Implied Volatility for Convertibles with Conversion and Redemption Terms
Summary
The document considers how to infer the volatility implied by a convertible note’s issue price when the instrument includes early conversion, issuer redemption, and capped calls. A simple decomposition into bond value and option value may be inadequate once these contractual features affect when the note can be converted or redeemed. The proposed approach is to price the specific convertible under a model and solve for the single volatility that reproduces its market price.
For more consistent valuation across instruments, the answer suggests calibrating a volatility curve or surface to existing convertibles, then using it to assess the new issue. A numerical method such as a tree or partial differential equation can work backward from maturity while checking conversion and redemption conditions at each step. Stochastic interest rates are also possible, at greater implementation complexity. The discussion does not give a complete model, calibration procedure, or treatment of the capped calls’ separate cost, so those choices remain instrument- and model-dependent.
Key ideas
- A convertible’s implied volatility should be inferred from a model that represents its specific contractual terms.
- Early conversion and issuer redemption features require valuation methods that can evaluate exercise decisions over time.
- A tree or partial differential equation can work backward from maturity and check those conditions at each step.
- A volatility curve or surface calibrated to existing convertibles can provide a broader comparison than a single-instrument estimate.
- Stochastic interest rates may be included, but add implementation complexity.
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# Determining IV priced into convertible notes # Determining IV priced into convertible notes I'm trying to determine the IV priced into convertible notes at the time they were issued. Ideally, using python. Here are a list of the terms: It is my understanding that for a simple convertible note (coupon, and maturation date), I can assume: > par value ($1,000) = <bond value> + <option value> The bond value would be the principal discounted at some rate, as well as the sum of the coupons (also discounted at some rate). Which then allows me to solve for sigma in the option value: > <option value(sigma)> = $1,000 - <bond value> That's easy enough, except the convertible notes I'm looking at have a few other terms which may or may not alter the bond value and/or the option value, and so it's out of my depth. - Holder can convert earlier than maturity date - Company can request redemption, if stock trades at 130% the conversion price, for 20 out of 30 days. The redemption request would result in the note holders converting. There is also one last twist to this: - Capped Calls, and their cost. I'm not sure if it makes sense to compute IV with this factored in, or to compute the imputed IV of the capped calls after the fact. It is my understanding that they are essentially bull call spreads, long at <conversion price> and short at <cap price>.. but the above terms make this more complicated. ## Answer by marie albertini (score 1) https://quant.stackexchange.com/a/81630 you need to have a model to price the convertible with its specificities using a "single" volatility and then solve for the volatility that gives you the right price. in this case you get a so called "implied vol" per instrument a better way is to build a volatility curve or surface that prices correctly existing convertibles and then see how the new one prices using these vols ## Answer by marie albertini (score 0) https://quant.stackexchange.com/a/81651 in order to incorporate the specificities of the convertible, you need to have a numerical method that allows you to do that....some tree or pde...then work you way backwards from matutity and check at each step for your early redemptions, conversions... you can incorporate stochastic rates or not (this is obviously better, but nore complex to implement !) have a look at (section 3.1)....https://arxiv.org/pdf/2411.05425
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