Pricing Equity-Linked Coupons as Bonds Plus Options

Summary

The document considers bull and bear equity performance bonds that pay periodic coupons linked to stock returns, subject to a minimum coupon. It raises a modeling ambiguity: the coupon might be a minimum plus a positive performance component, or the maximum of a floor and a multiplicative return-based amount. The answer questions the multiplicative interpretation and frames either payoff as a bond combined with a forward-starting option.

For a simple valuation, it suggests using Black–Scholes with volatility over the relevant forward period. This offers a tractable way to price the option component, but it does not account for changes in implied volatility across strikes and maturities. A more complete approach would use a persistent implied-volatility skew surface. The response expects the simplified method may suit an introductory assignment, but supplies no worked valuation, calibration details, or treatment of issuer credit risk and other note terms. The payoff definition should therefore be settled before choosing a pricing model.

Key ideas

• The coupon definition must be clarified because alternative formulas produce different payoffs.
• The note can be decomposed into a bond component and an option linked to equity performance.
• Black–Scholes with forward-period volatility provides a simplified pricing approach.
• A persistent volatility skew surface can better represent the underlying option market.

Tags

# equity linked notes (bull/bear equity performance bonds)


# equity linked notes (bull/bear equity performance bonds)












I have to price what my lecturer calls "Bull and Bear Equity Performance Bonds". Basically there's dates $t_i \in [0,T]$, where $t_i - t_{i-1}$ is the same for all choice of $i$. On each date the bull bond will pay coupon $C_i := max\{C_{min},C_{min}(1+R_i)\}$, where $R_i = \frac{S_t}{S_{t-1}}-1$, $S_t$ is the stock price process. The coupon for the bear bonds are similarly defined.

Do any of you know any references/books that deal with the pricing of this derivative? I went through the indexes of 20+ derivatives books in my library and could find 0 mentions of equity-linked notes, bull/bear performance bonds. Best I found was a 1 page qualitative dicsussion of "notes".

Thanks.

## Answer by Strange (score 1, accepted)

https://quant.stackexchange.com/a/4233

Just to clarify, the periodic coupon is $C_{\min} + \max(0, \text{perf}_i -1) $ or is it actually $\max(C_{\min}, C_{\min}\cdot(1+\text{perf})) $? I don't think the multiplicative version makes sense.

In either case, it's a bond plus a forward striking option. The simple solution is to price it using Black Scholes with forward volatility $\sigma(t_i,t_j)$. This way, however, you will ignore the forward skew issue, but given that it's ATM, the correction is going to be fairly small. The "proper" way is to build a persistent skew surface for your underlying and price it using persistent skew, but I doubt your lecturer is actually asking for that level of detail.

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