# Adding Convexity with options

I understand when we add convexity we:

The bond with higher convexity will have higher price than the one with lower convexity. In a volatile market, this is beneficial and we want to add convexity.

An example, is adding convexity by Buying a call option on a bond.

If interest rates decrease –> we’re in better shape than the straight bond –>because we can call –> **Higher price using the higher convexity option.**

If interest rates increase –> How is that this same higher convexity bond has a higher price than the straight bond? I just see it as the options being worth nothing and approaches the same price as straight. **More specifically **what they get at is they say *total performance *of the portfolio with options is greater than portfolio without options. Why in rising environment would total performance be greater? I think it would just match the portfolio with options because the option is worthless.

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Total performance with bond with higher convexity will be higher because the price of bond with convexity decline less (and increases faster) than the price of a straight bond when rates increases (rate decreases). This is how higher convexity bond adds value.

Hope that helps.

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Nicely put Olajideanuouluwa!

It may be easier if you remember it as positive convexity as it was referred to in previous levels..

positive convexity = when rates drop price goes up

more, when rates go up price declineslessnegative convexity = when rates drop price goes up

less, when rates go up price declinesmoreBe as you wish to seem. Socrates

Your response Olajideanuouluwa

“Total performance with bond with higher convexity will be higher because the price of bond with convexity decline less (and increases faster) than the price of a straight bond when rates increases (rate decreases)”I understand the wording but not the logic. If we add call options –> WHY does the higher convexity bond (due to adding this call option) lead to the price of the bond declining LESS? I understand that higher convexity leads higher prices in a volatile interest rate environement.

A call option comes into play if yields drop and we can call it so performance is better. But with a YIELD increase –> what does this call option have to do with? The higher convexity bond with this option will perform the same as a straight bond. So where does this better performance come from in a increasing yield environment for this higher convexity bond with an option?

Your response FrankLiving

It may be easier if you remember it as positive convexity as it was referred to in previous levels..positive convexity = when rates drop price goes up more, when rates go up price declines lessAgain, I’m confused. I don’t know how this relates to my inquiry with options increasing convexity and how this works?

THanks guys,

when yield increase, the bond with call option will fare better than a vanilla straight bond, because you can call it and lend out your capital at higher rate.

when yield decrease, you do not want to call your bond with call option. the call option does not add value in this case.

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Victoryoe1984 is on point. Practically, if the yield declines, call option on bond with convexity increases more in value than low convexity option free bond and the bond could be called at the lower strike price leading to an increase in portfolio BPV. On the other hand, increasing rate will lead to lower bond value for both higher and lower convexity bond, but the decline in value of higher convexity bond will be lower. However, due to the decline in bond value, call option will expire worthless, and the portfolio value will only decline by the premium paid for the call option (if straight bond were purchased instead of the call option, the portfolio will suffer the full price decline).

Why do Barbell portfolios have higher convexity?

It was my understanding that the higher volatility of short-term maturities adds positive convexity to a barbell structure. Until I read Vol. 4, page 164:

“Because the convexity of shorter maturities is relatively small…”

Confused. So interest rates on the long-end are relatively stable, but they add convexity to a portfolio?

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If you look at the formula for convexity I’m pretty sure that you won’t as a factor P(Δ

y) (the probability of a yield change).Simplify the complicated side; don't complify the simplicated side.

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What I’d like to know (and maybe I’ll do some calculations on this myself) is whether you can change the convexity with options

without changing the duration.Simplify the complicated side; don't complify the simplicated side.

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From 2010

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