11
votes
Self study references for a Mathematician
In general, quantitative finance requires mathematics, finance, and numerical programming. The mix of the three and the areas of focus within the three will depend on the particular area you intend ...
- 191
11
votes
Tick Imbalance Bars - Advances in Financial Machine Learning
Question 1. Actually, the assumption of trade data format is that you have timestamp, size and price (not bid/ask) of trade. Sometimes, trades(ticks) are included to Level 1 data (also called BBO) ...
6
votes
Measure of a Brownian motion = normal distribution?
It is correct that
$$
\mathbf{P}(t^{-1/2}W(t) \in[a,b])=Φ(b)−Φ(a), \forall t\in(0,\infty)
$$
due to the stationary increments property of the Wiener process and the fact that you normalized the ...
- 71
5
votes
Accepted
Is this a poorly written example, or could volatility in fact be negative?
You seem to use the term "volatility" to describe two very different quantities: (1) the diffusion coefficient of your SDE and (2) the standard deviation of the log-returns under your modelling ...
- 14.5k
4
votes
Accepted
Why are the greeks for the underlying stock 0 with the exception of delta?
In the black scholes model, today's stock price, risk free rate and stock volatility are considered independent variables. They are inputs to the model. Hence the cross partial derivatives are zero. ...
- 16.2k
4
votes
Show a process is Martingale
Assuming standard BS dynamics for $S_t$, you have
$\frac{Z_t}{Z_0}\equiv(\frac{S_t}{S_0})^p = \exp{((rt-\frac{\sigma^2}{2}t+\sigma W_t)(1-\frac{2r}{\sigma^2}))}$
Now, this is a lognormally ...
- 1,356
4
votes
Accepted
Is the Non-discounted Bachelier call option price a Martingale?
Let $P(t,T)$ denote the time $t$ price of a zero-coupon bond maturing at time $T$ and $\mathbb{Q}_T$ be the associated equivalent martingale measure which uses $P(t,T)$ as numeraire. Then, for any $\...
- 15.1k
4
votes
issue with benchmarks in "standard securities calculation methods"
90.422798 is not the correct value for price for Benchmark 18A. If you were the original purchaser of the book in 1993 you would have received an errata correcting that benchmark. If you have a ...
- 41
3
votes
Accepted
How does this statement about the price of a prepaid forward on a stock follow?
It is a very badly worded question in my humble opinion.
There are three "prices" to contend with.
(1) If you want to buy a stock and pay for it now, you pay the current stock price S.
(2) If you ...
- 10.6k
3
votes
Accepted
What's the explanation for the formula for the volatility of a stock / volatility of the continuously compounded return of a stock?
The standard starting point with modelling a stock price process is to use the Black-Scholes model for the stock price. This simply asserts that the changes in the stock price are described by the ...
- 1,389
3
votes
Is there an error in this problem on pricing an asset using the true probability of an up move?
Your formula for $p$ is $$p = \frac{e^{{(\alpha - \delta})h} - d}{u - d},$$
where $\alpha$ is not expected return on stock but continuous risk free rate, i.e. 1%.
If you use $\alpha$ as 1%, you will ...
- 2,218
3
votes
Accepted
Understanding the relationship between the Black-Scholes formula and a replicating portfolio
It is my understanding that a replicating portfolio for a put involves short selling stock and lending money.
You cannot statically replicate an option. So this is not true in general, you'll need to ...
- 14.5k
3
votes
Self study references for a Mathematician
For general mathematical finance, you may start with the book Stochastic Calculus for Finance, and then the books Martingale Methods in Financial Modelling and Mathematical Methods for Financial ...
- 21k
3
votes
Difficulty understanding put-call parity for currency options
It costs 0.03 dollars for the option to (sell 1 pound/buy 1.5 dollars. Now divide everything by 1.5:
It costs 0.02 dollars for the option to (sell 2/3 pound / buy 1 dollar). Now convert to pounds ...
- 16.2k
3
votes
Accepted
Why is the statement "the volatility of a $T - t$-month prepaid forward on asset X is $\sigma$" the same as "the volatility of asset X is $\sigma$"?
Almost, indeed
The volatility of an asset over a horizon $[t,T]$ indeed refers to the standard deviation of the log-return observed over that period:
$$ \sqrt{\text{Var}\left(\ln\left(\frac{X_T}{X_t}\...
- 14.5k
3
votes
Accepted
What is a notation '1' in risk neutral probabilities paper?
The 1 you are referring to is a vector of ones
The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:
...
- 6,864
3
votes
Accepted
Is a wiener proces measurable? (exercise from Bjork)
I think you mixed several things up. I will try to help you out.
Everything started with your claim that $\Bbb E \bigl[W(T) \mid \mathcal F_t \bigr] = 0$ which is wrong!
if $W$ is a Brownian notion,...
- 1,436
3
votes
Show a process is Martingale
As a starting point: for price dynamics $dS(t) = rS(t)dt + \sigma S(t) dW^\mathbb{Q}(t)$, to show that $Z(t)/Z(0)$ is a positive mean 1 Q-martingale, use Itô's formula to get:
$dZ(t)=p \sigma Z(t)dW^\...
- 187
3
votes
Accepted
Risk neutral modelling of a stock
The risk neutral measure is used to price assets (e.g. derivatives) and not to base your investment decisions on.
In the first part of you question your simulation gives you the Risk-Neutral ...
- 1,617
3
votes
Accepted
Probability and statistics in Quantitative Finance
Imho that's more of a probability question than finance really.
If you take $N$ attempts, then the probability of at least one (or more) failures is the complementary probability of never failing on ...
- 1,651
3
votes
Tick Imbalance Bars - clarification on T index
There is an open source hedge fund project which is implementing the ideas contained in the book and which has a github where you can see their code implementation of tick bars. Personally I always ...
- 2,149
3
votes
How does buying a CDX and then taking a short CDS position generates alpha?
One can argue that the theoretical fair value (intrinsic value) of a credit index is just the sum of values of its component CDSs on the same names and with the same terms and conditions (maturity, ...
- 11.7k
2
votes
Accepted
Derive OIS rate from IRS rate and Fed Funds/Libor basis spread
If you're lucky enough that the payment schedules (start/end dates, frequency, day count, business day adjustment etc.) are the same between the fixed leg of the interest rate swap and the "spread" ...
- 46
2
votes
Accepted
Problem solving using the put-call parity
What a difficult problem. The first line gave $165 e^{-rt} -3 S e^{-dt} = 15$ [since 50+55+60 = 165]. In the second line we want to evaluate $110 e^{-rt} -2 S e^{-dt} $. We notice that this is exactly ...
- 9,272
2
votes
Computing $\gamma$ and $\mu$ at the efficient frontier
First of all I’ll work with column vectors because I find it easier than with row vectors as you did. I guess it’s a little bit easier if we modify your first equation a little bit. Notice that is ...
- 1,886
2
votes
Accepted
Difficulty understanding put-call parity for currency options
Let $\{X_t \mid t \ge 0\}$ be the foreign exchange rate rate from $£$ to $\$$. Moreover, let $C(X_0, K, T)$ and $P(X_0, K, T)$ be the prices of the respective call and put options with strike $K$ and ...
- 21k
2
votes
Self study references for a Mathematician
As you don't have any background in finance. I would recommend you following book :
Investment by Bodie
Options, Futures, and Other Derivatives by Hull
An Introduction to the Mathematics of ...
2
votes
Accepted
Understanding the necessary and sufficient conditions for rational early exercise of a call option
If $PV_{t, T}(\text{Divs}) \ge K\big(1-e^{-r(T-t)}\big)$, since $P_{Eur}(S_t, K, T-t) >0$, the identity
\begin{align*}
C_{Eur}(S_t, K, T-t) = P_{Eur}(S_t, K, T-t) + (S_t-K) -PV_{t, T}(\text{Divs}) +...
- 21k
2
votes
Accepted
Expected Utility
The best explanation I came across so far is the one in Gravelle and Rees (2003) chapter 17. I could exactly write here what they state, but that would be copying.
- 7,827
2
votes
Accepted
Clarification on this author's solution for this problem on lognormal stock distribution
Yes, your steps are valid
This is a wrong use of the term "quantile". Here you need to compute a probability (through the normal cdf) and not a quantile (i.e. the value of a random variable ...
- 14.5k
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