daily 'option' cost of having a stock limit order roughly right ?

Discussion in 'Automated Trading' started by bidask201, Feb 10, 2013.

  1. One way of looking at -a- key cost of making a market with passive limit orders is you are selling an option for free.

    A very rough calculation it seems is just to use Black Scholes with the strike price being the 1+spread/2. Lets say you are making a market all day (but keep in mind right now we are not looking at the number of times you earn the spread just that you are out there trying) I have read 2/3 of volatility is during the day. so t is roughly 2/3 * 1/250. Lets say volatility is 25% and interest rates are zero. I get a daily cost of 37 basis points on a stock with 67 basis points of spread, or 51bp on a stock with close to zero spread.

    I know that a discrete model should be used for this but is this number completely out of the ballpark ?
  2. This should be " ... * 1/(SQRT(250)) ...", I think ...
  3. I have seen this estimated another way, which might be of interest to you … (see “Trading & Exchanges”, Larry Harris, Ch. 14, section headed “A simple timing option example”) …

    I am rephrasing it for the case you gave, but essentially it goes as follows:

    Assume there are four traders (A, B, C and D) in a given market where the current price is PRICE, and where in time T, the asset's price can only do one of the following, go to PRICE – delta, stay at PRICE, or go to PRICE + delta, each of which it can do with equal probability (i.e. of 1/3).

    Trader A is a market maker who places a limit sell order at price PRICE.

    Trader B is an impatient trader who may show up and will use a market order to trade against A’s limit order; and B has a probability of PROB of showing up in time T.

    C is another market maker who places limit buy orders, and who is always willing to trade, but only at a price which is a price SPREAD below the market.

    During time T, A will trade with B if B shows up, and will otherwise trade with C.

    = (price if trades with B * probability of trading with B) + (price if trades with C * probability of not trading with B)
    = (PRICE x PROB ) + (1 - PROB) x (((PRICE – DELTA - SPREAD) x 1/3) + ((PRICE - SPREAD) x 1/3) + (((PRICE + DELTA - SPREAD) x 1/3))

    = (PRICE x PROB) + (1 - PROB) x (PRICE - SPREAD) -------------[XXXXXXXX]

    + + + + + + + + + + + + + + + +

    D is a fourth trader, who will trade opportunistically using a market order against A’s limit order if
    - B has not already lifted A’s order, and
    - pice has gone up (i.e. to PRICE + DELTA).

    … The difference is that A no longer has a chance to make a price of (PRICE + DELTA - SPREAD), because in that situation D will trade with a A at PRICE instead…

    So the expected price becomes …
    = (PRICE x PROB ) + (1 - PROB) x (((PRICE – DELTA - SPREAD) x 1/3) + ((PRICE - SPREAD) x 1/3) + (((PRICE x 1/3))

    = (PRICE x PROB) + (1 - PROB) x ((PRICE - (SPREAD x 2/3) - (DELTA x 1/3 )) -------------[@@@@@@]

    + + + + + + + + + + + + + + + +

    The difference between [XXXXXXXX] and [@@@@@@], namely (1 - PROB) x (1/3) x (SPREAD - DELTA), can be thought of as a measure of the cost to A of the option A is offering to D.

    Just to plug in some numbers… Assuming that you are looking at a time T that is long enough for DELTA to be say 2 x SPREAD, and using your spread of 67 bp, and assuming that PROB (i.e. the probability of B showing up) is say 50%, I get cost of option = (1/2) x (1/3) x 67bp = approx 10 bp.

    Obviously, the model is overly simplistic, and plug in different numbers and you get a different result (plus the above doesn't work for your case of "zero spread"), but … if you take the above as being an approximation to the option cost (value) over a time T that is long enough for price to move 2 x SPREAD, you ought to get a much, much bigger (several orders of magnitude, I'd have thought?) cost (value) if you left the limit order in place for a whole day (where price has the potential to move much more than just 2 x SPREAD.

    Therefore, the daily costs you have calculated seem too low to me (possibly even if you correct for 1/(SQRT(250)) factor …