I have seen this estimated another way, which might be of interest to you ⦠(see âTrading & Exchangesâ, Larry Harris, Ch. 14, section headed âA simple timing option exampleâ) â¦
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 â
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
Aâs 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.
What is Aâs expected trade price?
= (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 â
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
Aâs limit order if
-
B has not already lifted
Aâs order, and
- pice has gone up (i.e. to
PRICE +
DELTA).
Now that D is in the market, what is Aâs expected trade price?
⦠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 insteadâ¦
So the expected price becomes â¦
= (
PRICE x
PROB ) + (1 -
PROB) x (((
PRICE â
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 â¦