What Proportional Bid-Ask Spread Data Can and Cannot Determine
Summary
The exercise asks for distributions of proportional bid-ask spreads for two shares, given quoted bid and ask prices and a stated normal distribution for the dollar spread. The questioner calculates proportional spread as the dollar spread divided by the midpoint and then seeks the dispersion needed for a stressed liquidation-cost estimate. The included answer says the distribution cannot be derived from the supplied figures alone and illustrates a standard deviation by treating the two calculated proportional spreads as equally likely observations.
That calculation describes variation across the two quoted assets under an equal-weight assumption; it does not derive the time-series distribution of either asset's proportional spread. The document supplies no observations or model connecting the random dollar spread to a random midpoint, so the proportional-spread distributions and liquidation-cost inputs are underdetermined. The example highlights why the intended population, probabilities, and spread denominator matter. Its equal-probability calculation is conditional on an assumption, rather than an estimate supported by the exercise data.
Key ideas
- A proportional spread is computed by dividing the dollar spread by a reference price such as the midpoint.
- The stated dollar-spread distribution alone does not specify the distribution of proportional spread when the denominator can vary.
- The answer calculates dispersion across the two assets only by assuming they are equally likely.
- That cross-asset dispersion is not automatically the standard deviation for either asset's liquidation-cost model.
- A stressed liquidation estimate requires a clearly defined spread distribution and relevant dispersion inputs.
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Full text
# Distribution of proportional bid-ask-spreads
# Distribution of proportional bid-ask-spreads
I already asked this yesterday at "Economics Stack Exchange" but think this question might be better suited here. In the meantime i really tried to solve it by myself, but couldn't find anything what might help me. It's not just about the solutions, i do really want to understand how to solve problems like this.
"I am currently studying for my upcomming exams. There is an exercise i can't solve or even understand properly.
The full exercise is: "You bought 100 shares of company A and 200 shares of company B. The shares of A are bid \$50 and ask \$60, while the shares of B are bid \$25 and ask \$35. The bid-ask spreads of both A and B are normally distributed with mean \$10 and standard deviation \$3.
Determine the distributions of the proportional bid-ask spreads for A and B."
I already got the proportional bid-ask-spread for A and B by the formula $s_{p}(X) = \frac{ASK - BID}{MEAN}$. Therefore $s_{p}(A) \approx 0.18$ and $s_{p}(B) \approx 0.33$.
Now i need to calculate the distributions of those spreads. (The actual aim of this exercise is to calculate the cost of liquidation in a stressed market.)
I'm not quite sure what is meant by "distribution", so i assume it's the mean and standard deviation of those spreads. I just can't get my head around the standard deviation, since i need at least two values to calculate the standard deviation. (as far as i undersand) But i don't have more than one value for each spread.
How do i have to solve this exercise? Like, is there a general way of doing it?"
UPDATE: I need these results to calculate the "cost of liquidation in a stressed market". I read that one needs to use the following formula for this.
$\sum_{i=1}^{n} \frac{1}{2}(\mu_i + \lambda_i \sigma_i)\alpha_i$, where
$n=2$,
$\mu_i = s_p(X_i)$, so in my case $\mu_1 = 0.18$ and $\mu_2 = 0.33$,
$\alpha_i = \text{volume } X_i$, in my case $\alpha_1 = 100$ and $\alpha_2 = 200$,
$\lambda_i = \text{confidence-level}$, like $\lambda_1 = \lambda_2 = 2.33$ for $99\%$ confidence-level and finally
$\sigma_i = \text{that "distribution" (standard-deviation?) value i can't calculate}$.
Maybe this can describe my problem in more detail.
## Answer by Neeraj (score 0, accepted)
https://quant.stackexchange.com/a/24496
You can not derive the distribution of proportional spread with the information given in your question. You have given $S_p{(A)}$ and $S_p (B)$. By assuming equal probabilities for both, you can simply calculate standard deviation of proportional spread as: $$Var(S_p)=E[(S_p-\mu)^2]$$ So, $\mu = 0.5*0.18 + .5*.33 = 0.255$, and $$Var(S_p)=.5(0.18-.255)^2 + 0.5(.33-.255)^2=0.005625$$ $$\sigma=\sqrt{Var(S_p)}=0.075$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.