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Does it matter that Bachelier IV differs from BS IV for a given option price? |
In one sense, it’s just an accounting convention, so it doesn't matter. In another sense, the implied volatility can be interpreted as the minimum realised volatility which implies that your option price was ≤ fair value (realized via dynamic hedging/gamma scalping, see https://quant.stackexchange.com/questions/39619/g... |
While studying Brownian motion applied to mathematical finance, I came across these [lecture notes][1] by prof Steve Lalley. In the prologue, he gives this explanation for the occurrence of Brownian motion in the theory of asset pricing
> In equilibrium, the discounted price process of any tradeable asset,
> observ... |
Does the Lévy characterization imply that the discounted price process of any tradeable asset is a Brownian motion? |
Does the Lévy characterization imply that the price process of any asset is a Brownian motion? |
While studying Brownian motion applied to mathematical finance, I came across these [lecture notes][1] by prof Steve Lalley. In the prologue, he gives this explanation for the occurrence of Brownian motion in the theory of asset pricing
> In equilibrium, the discounted price process of any tradeable asset,
> observ... |
Suppose a (European) call option writer is hedging their risk by taking a long position in stocks (holding $\delta_C$ shares). The value of the portfolio is $V(S)=\delta_CS-C$. Then is the gamma of the portfolio $\Gamma_P=-\Gamma_C<0$? If so, is the plot of the profit of the portfolio against stock price a concave down... |
Negative gamma delta hedging (for a call option writer): how will the stock price affect the portfolio profit? |
Negative-gamma delta hedging (for a call option writer): how will the stock price affect the portfolio profit? |
Am I correct in saying that most stochastic vol models are meant to behave in a way that as atm vol goes up the smile comes down and risk reversals become "less stretched?" - by that i mean say if a 25d RR is trading at 5% for puts when vol is 15%, it would come down from 5% when atm vol goes up to say 20% and go up wh... |
I am currently running fixed effect regressions with multiple dummy variables. These dummy variables are created by a grid of '1' '0':
```
e <- c("1","0")
r <- expand.grid(e, e, e, e, e)
```
By creation, the correlation of each dummy with the other dummies is 0.
I regress (multivariate) a variable on these 5... |
How to calculate the yield of a perpetual bond that pays a floating coupon payment? |
I'm new to finance and crypto and this question is more of a thought experiment so I would like to hear both theoretical as well as practical considerations. Suppose I would like to short a particular cryptocurrency token and assume for this experiment that I know its price will drop in the near term, say 1 week. My go... |
What limits the maximum possible returns when shorting crypto? |
I'm currently trying to calibrate the SABR model. The question I have is that when I consider papers and other websites I only come across cases where the SABR parameters are calibrated to the implied volatility smile, thus for one specific time-to-maturity. However, I'm wondering if it is possible to just calibrate th... |
Under the assumption of the normal distribution, I'm trying to create a single stock mean reversion strategy. I took the log returns because they are stationary, I standardized them and used the zscore as trading signals(buy when zscore is -2, take gains when the zscore is 0, sell when zscore is +2 and take gains when... |
Let's say you have two investments, A and B. Both A and B have the same expected return. However A has low variance and is not correlated with any other investments. Whereas B has high variance and has correlation with other investments.
Which would you pick? How would you make the decision? |
Which investment do you choose? Same expected return but differing variances and correlation |
Suppose I have three assets: the market, factor A and factor B. The market is in excess returns of the risk free rate. The other two factors are long-short portfolios. I have net returns for these factors, since I deduct estimated transaction costs. The expected net returns for these factors are negative
The mean va... |
So I have the following short-rate model
$$dX_t = a_1X_tdt + \sigma_1dW_t$$
$$dY_t = a_2Y_tdt + \sigma_2dB_t$$
$$r_t = X_t + Y_t + f(t)$$
with $X_0 = Y_0 = 0$ where $W$ and $B$ are Brownian motions with correlation $\rho$ and $f(t)$ is some deterministic function chosen to match the current term structure.
I wa... |
Newbie here. I should say upfront that I'm not a quant, just someone trying to broaden his knowledge of fixed income investing. I apologise in advance if I'm mangling some terminology.
Imagine a rational investor who's trying to select from a set of risky bonds, all of which have the same maturity date yet different... |
Simple (?) question about expected bond returns |
Newbie here. I should say upfront that I'm not a quant, just someone trying to broaden his knowledge of fixed income investing. I apologise in advance if I'm mangling some terminology.
Imagine a rational investor who's trying to select from a set of risky bonds, all of which have the same maturity date yet different... |
> Is there a proof that shows that this is always the case when the correlations amongst your independent variables are 0?
This is a known result; see e.g. Kennedy "A Guide to Econometrics" (6th ed., 2008) section 3.1. There must be a proof for it; it is given as exercise 3.15 in Davidson & MacKinnon "Econometric Th... |
In [this][1] note/sketch, I derive among others a closed-form formula for an up and in put (UIP) in stochastic volatility models of the form
$$
dS(t) = \sigma(t) S(t) \left[ \rho dW(t) + \sqrt{1-\rho^2} dZ \right],\\
d\sigma(t) = a(\sigma(t)) dt + b(\sigma(t)) dW(t)
$$
where $dW(t)dZ(t) = 0$.
I argue that the p... |
In [this][1] note/sketch, I derive among others a closed-form formula for an up and in put (UIP) in stochastic volatility models of the form
$$
dS(t) = \sigma(t) S(t) \left[ \rho dW(t) + \sqrt{1-\rho^2} dZ \right],\\
d\sigma(t) = a(\sigma(t)) dt + b(\sigma(t)) dW(t)
$$
where $dW(t)dZ(t) = 0$.
I argue that the p... |
I was studying this question from "A Practical Guide to Quantitative Finance Interviews" and was having some trouble understanding one solution. Please advise if misunderstood anything or if there is any reference paper I could turn to.
Here is the question:
*Suppose that X is a Brownian motion with no drift, i.e... |
I am trying to create an indicator that will find all the divergences between 2 signals.[![enter image description here][1]][1]
The output of the function so far looks like this
[![enter image description here][2]][2]
But the problem is that is painfully slow when I am trying to use it with long signals. Cou... |
I'll mark this closed shortly, the comments from nbbo2 and fesman were both very helpful. If anyone else finds this topic interesting, I subsequently found a paper that attempts to separate out the "actuarial" default premium to arrive at estimates of the CRP. There are similar papers, but I found this one the clearest... |
I am trying to create an indicator that will find all the divergences between 2 signals.[![enter image description here][1]][1]
(A divergence being defined as t1, t2 such that one signal increases between t1 and t2 while the other decreases between t1 and t2).
The output of the function so far looks like this
... |
I am attempting to calculate Model-Free Implied Volatility for several equity indices (S&P500, NASDAQ100, CAC40, FTSE100, DJIA, EUROSTOXX50, NIKKEI225, NIFTY50). I wish to get historical data of expired options since Jan 2000. Can I obtain this data from bloomberg in some way?
Another question I have is that I am su... |
Suppose you have two highly correlated risky assets.
Correlation coefficient: 0.9
Volatility:
Asset 1 price varies 2.5% /day
Asset 2 price varies 5% / day
What can be done to do reduce the risk and increase the return?
Consider how to weigh each asset (ex. 50/50, 60/40, 80/20) and the possibility of leverage or sho... |
How to optimize two highly correlated risky assets? |
I am trying to **simulate correlation** in order to price a correlation swap (via Monte-Carlo). For simplicity, let's assume we have 2 assets, and everything is correlated with $\rho$, and there is no drift or anything. Say, the swap has $M$ observation periods, and we simulate $N$ paths per asset.
So we start by us... |
*> However, I'm wondering if it is possible to just calibrate the SABR parameters to the entire volatility surface*
No, this is not how it supposed to work. SABR model describes dynamics of a single forward $F$, i.e. the parameters of the process can only be calibrated to a single expiry and they are not time-depend... |
A `9 month` American option (underlying) is known to pay dividend of `USD 1 and USD 0.75` at the end of the `3rd and the 07th` month respectively. The `strike price` considered is `USD 45`. The `Risk Free Rate` is continuously compounded and is fixed at `6%` over the tenure. The `stock` is currently trading at `42`. Th... |
What is the optimal time for exercising American call and put option? |
I'm trying to figure out the way to value SONIA swap over the range of days based on the example in the QuantLib Python book (which has bonds in it).
```
import QuantLib as ql
rate = 0.02
calculation_date = ql.Date(10, 12, 2021)
ql.Settings.instance().evaluationDate = calculation_date
day_count = ql.Actual365... |
I am trying to create an indicator that will find all the divergences between 2 signals.[![enter image description here][1]][1]
(A divergence being defined as t1, t2 such that one signal increases between t1 and t2 while the other decreases between t1 and t2).
The output of the function so far looks like this
... |
I am trying to create an indicator that will find all the divergences between 2 signals.[![enter image description here][1]][1]
(A divergence being defined as t1, t2 such that one signal increases between t1 and t2 while the other decreases between t1 and t2).
The output of the function so far looks like this
... |
I would like to get 10Y and 5Y intraday historical government bond prices (best bid and ask) for different countries. I thought that I have found a solution on Bloomberg, but there are some issues.
I am using the following formula (here Sweden, 10Y is used as an example):
bid price
=BDH(“GTSEK10Y Govt”;”Open”;... |
A `9 month` American option (underlying) is known to pay dividend of `USD 1 and USD 0.75` at the end of the `3rd and the 07th` month respectively. The `strike price` considered is `USD 45`. The `Risk Free Rate` is continuously compounded and is fixed at `6%` over the tenure. The `stock` is currently trading at `42`. Th... |
I was going over historical prices in Yahoo Finance, when I noticed that AAPL and IBM have no prices for 1981-04-17. I have a couple of questions regarding this:
1. Why are the prices missing?
2. In the future, how can I answer such historical anomaly questions? Searching the date and "finance" or "market" in Goo... |
Why are there missing prices for 1981-04-17? |
I'll mark this closed shortly, the comments from nbbo2 and fesman were both very helpful (and thanks also to Dimitri V). If anyone else finds this topic interesting, I subsequently found a paper that attempts to separate out the "actuarial" default premium to arrive at estimates of the CRP. There are similar papers, bu... |
If one assumes the underlying return process is I.I.D, is there a solution to the question of the autocorrelation of squared returns?
Thanks. |
I.I.D log returns. What about their square? |
This might be a very trivial question, but I simply can't find a practical answer anywhere.
I am writing my Master's Thesis, and in this regard, i'm investigating the cross-sectional momentum effect in the cryptocurrency market, using the methodologies of Jegadeesh & Titman (1993). I am using past *J* (1, 2, 3 & 4) ... |
I have been keep hearing that gamma is a bet on realized volatility. That is, if we are long gamma then we need higher realized volatility to coming in the future. otherwise what happens ?
from other source :
```
If you are long gamma and delta-neutral, you make money from big moves in the underlying, and lose m... |
What is gamma to do with realized volatility? |
I am trying to long a security that is expected to outperform its peers after certain corporate actions, but want to hedge using the same group of peers (so short ~5 names). So the goal here is to hedge out any risk from the sector and left with the alpha from the event. What's the best way to figure out the hedge rati... |
I like to think about this problem graphically.
The pic below shows a call option value at some point before expiry as a function of the underlying. At the expense of stating an obvious fact, we note that the option value as a function of the underlying is **not** linear.
Imagine the option is on 100 units of th... |
I like to think about this problem graphically.
The pic below shows a call option value at some point before expiry as a function of the underlying. At the expense of stating an obvious fact, we note that the option value as a function of the underlying is **not** linear.
Imagine the option is on 100 units of th... |
I like to think about this problem graphically.
The pic below shows a call option value at some point before expiry as a function of the underlying. At the expense of stating an obvious fact, we note that the option value as a function of the underlying is **not** linear.
Imagine the option is on 100 units of th... |
Without doing portfolio optimization, how do long short portfolio managers size their positions? |
I do not have enough exposure to Markowitz and mean-variance optimization so please just take this more like a comment than a full answer but lacking of the latter i offer you my view: I remember that i saw an article using an numerical integral of the area below the efficient frontier to approximate the set of stochas... |
I have been keep hearing that gamma is a bet on realized volatility. That is, if we are long gamma then we need higher realized volatility to coming in the future. Otherwise what happens ?
from other source :
> If you are long gamma and delta-neutral, you make money from big moves
> in the underlying, and lo... |
I like to think about this problem graphically.
The pic below shows a call option value at some point before expiry as a function of the underlying. At the expense of stating an obvious fact, we note that the option value as a function of the underlying is **not** linear.
Imagine the option is on 100 units of th... |
I like to think about this problem graphically.
The pic below shows a call option value at some point before expiry as a function of the underlying. At the expense of stating an obvious fact, we note that the option value as a function of the underlying is **not** linear.
[![enter image description here][1]][1]
... |
as the question stated. I want to find a way to calculate the delta for a barrier option under the Heston model. Is there any closed-form solution?
All I can find is:
- Delta of a barrier, but under BSM
- Delta under Heston, but vanilla
If there's no closed-form solution, I guess I have to use Monte Carlo to calc... |
I will discuss only the single barrier up-and-in put (UIP) with barrier $B \geq K, S(t)$. Other single barriers treated similarly.
The answer is it depends: If correlation between the instantaneous stochastic volatility and the asset price is zero then the price of an UIP under Heston model (or any other stoch vol m... |
I will discuss only the single barrier up-and-in put (UIP) with barrier $B \geq K, S(t)$. Other single barriers treated similarly.
The answer is it depends: If correlation between the instantaneous stochastic volatility and the asset price is zero then the price of an UIP under Heston model (or any other stoch vol m... |
I will discuss only the single barrier up-and-in put (UIP) with barrier $B \geq K, S(t)$. Other single barriers treated similarly.
The answer is it depends: If correlation between the instantaneous stochastic volatility and the asset price is zero then the price of an UIP under Heston model (or any other stoch vol m... |
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 way to derive the closed-form solutions? Or at least numerically using ODEs. I know there are closed-form soluti... |
Is there a closed form solution to the following system of SDEs? |
Given historical asset prices at consistent time intervals, one can estimate annual volatility as:
SampleStDev(log(S<sub>i</sub>/S<sub>i-1</sub>)) / sqrt(interval)
What's the correct way to do this when the time intervals are inconsistent (e.g. S<sub>2</sub> was observed 1 day after S<sub>1</sub>, but S<sub>3</s... |
I have a question about local volatility models.
In a lot of articles it is stated that the implied spot vol correlation of this model is -1 and we usually compare this with stochastic volatility models where you can control this correlation.
Is there a proof demonstrating how this correlation is -1 ?
I tried ... |
I was creating an yield curve using zeroRate, when I read back the value from the created yield curve - it differs from expectation.
```
tod = ql.Date(5,5,2022)
ardates = [tod, tod+ql.Period(1,ql.Weeks), tod+ql.Period(1,ql.Months), tod+ql.Period(3,ql.Months),
tod+ql.Period(6,ql.Months),tod+ql.Period... |
I'm trying to confirm that I'm understanding this concept correctly: dealer gamma exposure. I can make sense of dealers / gamma in isolation:
- Dealers: make markets for certain securities, notching the bid/ask
spread in profit. Inventories can be positive or negative, possibly
based on assessments of adver... |
What is the consensus interpretation of index future dealer gamma? |
For Libor we have the following Convexity adjustment formula for payment delay (under normal model)
$$CA = P(0,T_e,T_p)\rho\sigma_e^L\sigma_p^L\Delta_e^p(T_s-t_0)$$
where
1. $T_s$ is the period start date
2. $T_e$ is the period end date
3. $T_p$ is the payment date (assuming here $T_p$ > $T_e$)
4. $P(0,T... |
First time posting on this stack exchange, but I was using Polygon to get some Market Data recently, and I was curious about one thing.
Where do they get their data from? Might be a dumb question, but what's stopping "normal" investors from making their own API? Thanks! |
You need a connection to the data providers (e.g. exchanges). You can see details on [Poligon's website](https://polygon.io/stocks?gclid=Cj0KCQjwsdiTBhD5ARIsAIpW8CI1oTHfE4dr_86WKDwpU0kqmT3ePmfAGwDOlY6XGOQi38ImQrryy_gaAm9aEALw_wcB).
[![enter image description here][1]][1]
A counter question may be, what prevents ... |
I would like to calculate returns for a plain-vanilla (fixed-for-floating) interest rate swap. Consider, that I am long in USD 5-year swap rate, i.e. I'm holding a receiver swap for 5-year swap rate with notional of 100\$. At initiation, the swap should be valued at 0\$, as both the fixed leg and the floating leg of th... |
Here are some resources that I found useful when learning about this subject, in which I'm very interested. (Some may be more general ESG than just just climate.)
* [Citigroup. Environmental and Social Policy Framework (March 2021)][1]
* [UBS. Suni Harford. Investing in an ESG world - A practitioner’s guide (202... |
I recently came across the single curve problem that states that we need differenct curves for discounting cashflows and projecting forward floating rates.
On google I am not able to find a proper article about it, so could somebody tell me what exactly the problem with having only one curve is?
What I found out ... |
Here are some resources that I found useful when learning about this subject, in which I'm very interested. (Some may be more general ESG than just just climate.)
* [Citigroup. Environmental and Social Policy Framework (March 2021)][1]
* [UBS. Suni Harford. Investing in an ESG world - A practitioner’s guide (202... |
Can anyone provide us with an empirical example (in Python) of market impact visualization, methods of its estimation (in the framework of a parametric model for impact functions) ? |
A multi-curve means that you observe the discounting instruments (such as fed funds) and projection (libor, swap curve) and solve for all of them simultaneously; as opposed to bootstrapping separately a projection curve and a discounting curve.
A simple paper with examples is [Numerix Model Calibration: The Multiple... |
I need help in understanding Quantlib's interpretation of yield curve and rates. The rate output retrieved from yield curve differs from expectation for non continuous cases.
Illustration:
Let's start by defining the yield curve ..
```
tod = ql.Date(5,5,2022)
ardates = [tod, tod+ql.Period(1,ql.Weeks), tod+ql.... |
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 way to derive the closed-form solutions? Or at least numerically using ODEs. I know there are closed-form soluti... |
This is an embarassingly basic question, but in the course of studying windmill adjustment in binary FX options, I unexpectedly got stuck figuring out the correct units. Say we have a domestic binary call ("cash-or-nothing") and we replicate with a tight call spread around the strike. Theoretically, we have
$$
C_b ... |
This is an embarassingly basic question, but in the course of studying windmill adjustment in binary FX options, I unexpectedly got stuck figuring out the correct units. Say we have a domestic binary call ("cash-or-nothing") and we replicate with a tight call spread around the strike. Theoretically, we have
$$
C_b ... |
Let's have Stock S at `$`100 on January and my hypothesis is S will be trading at $150 in July.
Is there any Python/R package that I can feed with option prices from my broker and it would return the optimal call option for my hypothesis? That is, the call option that would maximise return if my hypothesis becomes t... |
I am trying to use the MV(mean-variance) optimization framework. When I change the mean term using future-ground-truth return (I am not supposed to do it), it has a higher effect on the MV optimization than using the future-ground-truth covariance matrix.
Why the mean term has a higher effect than the covariance te... |
MV Optimization, why does the mean term has a higher effect than the covariance term? |
I am trying to use the MV(mean-variance) optimization framework. When I change the mean term using future-ground-truth return (I am not supposed to do so), it has a higher effect on the MV optimization than using the future-ground-truth covariance matrix.
Why the mean term has a higher effect than the covariance te... |
I am trying to calculate the change of value in a universe of bonds given a series of shocks to the credit spread of each bond. As a constraint, the initial dataset only contains the spread change for each individual bond and its duration/maturity.
Is there any simple way to approximate the change in value for each... |
Let's have Stock S at \$100 on January and my hypothesis is S will be trading at \$150 in July.
Is there any Python/R package that I can feed with option prices from my broker and it would return the optimal call option for my hypothesis? That is, the call option that would maximise return if my hypothesis becomes t... |
I have a mental math test coming up for a quantitative trading firm. Anyone knows good websites to practice? |
How to train mental math for trading? |
You should be able to re-use the rate duration as a measure of spread sensitivity as well:
Let's assume a simple vanilla fixed-coupon-bearing bond w/o embedded optionalities that pays at coupon rate $c$ annually. Let's further assume a continuously compounded flat risk-free zero rate curve $r$ and a continuously com... |
I have been implementing an, in my opinion, interesting finite difference method (Runge-Kutta-Legendre of second order) to price American options in the standard Black-Scholes model (see "Pricing American Options with the Runge-Kutta-Legendre Finite Difference Scheme, F. Le Floc'h, International Journal of Theoretical... |
I have a question regarding the use of the FF5 Factors in an industry-fixed effects model. In order to clearify my question I'll post an example of my dataset
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/s4x5C.png
Note that this is just an example, the values for the FF5 factors... |
I have 2 dates, let's say 2010-01-01 and 2020-01-02, and I am interested in calculating the year fraction between them according to the Act/365 time convention.
Would this be just
$$
\frac{\text{raw # of days between the dates}}{365}
= \frac{3653}{365}
= 10 \tfrac{3}{365}
$$
or will I have to convert the yea... |
The market impact is the influence of the pressure exerted by the flow of **metaorders** on price dynamics.
Metaorders are large orders issued in general by asset managers or investment banks; they are then split in small child orders according to an optimal trading strategy.
Nowadays, if you look at orderflows o... |
I have some problems with understanding delta hedging with the Heston model.
So far I've understood that you need both Vega and Delta to hedge with the Heston model. However, I don't know how to practically do it. I have used the Heston model to obtain both the delta and vega.
How do I practically use them to de... |
I am currently using the [FinRL](http://finrl.org) library to try to automate Trading using Reinforcement Learning. However, I wanted to understand how FinRL simulates the stock market using historical data. I read [here](https://arxiv.org/abs/2011.09607) that they "simulate live stock markets with real market data acc... |
What are common ways to realistically simulate the stock market using historical market data? |
I want to backtest VaR models which are applied to portfolios of products which have maturities (options and futures) and even schedules (bonds). I have a question which never came up when backtesting VaR models for equity portfolios: is it adequate to roll the contract schedules/maturities backwards as I move backward... |
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