6
votes
Accepted
Figure of Stopping and Continuation Region
The exercise boundary $B_t$ for a finite maturity American put option is not a constant function of time as in your plot. As mentioned in the excerpt, $B_T = K$ at maturity. But for $t < T$, we ...
- 5,959
4
votes
What are the main differences between discrete and continuous time models when modeling asset price dynamics?
Continuous time has a so-called elegance, but it is rarely correct. Most Q-measure people rarely care about correctness anyway, since they usually don't root their models in statistics. With no ...
- 1,068
3
votes
Accepted
Is Ornstein–Uhlenbeck process the continuous-time correspondence of AR(1) process?
This link looks very relevant to your question and probably an answer.
https://math.stackexchange.com/questions/345773/how-the-ornstein-uhlenbeck-process-can-be-considered-as-the-continuous-time-anal
- 1,082
3
votes
Which references would be useful as an introduction to econometrics as it pertains to CONTINUOUS TIME models?
This is seen as a bit of a niche field, which is likely why there are not so many books and these issues are not covered in standard econometrics texts. Options pricing models are usually fitted to ...
- 1,707
3
votes
Black Scholes in Practice: Delta Hedging
1.) in textbooks usually stochastic differential $d$ is used which is rigourous
2.) delta heding in Black Scholes worldis perfect as it's the only way to eliminate risk completely in non friction ...
- 851
2
votes
Accepted
Pricing Secured Barrier Call
The goal of this exercise is to replicate the payoff of the Secured Barrier Call by a linear combination of the known products: European up-out call (cost 12), digital strike 33 (cost 0.73) and ...
- 1,558
2
votes
Accepted
How to derive the relationship between log yield and log price?
This is a misnomer by Cochrane and Piazzesi. It should simply be called the continuously compounded yield.
- 2,137
2
votes
Black Scholes in Practice: Delta Hedging
The $\Delta$ just define the time difference you are considering. In order to formulate the differential equation correctly you need to send $\Delta$ to zero. i.e $lim_\Delta\rightarrow 0 $
In order ...
- 532
2
votes
Bjork exercise 7.6: Claim that depends on $T_1$ and $T_0$
The valuation formula for a contingent claim delivering a payoff at $T$, as seen of today $t$ knowing that the underlying is currently worth $s$ reads
$$ \Pi(t,s) = e^{-r(T-t)} \Bbb{E}^\Bbb{Q} \left[ ...
- 14.5k
2
votes
Accepted
Why can't/doesn't the Fed adjust the federal funds interest rate continuously?
The Fed (under the Yellen regime) has always stated that any adjustments to the Federal Funds rate are "data dependent." These data points (CPI inflation, inflation expectations, non-farm payrolls, ...
- 341
2
votes
What's a good book to learn computational finance topics?
The curriculum for this course (which I teach with David and Rolf) is my Modern Computational Finance book with Wiley: https://medium.com/@antoine_savine/modern-computational-finance-aad-and-parallel-...
- 618
2
votes
Accepted
Definition of continuously compounded yield for perpetual defaultable coupon bond
You could equate the value function with an infinite series of discounted cash flows, discounted at the yield. Assuming a continuous coupon rate and a continuous yield $r^d$:
$$
r^d:P(V) \stackrel{!}{=...
- 6,425
1
vote
Accepted
Risk-neutral pricing to determine no-arbitrage price
To find the $S$-dynamics under $\mathbb{Q}$ we have to use Girsanov's theorem:
$$dW_t^P=\varphi_t dt+dW_t^Q$$
Dynamics under $\mathbb{Q}$ is thus
$$dS_t=a(b-S_t)dt+\sigma S_t(\varphi_t dt+dW_t^Q)=abdt-...
- 835
1
vote
which method is the roubust method to estimate the Hurst parameter?
I would say that - from personal experience, when analyzing the stability of the estimates and their oscillatory behavior - the two most robust techniques are the generalized Hurst exponent, and the ...
- 94
1
vote
Compounding Equivalence
They're not equivalent, but you can use log identities to derive something similar after applying a log.
Eg,
$ln\left(\left(\frac{1+E_2}{1+E_1}\right)^{\frac{d}{360}}\right)$
$\frac{d}{360}*\left(1+...
- 1,643
1
vote
Accepted
What is the consumption constraint in writing the continuous version of Asset Pricing Model?
I guess you meant to write $c_t = e_t - \xi p_t/dt$. Think of $c_t$ and $e_t$ as the 'flow' of consumption and endowment, whereas $p_t$ is the price of good at time $t$. Within the span of time $dt$, ...
- 156
1
vote
Some aspects of the market price of risk
You can transform your process to the following:
$$dX_t= \left[(\mu-\frac{1}{2}\sigma^2) /X_t\right] \times X_t \times dt + (\frac{\sigma}{Xt}) \times X_t \times dW_t$$
So the market price of risk ...
- 852
1
vote
Accepted
Which are the practical implications that the continuously compounded rate of return can be smaller than the expected rate of return?
The key word in your question is compounded. The expected arithmetic return for each $\Delta t$ is $\mu$, but the growth rate is $\mu - \frac{\sigma^2}{2}$. As others mentioned, volatility reduces ...
- 509
1
vote
Can anyone explain to how Hull get from stock returns to continuously compounded stock returns?
The key is on the left hand side. Recall that the differential of log of x is:
$d \ln x =\frac{1}{x}dx$
So you get:
$\ln x_t-\ln x_0=at$
Which you will need to exponentiate to get rid of the log:
...
- 6,204
1
vote
Why Girsanov's theorem used here?
The first result you are alluding to is known as the martingale representation theorem. More specifically, what you say holds for continuous paths processes. For jump processes, there can and will a $...
- 14.5k
1
vote
Bjork exercise 7.6: Claim that depends on $T_1$ and $T_0$
The spot price process is driven by a constant coefficient geometric Brownian motion. Thus, the ratio $S \left( T_1 \right) / S \left( T_0 \right)$ is
independent of $\mathcal{F} \left( T_0 \right)$ ...
- 5,959
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