equity linked notes (bull/bear equity performance bonds)

Question

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 1

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.