12

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 ...


7

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 ...


6

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 ...


6

To answer, the assertion that volatility is easier to predict than expected return requires clarification. The phrase "easier to predict" is particularly ambiguous. To me this means that the estimation of volatility from a sample of returns is more robust than the estimation of expected return in the context of relative sampling error. Suppose over a time ...


6

We can obtain a closed-form solution for the expected return over an arbitrary holding period under some typical assumptions. Assuming geometric Brownian motion with drift $\mu$ and volatility $\sigma$, the stock price at time $t \geqslant 0$ is $$S(t) = S(0)e^{(\mu - \frac{1}{2}\sigma^2)t}e^{\sigma \sqrt{t} z},$$ where $z \sim \mathcal{N}(0,1)$, a standard ...


5

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

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^...


4

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 ...


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 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

The CAPM is an economic theory that expected returns in excess of the risk free rate should be linear in the regression beta on the market. $$ \operatorname{E}[R_i - R^f] = \beta_i \operatorname{E}[R^m - R^f]$$ Graphically, it would look like this: As market beta increases, expected returns increase. Testing the CAPM with a cross-sectional regression ...


4

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 ...


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

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

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

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

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

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

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 ...


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

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

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 ...


3

Renaissance Technologies averaged 66% a year since 1998. The worst funds will lose all their capital and then some. A small time investor like you can expect to make not more than than buy and hold before fees. I advise you to read up on the efficient-market hypothesis, when you have digested that and you've come to the conclusion that it doesn't hold for ...


3

I tried a MonteCarlo simulation with 10000 iterations and it seems to confirm a probability of about 0.0033 However I wonder if we are interpreting the problem correctly, we are assuming infinite capital, perhaps we should take into account the gambling has to stop when you hit zero capital? Any other issues I may have missed? With the vig taken into account ...


3

When using arithmetic returns the right way to calculate an average is via the geometric average. The reason is that there is a multiplicative relationship between the returns. Example: Let $P_t$ denote the stock price at time $t$, then the simple (arithmetic) net return is defined as: \begin{equation} r_t=\frac{P_t-P_{t-1}}{P_{t-1}}=\frac{P_t}{P_{t-1}}-1 \...


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

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

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 ...


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