# Tag Info

10

Because you can hedge. Once you have delta hedged, the pay-off is symmetric about up and down moves so drift doesn't matter. Also the delta-hedged call and the delta hedged put have to have the same value since they have the same pay-off. (Put-call parity) Yet any argument that the call should be worth more because of drift says that the put should be ...

6

I think to gain intution you have to understand that the same agents that value the stocks will value the options. And agents compensate for volatility by demanding higher expected returns. Therefore you should ask: Why are stocks priced as they are in the first place? In your example, the stock with higher volatility has much lower expected return. This ...

5

You are trying to apply the Kelly Criterion, supposedly to maximize how aggressively to bet, and you are having trouble when the Kelly Value turns negative. The naive answer to your question is that when your kelly value turns negative, then $f=\frac{bp-q}{b}$ turning negative means the instantaneous expected return is negative, which means you should not ...

5

If you don’t have any specific model which describes the behavior of the asset being traded, you can estimate the empirical distribution of returns by backtesting your momentum strategy. Then you can adjust this estimate during your strategy’s lifetime from your trading results. Additionally you can enhance this by accounting for different market regimes (e....

4

It seems that your real question is: is the PFP (Price Formation Process) diffusive from intraday to weekly sampling rate? It is a very good question since on intraday, some academics found some multifractal features into intraday returns, meaning that the PFP is not a Geometric Brownian Motion at small scales (even considering stochastic volatility). You ...

4

Trying to shed some light here: What we also see using this here, is that if returns are log-normally distributed, ie. $$1 + r = \exp(\mu + \sigma Z),$$ with $Z$ standard-normal, then $$E[1+r] = \exp(\mu + \frac 12 \sigma^2)$$ holds. But the geometric mean $GM$ is given by $\exp(\mu)$ and we have $$\log(GM) = \mu = \log(E[1+r]) - \sigma^2 /2$$ and ...

4

The answer to the original question is simple: the Chopra-Ziemba paper is highly flawed and unreliable. Note that the framework is in-sample and based on a utility function. It has nothing to do with out-of-sample behavior of the mean vs. the covariance in an optimization. Estimation error grows linearly in the mean but quadratically in the covariance. At ...

4

The price of a derivative does not explicitly depend on the expected return of the underlying, however the price change or return of the derivative depends on the return of the underlying. Hence the expected return of the derivative depends on the expected return of the underlying, which is what matters. Also remember that the price is a function of the ...

4

I prefer thinking in terms of well measured vs. poorly measured rather than significant vs. insignificant: arbitrary p-value cutoffs and ignoring sensible priors can both be problematic. On the question, "can poorly measured betas from time-series regressions give rise to well measured factor premiums from cross-sectional regression?" The abstract answer is ...

4

Expected returns are very difficult to estimate reliably without incurring estimation error as found out by Merton (1980) "On estimating the expected return on the market". This is why estimating volatility/the covariance matrix has become the default approach in the mean-variance model because volatility is easier to predict than returns. Even the global ...

4

One of the best ways I came across to estimate the expected return of a stock (even with limited time-series data), is Martin and Wagner (2019): What is the expected return on a stock?. From their paper: Second, our formula provides conditional forecasts at the level of the individual stock. Rather than asking, say, what the unconditional average ...

3

Since $Y=e^{(r-\frac{\sigma^2}{2})\tau + \sigma \sqrt{\tau}Z}$, then \begin{align*} xY > K \Leftrightarrow Z > -d_2, \end{align*} where \begin{align*} d_2 = \frac{\ln \frac{x}{K} + (r-\frac{\sigma^2}{2})\tau}{\sigma\sqrt{\tau}}. \end{align*} Consequently, \begin{align*} e^{-r\tau}\mathbb{E}\big(Y \mathbb{1}_{\{xY >K\}} \big) &= e^{-\frac{\sigma^...

3

Yes and No. In the absence of arbitragers, the price of the option will be different for each speculator based on their drift expectations (and each speculator has a risk in his position and will limit his ability to trade large sizes to avoid bankruptcy) and the option price will converge to priced off a supply-and-demand driven drift expectation. ...

3

What a great question -- it touches on many issues at the core of quantitative finance. This answer might be a lot more than you bargained for, but it's too interesting to pass up. References Mostly, this subject falls somewhere at the intersection of these three highly-interrelated topics: risk-neutral valuation, rational pricing and the fundamental ...

3

Another possible approach is taking views a la Black-Litterman. There is a 2006 paper "Incorporating Trading Strategies in the Black-Litterman Framework" that discusses the methodology in more detail. There are several practical issues that one should consider when implementing a momentum strategy with optimization. I would pay careful attention to the ...

3

Isn't this a simple mathematical rule? $$\Delta r_{t}=r_{t} - r_{t-1} = ln(p_{t}) - ln(p_{0}) - ln(p_{t-1}) + ln(p_{0})=ln(\frac{p_{t}}{p_{t-1}})$$ i.e. logarithmic or continuously compounded return. As a result: $$E(\Delta r_{t})=\frac{1}{T}\sum_{t=1}^{T}\Delta r_{t} = \frac{1}{T}\sum_{t=1}^{T}ln(\frac{p_{t}}{p_{t-1}})=\frac{1}{T}ln(\prod_{t=1}^{T}\frac{p_{... 3 In effect, you are wondering whether to price this option on risk-free probability distributions (B-S drift r_f), or real-world ones (B-S drift \mu, however calibrated) One cannot short the mutual fund, so the argument for using risk-free is weakened. But, there are various economic equilibrium arguments why using it may still be OK. If you use the ... 3 Assume the weights of the two assets are w,1-w respectively;the expected returns and standard deviations are denoted by \mu,\sigma with subscripts 1,2,p(for portfolio),i.e,we have \mu_1,\mu_2,\mu_p,\sigma_1,\sigma_2,\sigma_p.The correlation coefficent is \rho Then$$\sigma_p^2=w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_1\sigma_2\rho ...

3

The answer is that you may use an approach that includes IRR, but that's not a necessary component of what I would consider a good model. I have seen commercial tools that include them and those that don't. I have also seen practitioners set the variables in packages that include this approach, so that they were not a relevant component of the resulting ...

3

This will depend on the definition of "return on the long run". If we define the annualized return on the long run by $\frac{1}{T}\ln \frac{S_T}{S_0}$ for a certain time $T$ in the future, then \begin{align*} E\left( \frac{1}{T}\ln \frac{S_T}{S_0} \right) = \mu-\frac{1}{2}\sigma^2, \end{align*} as claimed. Note that $\mu$ is the instant, or instantaneous, ...

3

Financial markets & Corporate Strategy - Grinblatt & Titman The book is very intuitive, but as a consequence less comprehensive than ex. Options, Futures, and other Derivatives by Hull (which is seen as the basic foundation of everything quant in some parts of the industry.) A great entry level book to finance, and is publically avaliable here: ...

3

Note that \begin{align*} \frac{S_T-S_t}{S_t} &= \frac{S_T-K +K-S_t}{S_t}\\ &=\frac{(S_T-K)^+-(K-S_T)^+ +K-S_t}{S_t}. \end{align*} Then, \begin{align*} E\left(\frac{S_T-S_t}{S_t} \mid \mathcal{F}_t \right) &= \frac{e^{rT}}{S_t}(C_t-P_t)+ \frac{K-S_t}{S_t}. \end{align*} where \begin{align*} C_t &= e^{-rT} E\left((S_T-K)^+ \mid \mathcal{F}_t \...

3

Under GBM $$\frac {dS_t}{S_t} = \mu dt + \sigma dW_t$$ we get $$S_T = S_0 e^{(\mu - \frac{1}{2}\sigma^2)T + \sigma W_T}$$ suggesting that $$S_T \sim \text{ln}\mathcal {N} ( \tilde {\mu}, \tilde {\sigma})$$ where \begin{align} \tilde {\mu} &= \ln S_0 + (\mu - \frac{1}{2}\sigma^2)T \\ \tilde {\sigma} &= \sigma \sqrt {T} \end{align} Now if $X \... 3 Well, you need to know what is the stochashtic model you are using for$y_T$, if you assume it's a geometric brownian motion you have this process :$y_T = y_0 e^{\sigma W_T - \frac{1}{2} \sigma^2T} $If you compute the expectation and variance you get$ \mathbb{E}(y_T) = y_0$and$Var(y_T) = {y_0}^2( e^{\sigma^2 T }-1)$As$y_0 $is constant you ... 3 For a simple example, say you start with \$100 in an account. In the first year, it makes 50% gain (+50% interest) => \$150 In the second year, it makes 50% loss (-50% interest) => \$75 The arithmetic mean is (50% - 50%)/2 = 0% The geometric mean is (150% * 50%)^0.5 - 1 = 86.6% - 1 = -13.4% pear year You know that you go from \$100 to \$75 over 2 ...

3

There does not seem to be a clear relationship between interest rates and equity risk premiums. Damodaran (2019) has a great paper that goes into details of equity risk premiums. In this work, he writes: In much of valuation and corporate finance practice, we assume that the equity risk premium that we compute and use is unrelated to the level of ...

2

IEX is an ATS. The ECN/ATS business is dominated by rampant and well known conflicts of interest. A part of the IEX value proposition from the beginning was to offer an alternative to traders who were disenfranchised by this market structure. If maker-taker rebates are part of your trading business model or if you engage in any strategy that could be deemed ...

2

The concrete (general) answer to part (ii) of my question seems to be contained in Equation 8 of the following link: http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-portfolio-I.pdf In particular, interpreting $\sigma$ as volatility, take for example $E_A=0.10,\sigma_A=0.15,E_B=0.25,\sigma_B=0.40$ and $\rho =−0.2$. I get that about 83 percent of the ...

2

Singer and Terhaar original paper can be found at this link. They do not provide an explanation about how to estimate this factor and just mention that both values provide a boundary. The CFA curriculum mentions that " For example, it has been observed that developed market bonds & equities are approx 80% integrated and 20% segmented.", however the ...

2

Practically, it is very difficult to get a measurement of a stock's true drift while there are very well-documented processes to estimate volatility. It is therefore very convenient mathematically to select the risk neutral pricing measure that eliminates idiosyncratic drift. At its heart, Black Scholes constructs a dynamic, replicating portfolio for an ...

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