###### Implied Volatility

Implied Volatility We have seen that both the BSM model and Black model... **Read More**

Trade prices are compared to the benchmark price to compute the implicit transaction costs. The benchmarks used are the effective spread, implementation shortfall, and VWAP techniques of cost evaluation.

The effective spread is a measure of trading costs. It is taken as the difference between the price at which a market order is executed and the midquote price. It gives an overall estimate of the cost of trading using the midquote price as the benchmark price. It is expressed as:

$$ \begin{align*} & \text{Effective spread transaction cost estimate } = \\ & \text{Trade size} \times\left(\text{Trade price}-\left(\frac{\text{Bid}+\text{Ask}}{2}\right)\right) \end{align*} $$

And,

$$ \text{Trade size} \times\left(\left(\frac{\text{Bid}+\text{Ask}}{2}\right)-\text{Trade price}\right) $$

for buy orders and sell orders, respectively.

When a buy order is filled at the ask, the approximate implicit cost of the transaction is half the market spread. Therefore, \(\text{Ask}-\left[\frac{\text{Bid}+\text{Ask}}{2}\right]=\left[\frac{\text{Ask}-\text{Bid}}{2}\right]\). Further, the effective spread can be obtained by multiplying the midquote price benchmark transaction cost estimate by two.

**Price improvement** is realized when a buy order fills at a price lower than the ask price. In this case, there is a small spread.

Note, however, that the effective spread can be an inefficient measure of the transaction cost. This arises when traders divide large orders into smaller portions to fill in due time. Delay costs are, in such cases, not computed from the effective spread.

Consider two trades, X and Y, which fill a buy order of 6,000 shares. The following table gives a summary of the trade prices, trade sizes, the prevailing bid, and the offer of the trade.

$$ \begin{array}{c|c|c|c|c} \textbf{Trade} & \textbf{Trade} & \textbf{Trade} & \textbf{Prevailing} & \textbf{Prevailing} \\ {} & \textbf{Price} & \textbf{Size} & \textbf{Bid} & \textbf{Offer} \\ \hline X & 12.22 & 3,500 & 12.18 & 12.22 \\ \hline Y & 12.21 & 2,500 & 12.16 & 12.21 \end{array} $$

The effective spread transaction cost for a buy order is *closest *to:

$$ \begin{align*} \text{Effective spread transaction} &=\text{Trade size} \\ & \times \left(\text{Trade price}- \left(\frac {\text{Bid}+\text{Ask}}{2} \right) \right) \\ \text{For } X & \text{: } 6,000\times\left(12.22-(\frac{12.18+12.22}{2})\right) \\ & = 120 \\ \text{For } Y & \text{: } 6,000\times\left(12.21-\left(\frac{12.16+12.21}{2}\right)\right) \\ & = 62.5 \end{align*} $$

The volume-weighted average price (VWAP) is a measure of the average price at which a security is traded. It is computed as the sum of the total base currency value of the benchmark trades divided by the number of trades.

$$ \begin{align*} & \text{VWAP transaction cost estimate} \\ & = \text{Trade size}\times\left(\text{Trade VWAP}-\text{VWAP benchmark}\right) \end{align*} $$

And,

$$ \text{Trade size}\times(\text{VWAP benchmark}-\text{Trade VWAP}) $$

For buy orders and sell orders, respectively.

VWAP is easy to interpret. A trader either has a better or worse average price compared to other traders. Also, this measure is widely used because in case a large trader is the only buyer for a particular trading period, it would remain zero despite the market impact.

Nonetheless, the VWAP transaction cost estimates are inefficient when the trades took place at the same rate as other trades in the market. Similarly, VWAP transaction cost estimates will be inefficient when trades being executed are a fraction of all deals in the VWAP benchmark.

A company’s manager wishes to sell 9,000 shares of an asset. An order to sell 3,000 shares executed at $15.12 is submitted. At the time the order is submitted, the price is a $15.11 bid for 4,000 shares, and the offer is made at $15.16 for 3,000 shares. Further, another decision to sell 5,000 shares executed at the cost of $15.14 is made. At the time the order is submitted, the price is $15.15 bid for 4,000 shares, and the offer is made at $15.18 for 3,000 shares. Calculate the VWAP.

$$ \begin{align*} VWAP & =\frac{\sum\left(\text{Price}\times \text{Volume}\right)}{\sum \text{Volume}} \\ \text{Total volume} & = \left(3,000\times$15.12\right)+\left(5,000\times$15.14\right)=$121,060 \\ VWAP &=\frac{$121,060}{8,000}=$15.13 \end{align*} $$

## Question 1

Consider a sell order of 6,000 shares to two clients. The following table gives a summary of the trade prices, trade sizes, the prevailing bid, and the offer of the trade.

$$ \begin{array}{c|c|c|c|c} \textbf{Trade} & \textbf{Trade} & \textbf{Trade} & \textbf{Prevailing} & \textbf{Prevailing} \\ {} & \textbf{Price} & \textbf{Size} & \textbf{Bid} & \textbf{Offer} \\ \hline X & 12.19 & 3,500 & 12.23 & 12.22 \\ \hline Y & 12.18 & 2,500 & 12.22 & 12.23 \end{array} $$

The effective spread transaction cost for a sell order is

closest to:

- X=210; Y=270.
- X=122.5; Y=112.5.
- X=3500; Y=2500.
## Solution

The correct answer is A.$$ \begin{align*} \text{Effective spread transaction} &=\text{Trade size} \\ & \times\left(\left(\frac{\text{Bid}+\text{Ask}}{2}\right)-\text{Trade price}\right) \\ \text{For } X & \text{: } 6,000\times \left(\left(\frac{12.23+12.22}{2}\right)-12.19\right) \\ & =210 \\ \text{For } Y&\text{: } 6,000\times\left(\left(\frac{12.22+12.23}{2}\right)-12.18\right) \\ & =270 \end{align*} $$

## Question 2

A company’s manager wishes to sell 9,000 shares of an asset. An order to sell 3,000 shares executed at $15.12 is submitted. Upon order submission, the price was $15.11 bid for 4,000 shares, and the offer was made at $15.16 for 3,000 shares. Another order to sell 5,000 shares was executed at the cost of $15.14. Upon submission of the order, the price was $15.15 bid for 4,000 shares and the offer was made at $15.18 for 3,000 shares, and a final order is made for the remaining stock at $15.14 after the trading period.

The VWAP transaction cost estimate is

closest to:

- $15.13.
- $7.67.
- $7.20.
## Solution

The correct answer is C.$$ \begin{align*} \text{VWAP } & \text{transaction cost estimate} \\ & = \text{Trade size} \times(\text{VWAP benchmark}-\text{Trade VWAP}) \\ VWAP & =\frac{\sum\left(\text{Price}\times \text{Volume}\right)}{\sum \text{Volume}} \\ \text{Total Volume}& =\left(3000\times$15.12\right)+\left(5,000\times$15.14\right) \\ & +\left(1,000\times$15.14\right) \\ & =$136,200 \\ VWAP &=\frac{$136,200}{9,000}=$15.1333 \end{align*} $$

Using the sales sold by the company only, the VWAP is:

$$ \begin{align*} \left(3,000\times$15.12\right)+\left(5,000\times$15.14\right) & =$121,060 \\ VWAP & =\frac{$121,060}{8,000}=$15.1325 \end{align*} $$

VWAP transaction cost estimate is:

$$ 9,000\times\left($15.1333-$15.1325\right)=$7.20 $$

Reading 46: Trading Cost and Electronic Markets

*LOS 46 (b) Calculate and interpret effective spreads and VWAP transaction cost estimates.*