Understanding the Greeks

April 29, 2021

9:39 am

For our final discussion in this series, we hope to demystify the Greek letters that are key to understanding convertibles. Don’t worry, we will take it easy on the math, and there won’t be a test!

In our earlier discussions, we covered how the Black-Scholes option pricing formula revolutionised finance. The formula uses only a few inputs but is robust enough to generate consistent and predictable results. Based on this formula, investors can calculate sensitivity analysis to the price of their option.

We follow the practice of using Greek letters to stand for a variable, basically, how sensitive an option, or a convertible bond, would be to those key inputs. The Greeks can therefore be used to help assess both risk as well as potential reward.

Of course, these measures can also be calculated for a portfolio of convertible bonds, as well as for a single holding. There is some more math involved to get to the precise definition of each measure, but we will try to present a more intuitive view of what each of the Greeks means for investors.

Our review will cover:

  • What are Delta and Gamma?

  • What are Rho and Omega?

  • Are there other Greeks?

  • Why can convertibles have a negative yield to maturity or put?
What are Delta and Gamma

For most investors, understanding these two letters is about as much Greek as you need to know to understand convertibles. That is because they capture the sensitivity of a convertible to the price of the stock into which the bond converts, which usually drives most of the price change of a convertible itself.

You may also remember from our earlier discussions that we prefer mid-delta, or “balanced” convertible bonds with high gamma. That is because these convertibles usually offer the best ratio of upside capture versus downside risk, using a scenario where the underlying stock price moves both up and down an equal percentage change.

Let’s look at what each of these Greek letters means:


Expressed as a percentage from 0% to 100%, delta captures the change in the price of the convertible bond for a percentage move in the underlying stock price. A higher delta means a higher sensitivity, whereas a lower delta convertible bond will not move much in price as the underlying stock price changes.

With that said, delta is not static, and it will change as the underlying stock price changes. It is accurate for small changes in the underlying stock price, but you need the next Greek (gamma) to work out the price change for a large move.


This measures the rate of change of delta itself. Basically, for a long-only convertible investor, as a stock price moves higher, a higher gamma convertible bond should see its own delta increase quickly. This means that the bond can earn more from potential equity upside than a lower gamma instrument might.

Also, gamma can serve as a proxy for convexity, which is the ratio of upside vs. downside for a convertible bond if the underlying stock price were to move up and down by a similar percentage. For a long-only investor, more convex convertibles are generally preferable to those that are not. Gamma is shown as a whole number. It is enough to know that for potential convexity, a gamma of more than 0.75 would be considered good, and more than 1.0 would be quite good.

What are Rho and Omega?

Rho and Omega help us to understand the price sensitivity of a convertible bond to interest rate risk (duration) and the credit risk of an issuer.

Convertibles have an embedded option, but they are also bonds. And while the price of an option changes (usually only slightly) when interest rates move, the calculation in the Black-Scholes formula only needs a risk-free rate and doesn’t factor in expected credit risk of an issuer.

Here is how these Greeks help us to understand bond-specific sensitivities:


Interest rate changes affect the price of both the bond component and the option component. Rho shows the total sensitivity of a convertible if all interest rates were to shift equally, that is, by the same amount across the yield curve.

Generally, higher interest rates have the greatest impact to a convertible in driving down the value of the bond component, but with lower interest rates pushing up that value. Higher duration bonds are affected to a greater degree by changes in interest rates vs. short duration bonds. The longer you must wait until maturity, the higher the duration of your bond.

Rho is shown from -1 to 1, which represents its sensitivity/participation rate to shifts in the yield curve. Rho itself shows the change in convertible price for a 1bpsparallel shift on the yield curve.


As a credit spread of an issuer changes, the value of the bond component will change. This measure shows the effect on a convertible’s price if credit spread alone were to change by a standard amount. Omega can also be referred to as Omicron.

Omega is shown from -1 to 1, which represents its sensitivity/participation rate to moves in credit spreads. Omega is the change in convertible price for a 1 basis point move in credit spread.

Are there other Greeks?

There are, and they are good to know, but less critical to understand. The majority of price moves for a long-only holding in a convertible bond will be explained via delta and gamma, as well as rho and omega.


Remember that the option to convert into equity will rise in value as volatility increases, but it will fall if volatility were to decrease. Vega shows the forecast change in the convertible price as volatility changes by a standard amount. Higher vega convertibles are more sensitive to changes in volatility than a lower vega convertibles.

Vega is the change in convertible price for a 1% move in volatility.


Imagine if the value of the option to convert was divided among the number of days remaining in its life. If there was a trading day where the underlying stock closed below the strike price, you wouldn’t choose to convert, and your option therefore lost that day’s value.

Theta is a standardised measure that if negative shows the daily loss in value to the convertible bond and is also referred to as time decay.

Generally, all convertibles with a call option have a negative theta. Theta changes as the other Greeks change, and as the time to maturity draws nearer.

Theta is shown from -1 to 1, which represents sensitivity/participation rate to time decay. Theta is shown as the change in convertible price per day based on time decay.

There are two additional Greeks that show sensitivity to changes in the underlying stock dividend (phi) and recovery rate on the bond component (upsilon). They matter, but are less crucial to know than the other Greeks that we have reviewed.

Why do some convertibles have a negative yield?

It used to be that negative yields were uncommon for bonds. But while having a negative yield is less of a novelty for a convertible bond, let’s look at why this happens, and what the implications may (or may not be) for investors.

In general, bond investors receive coupons with a final repayment of principal. The final return from a bond depends on when they get their principal back from an issuer, and bonds have different features that can impact the timing and value of repayment. For example, not all bond investments are held to maturity, especially if an issuer or an investor can force an early repayment of principal through a call or put option.

But for convertible bond investors, there is another difference to consider, and that is because holders have the right to convert a bond into shares of stock. Because of that conversion option, the value of the bond if converted into stock can be higher or lower than the par amount of the bond that the issuer must repay at maturity. It just means that some of the traditional yield measurements for fixed income might not fully reflect the value of that option.

As an example, we may hold a convertible where the underlying stock price has risen in value by a moderate amount, and we still have a positive view on that underlying stock. If the yield to maturity or put of the convertible has become negative, does that mean that our investment is now going to earn a negative return? Do we need to convert the bond to lock in our gains, which is what the negative yield seems to suggest?

Looking at the Greeks, if we see that the delta of the convertible remains moderate (say, no more than 0.75) and that gamma of the convertible is still good (above 0.5), we still have a good ratio of reward to risk. So long as we believe that it is likely that the underlying stock stays above a level where it makes more sense to convert than to wait for repayment at maturity, that would still be an attractive convertible, even if the level of yield is negative.

The negative yield simply tells us the received return if we waited for maturity and chose not to convert the bond into stock. There is also no need to choose to convert before maturity – options always have time value! We can always choose to sell our holding in this hypothetical bond before it matures.

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