Ratio Call Writing
a topic suggested by Robert P.

Here's an interesting strategy, suggested to me via e-mail and ...

>And you didn't know nothin' about it, eh?
Well ... uh, no, but I'm willing to learn. Now pay attention:

  • We buy some stock, say, Sh = 100 Shares.
  • Suppose that we pay S = $40 for the Shares, so our investment is S *Sh = $4000.
  • Now we write N = 2 Call Option contracts, with a striKe price of K = $30.
  • The Call premium for these options is, say, Cp = $11.00 so, selling the contracts, we get
    100*N*Cp = 100*2*11 = $2200 (where that 100 is because each contract involves 100 shares of stock).
>But if your options are called, you have to provide 200 shares of stock, but you only bought 100 and ...
Pay attention:
  • So far, we've spent S *Sh = $4000 on stock and received 100*N*Cp = $2200 for our Calls.
  • Our total out-of-pocket Cost = S*Sh - 100*N*Cp = $4000 - $2200 = $1800.
  • Our options may be called at any stock price greater than K = $30 (and will definitely be called if the stock price increases to K + Cp = $30 + $11 = $41   or above).
  • Suppose that, at some time in the future, the STock increases to, say, ST = $45 and our options get called.
  • We now buy enough additional stock to cover the Call, namely 100*N - Sh = 200 - 100 = 100 shares.
  • Unfortunately, we must now buy these 100 additional shares at the current price, namely ST = $45,
    at a cost of (100*N - Sh)*ST = $4500.
>Wait! So far you're out-of-pocket by ... uh ...
  • So far it's cost us S*Sh - 100*N*Cp = $1800
    plus this additional cost of (100*N - Sh)*ST = $4500,
    for a total of S*Sh - 100*N*Cp +(100*N - Sh)*ST = $1800+$4500 = $6300.
  • We then sell the 100*N = 200 shares (called for by the options) at the strike price of K = $30,
    and that gives us 100*N*K = 200*30 = $6000.
>And you're down by $300, right?
$6300 - $6000 = $300. Right.

>And you're suggesting this strategy? I mean ...
Patience.

The final Gain or Loss is: 100*N*K - {S*Sh - 100*N*Cp +(100*N - Sh)*ST}

which can be written:

(1)     Gain (or Loss) = Sh*(ST - S) + 100*N*(K + Cp - ST)   if Options are called

>And if the option isn't called? If the stock drops below ...?
Patience. I was getting to that.

  • If the Option isn't called and the STock price ends up, at expiry, at ST = $25, for example, then our Sh = 100 shares are worth just Sh*ST = 100*25 = $2500.
  • Remember our Cost? It's Cost = S*Sh - 100*N*Cp = $1800.
  • Our Gain is then Sh*ST - Cost = Sh*ST - {S*Sh - 100*N*Cp} = $2500 - $1800 = $700.

>It's about time you made some money!
Okay. Our final result is this:

(2)     Gain (or Loss) = Sh*(ST - S) + 100*N*Cp   if Options are NOT called

>And what if ...?
Here's a spreadsheet. You can answer your "what if" questions ... all by yourself:

After you type in your parameters, the spreadsheet provides the Black-Scholes Call Premium, if you want to play.
(It's the Black-Scholes premium which requires the Volatility V and Risk-free Rate Rf ... else you won't need these parameters.)

>What's that "Winning Width" chart?
We'll get to the Winning Width later.

>Yeah, so, how do I get the spreadsheet?
Ah, yes, You RIGHT-click on the picture above ... and Save Target.

>I assume the spreadsheet does what you did above, eh?
Yes, except that ...

instead of using 100*N - Sh as the additional shares needed to cover the N Call Options, we use MAX(100*N - Sh,0) because we may have bought Sh = 500 shares (instead of 100, as we assumed above) so we don't need to buy any more to cover the Call and, in fact, after covering our Call with 100*N shares, we'd still have Sh - 100*N left over and, if the STock Price is, say, ST = $45, then these leftover shares are worth (Sh - 100*N)*ST.

Also, we set the leftover shares as MAX(Sh - 100*N,0) so that, unless Sh is larger than 100*N, we won't have any leftovers and if there are leftovers, they'd then be worth MAX(Sh - 100*N,0)*ST and that'll change our Gain formula (1), above (if the Options are called) to:

(1A)     Gain (or Loss) = 100*N*(K +Cp) + MAX(Sh - 100*N,0)*ST - {S*Sh +MAX(100*N - Sh,0)*ST}

However (surprise!) the two terms MAX(Sh - 100*N,0) - MAX(100*N - Sh,0) turn out to be simply Sh - 100*N so we have (finally!):

(1B)     Gain (or Loss) = 100*N*(K +Cp) + (Sh - 100*N)*ST - S*Sh

>Mamma mia! Am I supposed to follow all that stuff?
Of course not. Just look at the pictures.

>But the picture of the spreadsheet says you lose if the stock goes up or down.
I guess this neutral strategy is for those who think the stock won't move too much, that the distribution of future stock prices is centred on the current price.

Altogether now:
Total Gain/Loss = Sh*(ST - S) + 100*N*Cp     if Options are NOT called
= Sh*(ST - S) + 100*N*(K + Cp - ST)   if Options are called
where   Sh = Number of shares purchases
S = Price of shares purchased
N = Number of Call Options written
K = Strike Price of Call options
Cp = Call Premium
ST = Stock Price (if and when Options are called)

and the Option will NOT be called for ST < K

Note the slope of the chart of Gain/Loss versus ST ... the Slope is the coefficient of ST:

Slope = Sh if the Option is NOT called ... so ST < K.
Slope = Sh-100*N if the Option is called ... with ST > K.
The Slope changes at ST = K where the Gain is Sh*(K - S) + 100*N*Cp.

For our example, the slope changes from +100 to -100 at ST = $30
where the Gain is 100*(30-40)+100*2*11 = $1200.

We'd like to arrange things so the range of winning stock prices is large.
That is, we make money for a large range of stock prices.
That is ...

>So do it!
Okay. Look at Figure 1.
It's a picture of the Gain/Loss chart with H = Sh*(K - S) + 100*N*Cp and Slope A = Sh and Slope B = Sh - 100*N
and we have, identifying the Slopes:
H/(K - Min) = A so Min = K - H/A, and
H/(Max - K) = -B so Max = K - H/B, and
Max - Min = H (1/A - 1/B) or
Max - Min = {Sh*(K - S) + 100*N*Cp} {1/Sh - 1/(Sh-100*N)}
That's the width of the Stock Price interval where we'd win:
the Winning Width.

For our example, this would be:
{100*(30 - 40) + 100*2*11} {1/100 - 1/(100-100*2)} = $24.

Now suppose we write N = 2 contracts and buy Sh = 100 shares.
We'd have:
Max - Min = {100*(K-S) + 200*Cp}{1/100+1/100} = 2(K-S)+4Cp

If we check here, we find that the Black-Scholes Call Premium is given by:

Cp = S* N[d1] - K* EXP(-Rf*T)* N[d2]

where d1 = {log( S/K)+ (Rf+ V2/2)* T} / {V*SQRT(T)}
and d2 = d1 - V*SQRT(T)
and N[ ] denotes the Cumulative Normal Distribution
... as in Figure 2


Figure 1


Figure 2


>What's that T thing?
Uh ... T is the number of years to expiry ... for the option.

If we use Black-Scholes to calculate Cp, then, for a fixed T (time to expiry) and Volatility and Rf (the Risk-Free Rate) and current Stock Price, we can run through a bunch of Strike Prices, K (as a percentage of the current Stock Price) and see what's that Winning Width.
In fact ...
>In fact, that's that extra chart on the spreadsheet, right?
Right. You'll notice (from the picture of the spreadsheet), that although the Gain is a maximum when the Stock Price equals the Option's Strike Price, it also the gives the narrowest Winning Width.


>How 'bout a real, live example that ...
Here's a real example ... an old one, taken from an old tutorial:

The stock is selling for S = $65 and we buy 100 shares and sell 2 contracts with a strike price of K = $5 3/8, let's say $5.38 and ...
>What's Black-Scholes say?

If we pick appropriate V and Rf (namely 31% and 6%, respectively) we also get $5.38, from Black-Scholes.
The maximum Gain is $577 (with an initial Cost = $5423) ... if the stock ends up at the Strike Price of $60.

For the above real, live example, the option expires in 17 days and the Width is 18% of the stock purchase price of $65 ... that's a Width of about $12 which gives a winning range of Stock Prices from $54 to $66.
>Wait! Where are all these numbers coming from?
From the spreadsheet. Whatchya think?

Anyway, if we change to, say, 150 days to expiry, we'd get a smaller initial Cost = $4744, a larger maximum Gain of $1256 and a larger Width of 38.6% of $65 (that's a Width of about $25, in dollar terms) with a winning Stock Price range from $47.44 to $72.56 and ...

>If the stock price really takes off, you could lose big time, eh?
Yes, see Figure 3 for an example.

Of course, you could buy more shares, then you'd have a covered call and maybe you'd buy the extra shares - to cover the call - when the stock price starts getting too high so that ...


Figure 3

>Your example is from 1999. Can't you find a more recent ...?
Okay. Let's look at Microsoft, on July 9/02 and consider options which expire in October - 101 days - and we see the following:

Let's look at options with a Strike Price of $45, with a premium of, say, $11 and suppose that ...

>Why not just show a picture?

Here's a picture of the spreadsheet:

>You're assuming a 45% volatility? You kidding?
Well ... uh, that's just to see what volatility will give a Black-Scholes close to $11.
But I didn't use the $10.97 Black-Scholes estimate suggested in the spreadsheet (at cell H3).
I used $11.00 (in cell C4) and ...

>Wait! Can I ask a question?
Sure.
>This strategy is called Ratio Call Writing?
Yes.
>Why?
I have no idea.
>I'd say it's because of the ratio: options sold and stock bought.
Sounds good to me.

    See also Price Distributions