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Been trying to understand the yield curve for a while now. This is what I collected so far,
There is a relation between short rates and long rates that goes via the forward rate, and so by the expectation hypotheses one could infer that the long rate is given via the forward rates+term premium. Is this correct?
G... |
Been trying to understand the yield curve for a while now. This is what I collected so far,
There is a relation between short rates and long rates that goes via the forward rate, and so by the expectation hypotheses one could infer that the long rate is given via the forward rates+term premium. Is this correct?
G... |
Been trying to understand the yield curve for a while now. This is what I collected so far,
There is a relation between short rates and long rates that goes via the forward rate, and so by the expectation hypotheses one could infer that the long rate is given via the forward rates+term premium. Is this correct?
G... |
I've got a question about theory which is probably a one line answer. I use to understand it but I'm stuck right now.
In the Binomial model, we define the progression of the price as:
$$
S_k = S_{k-1} e^{\alpha X_k}
$$
where $P(X_k = 1) = p$ and $P(X_k = -1) = 1-p = q$
Now, to determine $\alpha$ and $p$ u... |
This is usually called the "Linking problem" or multi-period linking in Performance Attribution.
Several methods have been proposed and there is some controversy as to which is best (in fact a great deal of discussion goes on in performance journals seemingly without coming to a resolution).
For what it is worth,... |
This is usually called the "Linking problem" or multi-period linking in Performance Attribution.
Several methods have been proposed and there is some controversy as to which is best (in fact a great deal of discussion goes on in performance journals seemingly without coming to a resolution).
For what it is worth,... |
I've been doing that manually for 2 months successfully (40% ROI) with SPX 0-1 DTE (Days To Expiration) options, both puts and calls. I might be just lucky so I purchased some data to do backtesting with 15m timeframe. I didn't complete the test yet but I can calculate probabilities of certain events happening that con... |
Can one successfully daytrade 0dte options based on RSI? |
For whoever stumbles upon this, as I did.
The dimensionality of the problem is so much larger. If someone tells you backtesting options is the same as backtesting stocks or any delta-1 underlying, they are entirely missing the point.
1. Data - Option data is expensive. The biggest favor you can do yourself is to... |
Suppose we have an interest rate model $R(t)=\alpha(t)d(t)+\sigma d\tilde{W}(t)$, where the brownian motion is under the risk neutral measure. Suppose $S(t)$ is the price at time $t$ for a contract that pays $R(T)$ at time $T$, where $0\leq t\leq T$. Here is how we price this contract:
$$S(t)=\tilde{\mathbb{E}}_t[e^{-... |
How to replicate a zero coupon bond? |
Let's consider the Option Chain for the Stock. There are two 3D surfaces representing the probability of the future stock price and the option prices. **I wonder if they are representing the same thing?**
**3D Option Surface**: there's the set of PUTs and CALLs options, with different expiration, strikes, and premiu... |
Is option surface same as future price probability surface? |
The definition of micro-price is
S = Pa * Vb / (Va + Vb) + Pb * Va / (Va + Vb)
where `Pa` is the ask price, `Va` is the ask volume, `Pb` is the bid price, and `Vb` is the bid volume.
The typical explanation for micro-price is that a larger quantity of shares on the bid than on the ask indicates greater buy... |
How to understand micro-price (aka, weighted mid-price)? |
Q: Does that means that option pricing methods are kinda the same as methods for future price prediction?
A: Yes. Option price is equal to the expectation of discounted payoff.
Q: And thus the ML technics like RNN for future price prediction are also could be used in Option Pricing.
A: Yes. But whether it is u... |
This is a common feature of stochastic methods, in particular optimisation methods. See for instance section 3 in [A Review of Heuristic Optimization Methods in Econometrics][1]. In general, you cannot get rid of randomness; you
need to analyse it, by looking at and analysing distributions (e.g. of portfolios) inste... |
Stochastic solutions are an unavoidable property of stochastic methods, in particular optimisation methods. See for instance section 3 in [A Review of Heuristic Optimization Methods in Econometrics][1]. In general, you cannot get rid of randomness; you
need to analyse it, by looking at and analysing distributions (e... |
I have two time series of daily return calculated as [(Price T / Price T-1) -1]. One is the daily returns of a portfolio, the other the daily returns of the index (MSCI World). Period is 2020 YTD (so during COVID19 crash). There are 78 daily returns in each time series.
The portfolio (long only, no leverage) under p... |
CAPM and Beta: problem with regression (Beta is too low yet statistacally significant?) |
How to replicate the future instantaneous short rate? |
I'm trying to calibrate bond option implied volatility from black Scholes model to market volatilities, I tried calibration in python but the smile isn't correctly matching with market volatility?
Any help is appreciated. Thanks in Advance |
I have two time series of daily return calculated as [(Price T / Price T-1) -1]. One is the daily returns of a portfolio, the other the daily returns of the index (MSCI World). Period is 2020 YTD (so during COVID19 crash). There are 78 daily returns in each time series.
The portfolio (long only, no leverage) under p... |
The -0.05% intercept (typo assumed) is a daily rate, so for 78 days, that is a cumulative alpha of around -3.9%. But your 80% beta to an index down 10% would have saved you around 2%. So on the numbers given, your portfolio should be about World -2%.
Another way of thinking about Beta is as Correlation times Relativ... |
Looking for resources / explanation.
Creating a dollar neutral hedge is easy - stockprice-a/stockprice-b
How do I create a beta neutral hedge?
I find discussions but no explicit derivation methods.
Could someone explain or refer me to a resource?
Thanks in advance. |
I have two time series of daily return calculated as [(Price T / Price T-1) -1]. One is the daily returns of a portfolio, the other the daily returns of the index (MSCI World). Period is 2020 YTD (so during COVID19 crash). There are 78 daily returns in each time series.
The portfolio (long only, no leverage) under p... |
CAPM and Beta: problem with regression (Beta is too low yet statistically significant?) |
As far as I know, we usually use log returns( $ln\frac{p_{t+1}}{p_{t}}$ ) in quantitative finance.
For example, let's say we have lots of monthly log returns data, $R_m$.
Then, we can get the mean of monthly log return, $\mu_{month}=mean(R_m)$ and volatility of log return $\sigma_{month}=std(R_{m})$
From $\mu,... |
I am using a dynamic programming approach to price European options where formulate the pricing as a discrete-time continuous-space Markov Decision process. The MDP is risk-sensitive as in I don’t take expected value of hedge portfolio to be the option fair price but rather I add a “risk-premium” which is equal to dis... |
Does simulating price as GBM automatically implies risk neutrality? |
I'm trying to price European options with a GARCH volatility model. What I have is a program that calibrates the GARCH volatility process for a stock which I intend to use to value a derivative on the underlying. My initial thought would be to simple discretize the geometric brownian motion and simulate with a two fact... |
Do you need to simulate the entire stock path for option pricing with GARCH? |
I wish to verify my understanding and correctness of my methodology with this question, when calculating the sortino ratio for the SP500. My base data is the total returns SP500 index from Yahoo finance:
https://finance.yahoo.com/quote/%5ESP500TR/history?p=%5ESP500TR
I took the adj. closing price starting with th... |
A am currently reading a manual on how to use some actuarial modelling software to project the expected liability payments made under an annuity contract. In this guide, the following statement is made:
> In financial modelling we start at the point when everyone has died. This is the oldest age in our mortality tab... |
Why are cashflows "modelled backwards in time"? |
A am currently reading a manual on how to use some actuarial modelling software to project the expected liability payments made under an annuity contract. In this guide, the following statement is made:
> In financial modelling we start at the point when everyone has died. This is the oldest age in our mortality tab... |
Universa Investments run by Mark Spitznagel popularized the idea of portfolio insurance (also known as tail hedge) protecting the investor against *severe* market declines (tail risks). By using this tail hedge, the investor can increase their share in riskier assets (stocks) while bringing the total risk of the portfo... |
For example, the bet may be about someone giving me P dollar and I will pay that person 300 dollars if tomorrow Tesla gets above 860 dollars.
While assuming the Black-Scholes model hold, I can simply use the risk-neutral measure to calculate the fair amount of P.
However, is there a way for me to hedge the risk a... |
How can I theoretically hedge a bet, which is about whether or not price of a stock will get above a number? |
I know that the tangency point between the CAL line (drawn from risk-free asset's return) and the efficiency frontier is the optimal risky portfolio. But what if there is no risk free asset? Can I draw CAL line without risk free asset? How do I find the optimal risky portfolio on the efficient frontier? |
Which one is the optimalo risly portfolio in the efficiency frontier in the absense of a risk free asset? |
Which one is the optimal risky portfolio in the efficiency frontier in the absense of a risk free asset? |
When reading John Hull's book about perpetual options (Section 26.2: "perpetual American call and put options", 10th edition), I am a bit confused about the boundary conditions used here.
The author starts with discussion of the "one-shot derivative", which pays off a fixed amount $Q$ when stock price $S=H$ for the ... |
why does "one-shot derivative" boundary condition hold for perpetual American call options? |
I have access to some tick data and Bloomberg data. Outside of data mining and hoping to find an economic rationale after the fact, what do you usually do to generate ideas before you look at the data? I'm having trouble thinking of anything outside of arbitrage, trading pairs of similar assets, data mining correlate... |
How do you formulate trading ideas and strategies? |
I have access to some tick data and Bloomberg data. Outside of data mining and hoping to find an economic rationale after the fact, what do you usually do to generate ideas before you look at the data? I'm having trouble thinking of anything outside of arbitrage, trading pairs of similar assets, data mining correlate... |
OK, your framework on this is right. Long-term yields embed "expectations" about future short rates. That is: what do I think I'll get if I sat for T years with cash in the bank for swaps, or the same rolling Govvie Bills with respect to Bonds?
Plus a cherry-on-top called "term premium". One can rationalise the exi... |
The mean variance model of Markowitz that uses multivariate covariance matrix requires the length of each of the N assets return time series under consideration to be of equal length. Are there any techniques to do asset allocation without this requirement, with or without the covariance matrix? (This is not a question... |
I have a working (swing) trading strategy based on equity index futures in place. I enter and exit by giving market orders. The strategy generates roughly 40 trades per instrument per year.
I want to diversify my risk by using it for a couple of different markets.
Presently I am looking at the Nikkei 225 futures.... |
Which Nikkei225 futures contract to take? |
Financial models by default use time bars for input data. I use time bars to refer to both intraday (high frequency) and interday (low frequency data) since the sampling occurs at fixed intervals of time for both.
The alternatives are tick bars, volume bars and dollar bars which sample based on the occurence of a n... |
Volume bars, dollar bars from low-frequency data? |
Markowitz does not require time series of equal length. Markowitz does not veen require that the covariance matrix be based on time series. Markowitz just requires a covariance matrix. The covariance matrix could, for all Markowitz cares, be based entirely on judgment, on proxies, etc.
So your question really is, ho... |
In your case, there are five elements that aggregate into your portfolio return. Just treat the intercept as a fifth factor for now. We'll come back to what it means in a second. But you should be able to take your regression betas, multiply these by each style's period-return, to give you 5 style return contributions,... |
I am new to finance and I have been looking at market data in terms of prices, volumes and net volumes. From what I have gathered:
Volume - number of transactions (buying-selling) throughout the specified time.
Net Volume - difference between the uptick and downtick during that time, which essentially, in the exa... |
Has anyone come across and fixed calculation issues on boundaries using Bjerksund-Stensland 2002 (Hull, Haug or Raouh implementations) ?
Thanks in advance |
I built a zero-coupon curve out of a generic par swap rate curve (**Step 1**) and I am trying to recover the swap curve back from the zero-coupon curve (**Step 2**).
Step 1 works but not Step 2. I get close quotes but they do not exactly match. Anyone has any idea what's wrong in my Step 2?
My guess is that it do... |
**A simple question:** When exchanging currency in order to finance an investment, is it standard/best practice to adjusted for exchange rates when calculating the NPL of that investment?
**For example:** I, a European, purchase 100 USD of american securities on Day 0, costing me 90 EUR @ a rate of 0.9 USD/EUR. At D... |
Is "time value of currency" to be accounted for in returns calculation? |
I am trying to implement Black-Scholes delta hedging for a European call option using Python.
The following is my Python code:
```
# evolve stock prices under GBM SDE solution in N steps
# BS parameters
S0 = 120
K = 100
r = 0.05
d = 0
sigma = 0.2
T = 1
# number of discretization steps
N = 50
stock_prices = np.nd... |
I am trying to implement Black-Scholes delta hedging for a European call option using Python.
The following is my Python code:
```
# evolve stock prices under GBM SDE solution in N steps
# BS parameters
S0 = 120
K = 100
r = 0.05
d = 0
sigma = 0.2
T = 1
# number of discretization steps
N = 50
st... |
I am trying to implement Black-Scholes delta hedging for a European call option using Python.
The following is my Python code:
```
# evolve stock prices under GBM SDE solution in N steps
# BS parameters
S0 = 120
K = 100
r = 0.05
d = 0
sigma = 0.2
T = 1
# number of discretization steps
N = 50
st... |
Did I implement Black-Scholes delta hedging correctly? |
I am trying to implement Black-Scholes delta hedging for a European call option using Python.
The following is my Python code:
```
# evolve stock prices under GBM SDE solution in N steps
# BS parameters
S0 = 120
K = 100
r = 0.05
d = 0
sigma = 0.2
T = 1
# number of discretization steps
N = 50
st... |
Has anyone come across and fixed calculation issues on boundaries using Bjerksund-Stensland 2002 (Hull, Haug or Rouah implementations) ?
Thanks in advance |
If we have an option between two commodities to hedge jet fuel and the commodities have results as follows: minimum variance hedge ratio: **1.07** for commodity 1 and **2.53** for commodity 2
Optimal number of contracts: 27 for commodity 1 and 63 for commodity 2
Out of these two options which commodity ... |
If you want to do it formally, the most basic texts on the subject are Duan (1995) and Heston and Nandi (2000). In the later case, they actually propose a quasi-analytical formula for pricing European options under their GARCH model. The model has this form under the physical measure:
\begin{align}
ln S_{t+1} - l... |
I'm trying to calibrate bond option implied volatility from SABR model to market volatilities, I tried calibration in python but the smile isn't correctly matching with market volatility?
Any help is appreciated. Thanks in Advance |
While trying to implement Black-Scholes delta hedging for a European call option using Python, I came across the following phenomena:
> Given a portfolio consisting of longing a delta shares of stocks and shorting a call option, its value can never be greater than the strike price of the call option.
Is this true... |
Can a portfolio value consisting of longing a delta shares of stocks and shorting a call option greater than strike price? |
To answer the questions in your comment:
1. What is a 3M swap rate? Either it's a fixed rate vs a shorter tenor (ex:1m) or a fixed rate vs the same tenor but forward (in this case a FRA), or if it's starting spot then it's the same as a zero rate because it has to intermediate payments.
2. The floating rate conve... |
Vega hedging swaption with caplets - precisely, what will go wrong? |
I am trying to create a Simulated Portfolio Optimization based on Efficient Frontier on 50 stocks, which you can find the [csv here][1]. Yet it already takes me several minutes to get a suboptimal solution: I can't draw an accurate efficient frontier:
[![enter image description here][2]][2]
Whereas it should be s... |
my question may be very dumb but I computed daily returns as the log(Pt)-log(Pt-1) in order to get the volatility. But I do not want to present my results over a certain period of time (whether it's a week, a month or ayear). I just want to see the daily variations. Is it meaningful ?
Also, in that case, should I s... |
Do I have to annualize daily volatility? |
If $f^{T_1}(t)$ is the price of a forward and $F^{T_1}(t)$ is the price of a future on some stock, both maturing at date $T_1$ and with the assumptions:
- no dividend
- constant interest rates
- no arbitrage
My notes say that the "constant interest rate" assumption implies the following:
$$
f^{T_1}(t) = ... |
Is the forward price equal to the future price? |
[![enter image description here][1]][1]
I'm given the aforementioned parameters for a two-step binomial model where the underlying pays no dividend, $S_0=50$ and $T=2$. With this information I was able to calculate the risk-neutral probabilities ($p$) and I am able to set up the following binomial tree for stock price... |
Stochastic Volatility Models - are they complete markets? |
I am looking for a text similar to Glassermsn's Monte Carlo Methods in Financial Engineering, but with a focus on numerical methods for PDEs. Glasserman's book seems to cover a lot for what is required from a financial engineer in terms of Monte Carlo methods, in other words, the foundations, the topics that are absolu... |
I am looking for a text similar to Glasserman's Monte Carlo Methods in Financial Engineering, but with a focus on numerical methods for PDEs. Glasserman's book seems to cover a lot for what is required from a financial engineer in terms of Monte Carlo methods, in other words, the foundations, the topics that are absolu... |
I really enjoyed the harmonic analysis course that I had in graduate school decades ago. But is it useful for finance? I looked at the paper [Silke Prohl. Harmonic Analysis for **Mathematical Finance**.][1] and it sounds like fun (did not read it in detail, hope to do it some other time) but I don't see immediate finan... |
I'm reading Mark Joshi's Concepts and Practice of Mathematical Finance, where in section 2.5 he describes an example of arbitrage-free pricing (attached below). I have a pretty solid understanding of how we arrived at $K' = K\frac{1+d}{1+r}$, but I got a little lost when he mentioned the alternative rate $L$.
[![S... |
I wanted to ask a simple question in representing mathematical concepts/terms into the portfolio optimization utility functions.
I have never worked with these in production environment so I am very likely to be wrong, please feel free to disagree, provide some insights.
In Statistical Arbitrage, people are basic... |
I have two time series of daily return calculated as $\frac{Price_{t}}{Price_{t-1}} -1$. One is the daily returns of a portfolio, the other the daily returns of the index (MSCI World). Period is 2020 YTD (so during COVID19 crash). There are 78 daily returns in each time series.
The portfolio (long only, no leverage)... |
Is it valid to apply GRS-test on non-normal and autocorrelated residuals?
I got residuals using 10 test-assets regressed on 3-factor and carhart.
If it is valid, how should I do it?
and if it is not valid, what should we do? |
Is it valid to apply GRS-test (Gibbons, Ross and Shanken 1989) on non-normal and autocorrelated residuals?
I got residuals using 10 test-assets regressed on 3-factor and carhart.
If it is valid, how should I do it?
and if it is not valid, what should we do? |
Black-Scholes with its assumptions results in a unique price for the call. If we introduce stochastic interest rates, would this still remain the case? |
I am starting to explore this area. My ultimate aim is to build a 3 month LIBOR forward curve.
1) I wish to know what exactly 'Dual Curve Bootstrapping' is (If someone could explain it in clear words).
2) What are the 2 curves I will be using for coming up with 3Month Libor forward curve
3) Can this process be do... |
What is Dual Curve Bootstrapping? And how to do it, with an example? |
A multi-curve meants 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 Multipl... |
I am trying to derive a risk-neutral density from European call option prices using a second order finite difference scheme. Let $C(K,T)$ be the price of a European call with strike $K$ and expiry $T$ then the risk-neutral density of the log price returns can be computed using Breeden-Litzenberger theorem.
I have gi... |
I have a fairly short and straightforward question. I am running a dynamic optimisation strategy and therefore need to construct the FF5 characteristics. I am using COMPUSTAT quarterely accounting data.
Now my question is why are all accounting variables such as BTM/SIZE etc constructed using at least 6 months follow... |
I have a fairly short and straightforward question. I am running a dynamic optimisation strategy and therefore need to construct the FF5 characteristics. I am using COMPUSTAT quarterely accounting data.
Now my question is why are all accounting variables such as BTM/SIZE etc constructed using at least 6 months follow... |
I have a fairly short and straightforward question. I am running a dynamic optimisation strategy and therefore need to construct the FF5 characteristics. I am using COMPUSTAT quarterely accounting data.
Now my question is why are all accounting variables such as BTM/SIZE etc constructed using at least 6 months follow... |
For example [![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/0cL1x.png
Can u provide with a detailed example please if i have ( maturity, issue date, coupon, frequency, days_countbase, (price or yield) what is the (yield or price given this information.
For example if i have
matu... |
how can i see the cashflows of a specific bond created in quantlib in Python? this is the code i have, how should i change it |
Is convexity with respect to the strike a necessary arbitrage free condition for the price of an American put on an underlying with value with out dividend? |
Is the price of an American put on an underlying without dividend convex with respect to the strike? |
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