(\frac{1+E_2}{1+E_1})^{\frac{d}{360}}$. For context, $E_2, E_1$ are interest rates. ...
vpy's user avatar
  • 187
0 votes
1 answer
264 views

Some aspects of the market price of risk

I am a little confused about the market price of risk. Take the following geometric Brownian motion: $$dS_t = \mu S_t dt+\sigma S_t dW_t$$ The market price of risk is defined as: $$\frac{\mu-r}{\...
Sandu Ursu's user avatar
0 votes
0 answers
32 views

What exactly is the 'continuous asset price model'?

I am reading An Introduction to Financial Option Valuation by Higham. In Chapter 6, the book covers two asset price models, a discrete one and a continuous one. In Section 6.3 (Continuous asset model) ...
herbhofsterd's user avatar
0 votes
1 answer
133 views

Relationship of par-curve and zero-curve/spot-curve

I've been trying to bootstrap the zero-curve from a swap curve composed of ESTR OIS swaps. Theory says when the par-curve is upward sloping, the zero-curve will be above the par-curve and vice-versa. ...
Energy Media's user avatar
0 votes
1 answer
85 views

Stock price modelling under binomial tree model?

In binomial tree model, the stock price is modelled in the form of $S_{k\delta}=S_{(k-1)\delta}\exp(\mu\delta+\sigma\sqrt\delta Z_k)$, where $\delta$ is time invertal between two observations $S_{k\...
Clay ZHAI's user avatar
0 votes
0 answers
98 views

Cost of carry proof

I'm currently review Arbitrage Pricing in Continuous time by Bjork and am stuck on this concept: Honestly I'm not too sure where to start as this chapter makes no mention of the Cost of Carry formula ...
Sushiix's user avatar
  • 17
0 votes
0 answers
101 views

Can Merton's continuous-time portfolio model be reformulated without a utility function?

Under the standard Merton optimization problem the agent maximizes expected utility $$J(\pi,c) =\mathbb{E}\Big[\int_0^TU(c_tX_t) dt + U(X_T)\Big],$$ where the dynamics of wealth of the agent satisfy $...
develarist's user avatar
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