instruction stringlengths 15 26.1k |
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Assume that dividend = 0, then the price of call option is
$$
C = S\cdot P_{s}[S(T) > K] - e^{-rT}K\cdot P_F[S(T) > K] = SN(d_1)-e^{-rT}KN(d_2)
$$
where
$P_s[S(T) > K]$ = Probability of ITM when $S(t)$ is set to be a numeraire and
$P_F[S(T) > K]$ = Probability of ITM under Forward measure
When $T \righ... |
I work in model validation and I am familiar with classical investment strategies, economic theories and pricing models of options and I am curious about algorithm trading. What is it driven by? What theory does it believe in? For example does it rely on economic theories like efficient market hypothesis, growth and in... |
What theory and philosophy does algorithm trading is based on? |
I'm reading recent research on Treasuries and to paraphrase, it says that long term 10y20y Treasury forward rates now have a positive term premium over the long run nominal funds rate.
It then says that, as the term premium of the 10y20y Treasury rate is unlikely to become higher due to certain factors such as the t... |
How does the term premium of the 10y20y Treasury forward rate relate to the 30y rate? |
I'm reading recent research on Treasuries and to paraphrase, it says that long term 10y20y Treasury forward rates now have a positive term premium over the long run nominal funds rate.
It then says that, as the term premium of the 10y20y Treasury rate is unlikely to become higher due to certain factors such as the t... |
I'm reading recent research on Treasuries and to paraphrase, it says that long term 10y20y Treasury forward rates now have a positive term premium over the long run nominal funds rate.
It then says that, as the term premium of the 10y20y Treasury rate is unlikely to become higher due to certain factors such as the t... |
I work in model validation and I am familiar with classical investment strategies, economic theories and pricing models of options and I am curious about algorithm trading. What is it driven by? What theory does it believe in? For example does it rely on economic theories like efficient market hypothesis, growth and in... |
I have to discount a cash flows of mezzanine and junior note of NPL's securization, so the discount curve have to include a zero spread for unexpected loss, could you suggest me a proxy to estimate quickly this spread?
I tried to use the differences between the
- CDS Benchmark CCC Western Europe 5Y and
- the spr... |
Let $S_u$ be the price of stock in the up-state one period from now. Let $S_d$ be the price of the stock in the down state.
Let $C_u$ be the payoff of a call option at time $1$ in the up-state and similarly for $C_d$.
If we define $\delta = \frac{C_u-C_d}{S_u-S_d}$, then if we purchase $\delta$ shares at time z... |
I found several filings having dozens of filers, here's a taste: https://www.sec.gov/Archives/edgar/data/1265235/000095013706009934/0000950137-06-009934-index.htm
I understand that some may have 2 filers: one filing agent and the actual subject but how is the example above possible, can somebody help me make sense o... |
Filings having multiple Filers on SEC Website, explanation? |
*"can someone please expand on how the Treasury supply affects the forward rate?"*
All else equal, more supply should mean lower bond prices (higher bond yields) as Attack68 noted. However, I think the Street has historically assumed that supply will affect spreads (e.g. swap spreads) more than the level of rates. |
Say I have an asset return time series:
Jan2020: -5%
Feb2020: +5%
Mar2020: -5%
Apr2020: +5%
May2020: -5%
Jun2020: +5%
Q3 2020: +20%
Oct2020: +5
Nov2020: -5
Dec2020: +5
Note that 3 months of data is an aggregate i.e., we don't have July, Aug, Sept. Is there a simple unbiased estimator... |
I am assuming that the weights of currency in the SDR were establishedby following a transparent process.
Are these weights the result of a vote or is there a mathematical algorithm/formula that generated the following weights:
U.S. Dollar 41.73
Euro 30.93
Chinese Yuan 10.92
Japanese Yen 8.33
P... |
What determined the weights of a currency in the International Monetary Fund SDR basket(2015)? |
I'm reading recent research on Treasuries and to paraphrase, it says that long term 10y20y Treasury forward rates now have a positive term premium over the long run nominal funds rate (neutral rate).
It then says that, as the term premium of the 10y20y Treasury rate is unlikely to become higher due to certain factor... |
I work in model validation and I am familiar with classical investment strategies, economic theories and pricing models of options and I am curious about algorithmic trading. What is it driven by? What theory does it believe in? For example does it rely on economic theories like efficient market hypothesis, growth and ... |
What theory and philosophy is algorithmic trading based on? |
Why do we need Kalman Filtering theory in dynamic models in finance when we consider an environment of asymmetric or incomplete information? I understand that this has to do with the update of the probabilities based on the beliefs that traders have wrt to the assets payoffs, however I can not understand how the mechan... |
This is the main part of the question, as I understand it:
"As I have just describe, the price is going to grow but, why should it change the relation between the expected return and volatility of the asset?"
There is only one way to price an equity security, and that's using the basic Dividend Discount Model. Th... |
This is the main part of the question, as I understand it:
"As I have just describe, the price is going to grow but, why should it change the relation between the expected return and volatility of the asset?"
There is only one way to price an equity security, and that's using the basic Dividend Discount Model. Th... |
If variance is stationary, specifically sample variance in Q3 was the same as in other periods:
1) Determine sample mean including missing observations as product^1/12: 1.9%
2) Calculate unbiased variance for known observations: -11.7%, or stdev 5.3%
This is the same as ignoring Q3 altogether. I'm curious for my... |
This is in response to the part of your question that asks about M1 versus M2, although it seems you've more or less answered parts of your own question. M1 is the simplest monetary aggregate and includes items most widely used as a medium of exchange (approximately 85% of household purchases are made using M1 balances... |
I am trying to create a Correlation Matrix of one particular industry (in the consumer goods sector) with around 15 securities across multiple stock exchanges for the year 2020.
However, I have a lot NaN values on my data frame due to holidays such as Berchtolds Day in Switzerland (January, 2nd, 2020), Lunar New Ye... |
Let me add a couple of points that I feel are missing.
In the CAPM world, we have the following three base features:
- We assume the existence of some "risk-free" asset (could be a government bond free of credit risk): the return of the risk-free asset is denoted $r_f$, and is assumed non-stochastic (that's why ... |
Are there any good papers/ references on the statistical distribution of Max Drawdown over a specified amount of time given a specified Sharpe? Assuming returns are iid normally distributed
I’ve been running some Monte Carlo simulations but wondering if there is any theory |
Let me add a couple of points that I feel are missing (basically, the rather long intro below aims to explain why it should never be possible for any one asset to be above the Capital Market Line in the CAPM model).
In the CAPM world, we have the following three base features:
- We assume the existence of some "... |
## Supply and Demand ##
That's a great question and your question has a simple answer. @Daneel showed you the ``maths'' behind it.
It boils down to: if an asset has too high returns than the CAPM predicts, then the asset is super attractive. An asset has a certainty riskiness (measured by its (market) beta) whi... |
## Supply and Demand ##
That's a great question and your question has a simple answer. @Daneel showed you the ``maths'' behind it.
It boils down to: if an asset has too high returns than the CAPM predicts, then the asset is super attractive. An asset has a certainty riskiness (measured by its (market) beta) whi... |
Let me add a couple of points that I feel are missing (basically, the rather long intro below aims to explain why it should never be possible for any one asset to be above the Capital Market Line in the CAPM model).
In the CAPM world, we have the following three base features:
- We assume the existence of some "... |
Your buggest problem is with computing the pairwise correlations of returns.
Suppose for simplicity that you have 2 assets A and B. For asset A, you have closing prices for all 3 days $t_0$, $t_1$, and $t_2$. If you also have dividends, you calculate A's total return from $t_0$ to $t_1$:
$$R_{A,t_0,t_1}=\frac{P_{... |
Let me add a couple of points that I feel are missing (basically, the rather long intro below aims to explain why it should never be possible for any one asset to be above the Capital Market Line in the CAPM model).
In the CAPM world, we have the following three base features:
- We assume the existence of some "... |
Let $S_u$ be the price of stock in the up-state one period from now. Let $S_d$ be the price of the stock in the down state.
Let $C_u$ be the payoff of a call option at time $1$ in the up-state and similarly for $C_d$.
If we define $\delta = \frac{C_u-C_d}{S_u-S_d}$, then if we purchase $\delta$ shares at time z... |
I've just started investigating the capabilities of the PortfolioAnalytics package. I've had it working for smaller considerations, around 20 stocks. These return all objectives I have registered (mean and ES). However I have now started to consider larger groups of stocks (~190) and now optimize.portfolio is not repor... |
I am trying to gather a time series data for Crude Oil Future prices (both Brent and WTI) to work on a project. I see that the **BCOMCO** (Brent) and **BCOMCL** (WTI) indices are constructed from the respective Futures prices with a rollover methodology.
I however, am failing to understand the difference between th... |
I was just wondering if someone could explain to me how an ETF market maker earns profit through the spread they collect while hedging the positions to be non-directional.
For example I read somewhere that:
Say that ETF market maker buys ETF at bid and hedges that by shorting the basket. He then manages to offloa... |
Your question has two parts (at least to me):
1. How do you make money hedging ETF market making vs some other product?
2. How do deal with the resulting positions?
For #1, a very similar question was asked about the trading process just recently. See here for details: https://quant.stackexchange.com/questions... |
Pricing a digital caplet using Hull White model
1 if R>K
0 otherwise
Why would you represent the payoff as a call spread, i.e.
$$Payoff = \frac{(R - (K+\epsilon))^+-(R - (K-\epsilon))^+}{2\epsilon}$$
rather than
$$Payoff = \mathbb{1}_{R>K} $$
What is the benefit?
|
Whenever we replace an order, we lost priority since we are added to the end of the queue. If we dont replace an order, there is an obvious chance that we might get picked. There are other situations possible, when we replace deeper back and forth in the book and keep losing priority. Is there any literature which disc... |
I have data containing 6 months returns of a stock.
+-------------+-------------+
| Date | Return |
+-------------+-------------+
| 31/Jan/2016 | -0.06861672 |
| 29/Feb/2016 | -0.02187975 |
| 31/Mar/2016 | 0.01420873 |
| 30/Apr/2016 | 0.01858721 |
| 31/May/2016 |... |
Pricing a digital caplet using Hull White model, which pays:
>$1$ if $R>K$, $0$ otherwise.
Why would you represent the payoff as a call spread, i.e.
$$\text{Payoff} = \frac{(R - (K+\epsilon))^+-(R - (K-\epsilon))^+}{2\epsilon}$$
rather than
$$\text{Payoff} = \mathbb{1}_{R>K}?$$
What is the benefit? |
In control variate technique we have to calculate
$$b=\frac{\text{cov}\{{X,Y}\}}{\text{var}\{{X}\}}$$
where $X$ is a payoff from standard call option and $Y$ is a payoff from for example barrier option. Why we have to estimate $b$ before we use this method and we cant use payoffs which we use during pricing? Or maybe... |
Let me add a couple of points that I feel are missing (basically, the rather long intro below aims to explain why it should never be possible for any one asset to be above the Capital Market Line in the CAPM model).
In the CAPM world, we have the following three base features:
- We assume the existence of some "... |
I was just wondering how the fair price of ETF with international holdings are accurately priced. Say we have an ETF that trades on domestic exchange but the holdings are international and the market is closed i.e. we do not have the spot prices of the underlying holdings. How then can the ETF market makers accurately ... |
**Edit**: my answer below assumed that when the OP asks "there might be assets that are above or bellow the market line", he's referring to the Capital Market Line. In fact OP's question seems to be about the Security Market Line, so the answer below is somewhat redundant. Regard the below as an "alternative" view of t... |
Question: Suppose you have two strategies. Strategy 1 gains 8% with probability p, and loses 5% with probability 1-p, where p = 0.53. Strategy 2 gains 8% with probability q, and loses 5% with probability 1-q, where q = 0.51. The outcomes of the two strategies are independent. You assign weights (which sum to one) f... |
I'm trying to compare the minimum cost of subscribing to primary sources of real-time market data on NYSE and NASDAQ for last sale and level 1 top-of-book quotes.
Use cases are log to files, store history in a database, process messages and historical data to submit orders via broker API.
I'm a non-professional u... |
I am trying to gather a time series data for Crude Oil Future prices (both Brent and WTI) to work on a project. I see that the **BCOMCO** (Brent) and **BCOMCL** (WTI) indices are constructed from the respective Futures prices with a rollover methodology.
I however, am failing to understand the difference between th... |
In control variate technique we have to calculate
$$b=\frac{\text{cov}\{{X,Y}\}}{\text{var}\{{X}\}}$$
where $X$ is a payoff from standard call option and $Y$ is a payoff from for example barrier option. Why we have to estimate $b$ before we use this method and we cant use payoffs which we use during pricing? Or maybe... |
Your biggest problem is with computing the pairwise correlations of returns.
Suppose for simplicity that you have 2 assets A and B. For asset A, you have closing prices for all 3 days $t_0$, $t_1$, and $t_2$. If you also have dividends, you calculate A's total return from $t_0$ to $t_1$:
$$R_{A,t_0,t_1}=\frac{P_{... |
Long time viewer, but first time poster, so excuse me if i'm in the wrong place please.
Anyway, I am working on a project that is pretty interesting. Through data mining, I am able to gather a ton of investment portfolios. Each portfolio has the obviously related statistics, including total revenue, total loss, resu... |
Best method to determine future success or to determine best linearity? |
You are forgetting about the actual cost to bring in your connectivity to the exchanges. You will have to spend a few \$k per month to bring in your own digital lines to bring the data back.
Also, you have some redundancy there why do you want CTA, UTP, and NYSE? If you just want the current best bid/offer for e... |
Pricing an ETF always involves estimation. Most ETFs have a regular trading market that is only a few bps wide. Even in the US we often have ETFs with relatively illiquid underlyers but with an ETF that is priced tight.
Market makers look for all kinds of proxies to model the fair value of ETFs. For internationa... |
I'm trying to compare the minimum cost of subscribing to primary sources of real-time market data on NYSE and NASDAQ for last sale and level 1 top-of-book quotes.
Use cases are log to files, store history in a database, process messages and historical data to submit orders via broker API.
I'm a non-professional u... |
in university's lecture notes, from what I understand using the replication of portfolio principle to price derivates, the forward price of a contract K should be: $K = P_0(1+r)$ where $P_0$ is the spot price of the underlying and $r$ is the risk-free rate. However, with this definition, I do not see how the delta of a... |
I think your error is in confusing forward contract and forward price.
The forward **price**, with continuous interest rates, is $K=P_0e^{rT}$. It is a **fixed** parameter of your forward contract.
The forward **contract value**, on the other hand, is $V_t=P_t-Ke^{-r(T-t)}$. Its derivative w.r.t. the underlying ... |
i've seen termsheets of callable accreting notional swaps where the accretion rate equals the fixed coupon rate. apparently these are used to hedge callable zcb's. but it doesnt seem to make sense! the fixed leg of the swap is paying coupons, whereas the zcb doesnt pay coupons. i guess maybe it is done like that to... |
how do traders typically hedge a callable zero coupon bond? |
I understand that with interpolation or bootstrapping one can determine spot rates given other spot rates, however how would you go about establishing what a hypothetical perpetual bond issued by, say, the US government given the current rates should yield? |
I would like some guidance with the following please.
Suppose I have two yields:
2020 bond --> yield = 2.5
2027 bond --> yield = X (to be interpolated)
2030 bond --> yield = 2.8
How could I interpolate the 2027 bond yield, considering that it should be closer to the yield from 2030? I am doing th... |
This is a real world problem and not a research one.
We are being proposed to buy an option that has to be exercised on a specific date T. So it is a European option.
This option has a strike price of K = 50. We are ITM if and only if the price of asset S1 is above K. So it is a call option.
However, if we cho... |
This is a real world problem and not a research one.
We are being proposed to buy an option that has to be exercised on a specific date T. So it is a European option.
This option has a strike price of K = 50. We are ITM if and only if the price of asset S1 is above K. So it is a call option.
However, if we cho... |
I got stuck in one exercise of binary options, it says that I have to consider a stock that does not pay dividends, the current price of the stock is 100, the volatility of it is 20%, the risk-free rate is 4%, consider that the option has a term of one year. Let S(1) be the price of the stock at the end of the contract... |
Long time viewer, but first time poster, so excuse me if i'm in the wrong place please.
Anyway, I am working on a project that is pretty interesting. Through data mining, I am able to gather a ton of investment portfolios. Each portfolio has the obviously related statistics, including total revenue, total loss, resu... |
I am using a Kalman Filter to estimate the return dynamics of a forwards curve on a particular commodity. My state space is the initial forwards values, and an initial guess of the drift functions for each forward (initially just the first observed price change). These are for OTC products, my estimates for uncertainty... |
The question: you have a portfolio of risky assets that with a 90% probability (normal state of the world) has an expected annual return of 10%. With a 10% probability (crisis state) the annual return would be -30%. What is the expected return and volatility of the portfolio?
I know the answer is 6% expected return... |
Is there another name for this strategy? A colleague told me that hedge fund manager Steven A. Cohen likes to squash volatility with this strategy.
>Cohen would create massive blocks of expensive puts and calls around a strike price, to eliminate vega and prevent gamma. Cohen would collect the premiums, while long... |
What are "call and put walls" around a strike price, to kill off vega and prevent gamma? |
This is a real world problem and not a research one.
We are being proposed to buy an option that has to be exercised on a specific date T. So it is a European option.
This option has a strike price of K = 50. We are ITM if and only if the price of asset S1 is above K. So it is a call option.
However, if we cho... |
Let your linear system be defined by the latent multivariate state variable $x_t$, progressing in an AR(1) fashion, and let your observation at time step $t$ be linear in the latent state:
$$
\begin{align}
x_{t+1}&=Ax_t+u_t\\
y_{t}&=Hx_t+v_t\\
\end{align}
$$
with $u_t\sim N(0,Q)$ and $v_t\sim N(0,R)$ time-and... |
Let your linear system be defined by the latent multivariate state variable $x_t$, progressing in an AR(1) fashion, and let your observation at time step $t$ be linear in the latent state:
$$
\begin{align}
x_{t+1}&=Ax_t+u_t\\
y_{t}&=Hx_t+v_t\\
\end{align}
$$
with $u_t\sim N(0,Q)$ and $v_t\sim N(0,R)$ time-and... |
The question: you have a portfolio of risky assets that with a 90% probability (normal state of the world) has an expected annual return of 10% plus a random variable with a standard deviation of 15%. With a 10% probability (crisis state) the annual return would be -30% plus a random variable with a standard deviation ... |
Let your linear system be defined by the latent multivariate state variable $x_t$, progressing in an AR(1) fashion, and let your observation at time step $t$ be linear in the latent state:
$$
\begin{align}
x_{t+1}&=Ax_t+u_t\\
y_{t}&=Hx_t+v_t\\
\end{align}
$$
with $u_t\sim N(0,Q)$ and $v_t\sim N(0,R)$ time-and... |
I'm trying to think about the right way to estimate the delta of a VX contract to the S&P 500. VX futures are on the VIX index, which is a basket of S&P 500 options. By extension, VX and ES (E-mini futures on S&P 500) returns have a strong contemporaneous relationship. Having a good estimate allows for construction of ... |
consider an interest rate derivative whose value $V$ depends on $n$ interest rates $r_1, \dots, r_n$.
Hence $V$ is a function in $n$ variables $V(r_1, \dots, r_n)$.
My question concerns the gamma $\gamma$ of this derivative with respect to parallel shifts.
Does it "always" hold that
$$
\gamma(r_1, \dots, r_n) =... |
I'm trying to think about the right way to estimate the delta of a VX contract to the S&P 500. VX futures are on the VIX index, which is a basket of S&P 500 options. By extension, VX and ES (E-mini futures on S&P 500) returns have a strong contemporaneous relationship. Having a good estimate allows for construction of ... |
consider an interest rate derivative whose value $V$ depends on $n$ interest rates $r_1, \dots, r_n$.
Hence $V$ is a function in $n$ variables $V(r_1, \dots, r_n)$.
My question concerns the gamma $\gamma$ of this derivative with respect to parallel shifts.
Does it "always" hold that
$$
\gamma(r_1, \dots, r_n) =... |
Forgive me for any violations of posting rules, I’m new to this forum.
I’ve written an algorithm that checks the order book for the price point that would completely fill my entire available balance, and then runs it’s analysis on that price point to determine if it is a signal to open an order.
I’ve been creepi... |
I am looking to passively invest/dollar cost average stocks. If I averaged the yearly returns of the stock, how many years would I need before I could say with 95% confidence that the averaged value would accurately predict the average yearly return for the next 10 years? If the answer is that "it depends", what factor... |
How much market data is required to confidently predict the yearly returns of a stock? |
I am looking to passively invest/dollar cost average stocks. If I averaged the yearly returns of a stock, how many years would I need before I could say with 95% confidence that the averaged value would accurately predict the average yearly return for the next 10 years? If the answer is that "it depends", what factors ... |
I have the Portfolio frontier below. Can someone provides me with details about how can I calculate the market portfolio from this efficient frontier? I know that I have to draw the tangent line from the risk free asset, but how? is there any specific formula to calculate the risk free asset? Any help will be appreciat... |
I know that ["There are other ways to hedge price risk other than buying the underlying. It's not uncommon but it doesn't "always" happen"](https://money.stackexchange.com/questions/83877/how-do-i-hedge-stock-options-like-market-makers-do#comment142595_83877). ["You can buy/sell other options to reduce price risk, or b... |
Presuming a Market Maker must delta-hedge, how can it still earn money? |
Investors who short a company might get into trouble if a group of passionate online day traders (who like the shorted company) decide to boost its stock value by coordinated investments.
How quickly can the investors who shorted the company back out of their short? If they can convert their short to a “long” immediat... |
How quickly can someone pull out of their short? |
The Wikipedia article on [naked short selling][1] says
> It is difficult to measure how often naked short selling occurs. Fails to deliver are not necessarily indicative of naked shorting, and can result from both "long" transactions (stock purchases) and short sales.[4][16] Naked shorting can be invisible in a liqu... |
Why is naked short selling "difficult to measure"? |
Carry is coupon income + pull-to-par - financing cost. Pull to par is derived as ytm-coupon. So carry can be rewritten as ytm - financing costs. Carry cash value is the current dirty price minus the cash flows in period x discounted at the current yield to maturity. So let's say you have a bond with annual coupon 2.5, ... |
It seems implicit in the question that you are happy to assume that the distribution of historical returns is an unbiased and consistent estimator of the distribution of future returns. Else "it depends" (on every variable that affects stock prices, ie all of them) ;-)
You can always come up with such an estimate; b... |
I am looking to passively/lump-sum invest stocks. If I averaged the yearly returns of a stock, how many years would I need before I could say with 95% confidence that the averaged value would accurately predict the average yearly return for the next 10 years? If the answer is that "it depends", what factors would it de... |
As @stans already said in the comments to your question, the existence of the market portfolio hinges on the existence of a risk free rate $r_f$, where *risk free*, in this context, means that its value can be perfectly contracted for the relevant return horizon, e.g. you will with probability one get that rate for 1 m... |
I struggle to understand the difference between Knightian uncertainty versus Black Swan event. If I understand at least the basic premises, both views say that uncertainty is different from risk, and that uncertainty becomes risk only with hindsight, i.e. after the event has happened.
Is this correct?
This quest... |
As @stans already said in the comments to your question, the existence of the market portfolio hinges on the existence of a risk free rate $r_f$, where *risk free*, in this context, means that its value can be perfectly contracted for the relevant return horizon, e.g. you will with probability one get that rate for 1 m... |
Investors who short a company might get into trouble if a group of passionate online day traders (who like the shorted company) decide to boost its stock value by coordinated investments.
How quickly can the investors who shorted the company back out of their short? If they can convert their short to a “long” immedi... |
**Let us first set the scene**. When a market participant $A$ is shorting a company $X$, he is effectively going to see some other participant $B$ who holds shares of company $X$ and borrows these shares from him (in exchange for some interest). $B$ delivers the shares to $A$ with the agreement that $A$ will eventually... |
It seems implicit in the question that you are happy to assume that the distribution of historical returns is an unbiased and consistent estimator of the distribution of future returns. Else "it depends" (on every variable that affects stock prices, ie all of them) ;-)
You can always come up with such an estimate; b... |
I'm sure this falls short of proper philosophical precision, but here goes. Hark back to a slightly modified rehash of Donald's Rumsfeld's infamous:
*Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also kno... |
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