# reading 20: Yield Curve Strategies: Call Option

Hi all,

Part of reading 20, section 4.4 Using Options

“The PVBP of the call option is substantially less than that of the 30-year bond. We will therefore need more par value of the option than of the bond it replaces if we are to maintain the portfolio duration. To derive the needed par value of the option, we multiply the par value of the bonds we are selling by the ratio of the bond’s PVBP to the option’s PVBP: 0.2113/0.149 = 1.418. Thus, we sell 6,800 par of the 30-year bond and buy 9,640 underlying par amount of the option (approximately equal to 6,800 × 1.418). The resulting portfolio is shown in Exhibit38; with equivalent market value and duration, it has an effective money duration equal to that of the pre-trade portfolio.”

I understand we want to keep the same Effective duration as initially (7.82) and same total MV (59,720).

How is the quantity (6800) of US 30Y bonds to be sold computed?

Anyone?

# of options / option delta

I should know that!

I agree.

Possible to elaborate on this? Am not able to see why it must be precisely 6,800 par of 30-year bonds sold. When I built the example out in excel, I was able to sell any amount of 30-year bonds and maintain an effective duration of 7.82. Maybe there is another constraint that we’re trying to satisfy that I’m not picking up on?

Am not sure what you’re referring to by “# of options”, as it looks like the # of options purchased is calculated from the amount of 30-year bonds sold.

The condition is to maintain the money duration of the portfolio pre- and post-trade (MV = 59,270 and EffDur = 7.82).

It is not just maintaining EffDur = 7.82.

In addition to EffDur, we also maintain market value regardless of how much 30-year bond we sell, because the proceeds go toward calls (which are purchased and listed at market value in the portfolio), with the left-over amount represented in the portfolio as cash.

For example, here’s the portfolio if we sell 5,000 par of 30-year bonds:

Your computation only involves purchasing 7,091 call options. But the objective is to increase the convexity of the portfolio, I guess your post-trade portfolio convexity is less than 5.952 (based on 9,640 call options)?

CFAI curriculum & learning replied to my query about the quantity of 6800 bonds, as I tried to understand how to compute it:

"The focus is on how portfolio convexity can be increased by adding call options and selling long (30-year) bonds. Please note that the amount of long-bonds to sell, here 6,800 par, should be taken as a “given”. The main point is that 6,800 x (0.2113/0.1490), the ratio of PVBPs of long bonds to call options, is the amount of calls, 9,640, that must be purchased to keep the portfolio’s duration (7.82) and market value (59,270) constant. The result is that portfolio convexity increases dramatically from 1.276 to 5.952. We hope this helps and good luck with your studies. "

I will need more than ‘good luck’ to pass L3…

Thanks Eugene - it’s what I suspected, the number is used to illustrate an example and shouldn’t be treated as the ONLY amount that must be sold.