Estimating Convertible Bond Value Changes with Delta and Gamma

Summary

The document asks how to estimate the price impact on a convertible bond when its underlying stock falls sharply. It describes a second-order approximation using the bond’s delta for the linear component and gamma for the curvature component, then questions why this calculation predicts a larger loss than observed market behavior or a pricing calculator.

The example highlights a key limitation: delta and gamma are local sensitivities, so applying them over a large equity shock can produce an unreliable estimate. The text frames the convertible holder’s embedded equity optionality as relevant, but gives no answer, alternative valuation method, or supporting empirical analysis. A practical estimate therefore requires repricing under the shocked market inputs with an appropriate convertible bond model, including its other material features; the document itself does not specify those inputs or model assumptions.

Key ideas

• Delta estimates the local linear sensitivity of convertible bond value to its stock price.
• Gamma captures local curvature and contributes a second-order adjustment to a price estimate.
• A large underlying stock move can make a local delta-gamma approximation inaccurate.
• The document raises the discrepancy with observed prices but does not provide a resolution or a validated alternative.

Tags

# Convertible Bond Price change estimation using Delta/Gamma


# Convertible Bond Price change estimation using Delta/Gamma












The holder of a convertible bond is effectively long an American call option on the underlying shares.

given the delta and gamma, and underlying stock price change, what is the best way to estimate the CB price change?

example: delta 80% gamma 0.4 underlying stock price: 100 underlying stock price shock: -25% current CB price: 120 CB price with if underlying equity is down by 25% ?

it appears that linear approximation would give the below result:

120*80% * (-25%) + 0.5 * 120 * 0.4 * 25% * 25% = -22.5

so the new CB price under the shock scenario would be 120-22.5 = 97.5

but this new CB price is too low comparing to what happens in real world or using Bloomberg OVCV calculator. I guess it's because of the linear approximation doesn't work if the equity shock is too large.

any one has any good ideas to estimate the CB price/PNL change in this case?

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