Subtlety about options that I've never seen before

CFA General
#1

I’m writing a series of articles on option strategies (bull spreads, butterfly spreads, and so on) for Level III, and I came across a subtlety about calculating the profit from an option position that I’ve never seen in print before.

Generally, when we discuss the profit on an option position, we take the payoff on that option position and subtract the (net) cost of the options; the difference is the profit.

The subtlety that’s missing is that we pay the cost of the options today, but receive the payoff in the future. As anyone who has gone through Level I quant knows, you cannot add a dollar today to a dollar in the future: time value of money. To compute the profit properly, we should compute the future value of today’s cost for the options and subtract that future value from the payoff.

Now, you’re probably saying to yourself, “Who cares?”

Well, it turns out that at Level III it makes a difference. I was writing an article about bull spreads, and, in particular, about how a bull spread can be constructed with call options, or with put options. The payoffs will be very different, but the profit has to be the same, lest there be an arbitrage opportunity.

I was using a BSM model to price the options (to lend a soupçon of verisimilitude to my example), and when I computed the profit for the bull spread with calls to that with the bull spread with puts, there was a 10¢ difference. I was puzzled by this for a short while (I’m ashamed to admit), and then it dawned on me that there was a time value of money factor at play. Sure enough, when I adjusted the costs for time until expiration of the options (the bull call spread had a positive cost while the bull put spread had a negative cost), it increased the call cost by 5¢ and decreased the put cost by 5¢, and the profits were equal.

Now, you may say that 10¢ isn’t enough to worry about. But if you could do that with bull spreads on, say, 10,000,000 shares, that’s a cool $1 million.

Anyway, as I say: I don’t recall this subtlety appearing in print anywhere; in particular, I don’t recall seeing it in the CFA curriculum. I’ll pore over the Level III Applications of Derivatives book and see if it’s mentioned; if not, I’ll shoot CFA Institute an e-mail to broach the subject.

Just a head’s-up for y’all.

#2

Very interesting. Thanks for sharing that!

#3

Apparently you and I are the only two who think that it’s interesting.

Sigh.

#4

I’ve never seen anybody mention that either. Interesting stuff.

#5

It’s definitey interesting. I’ve heard it before in grad classes-- professors have pointed it out (and in general about people not accounting for TVM in situations where they probably should, at least theoretically). Keep us updated on what the institute thinks!

#6

I shot them an e-mail this weekend.

We’ll see what transpires.

#7

Here’s the reply I got:

Thank you for your query. Recall that profit generally has an inflow at a time different than the related outflow (sales less cost of goods sold even does this). You are correct that to figure out if it is worth doing, you should factor in the costs of financing the outflow or present value of the inflow.

If I were a bettin’ man, I’d say that they’re unlikely to mention this subtlety. Ever.

#8

If you use binomial pricing technique you explicitely discount the payoff.

Using BSM you also take into consideration the TVM but at risk-free. One of the assumptions is that you can borrow and lend at risk ree.

#9

Nice work S2000!

Feeling gutted for the test takers though as it’s something extra to remember when calculating P&L on options…

#10

As CFA Institute doesn’t seem willing to recognize it, it’s not something extra for test-takers to remember.

But it’s sure-as-heck something for options traders to remember. Before and after the test.

#11

Wouldn’t the discount rate already be reflected in the price of the option at the time of purchase though?

#12

Not in the way you’re thinking of it.

Consider this analogy:

You buy $100 worth of bubblegum today, hang onto it for a year (at which time bubblegum has become scarce, and precious), and sell it for $125. You want to compute your profit.

You might say that it’s $25 (= $125 − $100).

But if the risk-free rate were 5%, then you would have had $105 in a year if you hadn’t bought the bubblegum, so the improvement in your wellbeing from buying the bubblegum is only $20 (= $125 − $105).

CFA Institute says that on our income statement, we show a profit of $25.

I say that on our income statement we show a (net increase in) profit of $20: $25 from sales less COGS, and −$5 from the lost interest income.

#13

^thats good stuff s2k

#14

academic literature is always about making simplifications and generalizations… in reality/practice, many of these secondary effects net out.

the same could be said that you lose interest when waiting for a receivable…

#15

S2k youre a freaking freak of nature

#16

On the last line, I’m not sure I understand. Opportunity and implicit cost aren’t recognized in accounting, whether it’s options, inventory, or cash sitting in a vault, so why should they be in this case? For the option, it should show $25 gain, for $100 invested at the risk free rate of 5%, $5 interest income and for $100 in a vault, $0.

Otherwise, you’d get charged $5 COGS for letting cash/inventory/ect sit.

#17

If it’s not counted, then you’ll mistakenly believe that there’s an arbirtage opportunity between identical bull call spreads and bull put spreads, for example: the spread constructed with calls costs you money today and has a positive (or zero) future payoff while the spread constructed with puts generates cash today and has a negative (or zero) payoff. If you ignore the time value of money, you should be making money by the bagfull by buying the spread with the higher profit and selling the spread with the lower profit. In fact, all you’ll do is earn the risk-free rate.

#18

I think the cost should be considered when doing a dcf in the discounting but only on the income statement if actually borrowed money and incurred an interest expense.

if not then the guy with $100 in risk free treasuries, are you going to charge him a $5 for opportunity cost as well and show he has a net profit of $0? What about his accounting and tax books? Is the extra $5 COGS going to be tax deductible?

If so, if I’m in treasuries, do I get to deduct a cogs expense and pay no taxes? If I have a warehouse full of extra inventory, could I deduct a cogs since i could have instead invested the money in treasuries or options?