# Forward Equity Curve Computation

I have been thinking about how forward equity prices are usually computed.
For the purpose of simplicity, let us take a share paying deterministic discrete dividends $$(D_i)$$ at times $$(T_i)$$ with a term structure of "zero coupon repo" factor denoted by $$R_t(.)$$.
To be clear:

$$R_t(T) = e^{-\int_{t}^{T}q(s)ds}$$

Of course I could use the below euation to answer my question:

$$P_t(T)F_t(T) = \left(S_t - \sum_{T_i \leq T} D_i P_t(T_i)\right)R_t(T)$$

• $$S_t$$, the spot price, is of course quoted in the market.
• $$(D_i)$$ are supposed to be deterministic.

The problem is:

• Where can I implicit the term structure of repo factor $$R_t(.)$$ ?
• Which yield curve to use for $$P_t(.)$$ ?

I called the "Call/Put parity" for help:

$$C_t(T, K) - P_t(T, K) = \left(S_t - \sum_{T_i \leq T} D_i P_t(T_i)\right)R_t(T) - KP_t(T)$$

The idea is to make a linear regression of $$C_t(T, .) - P_t(T, .)$$ for all available maturities in the market, we obtain:

$$\forall K, C_t(T, K) - P_t(T, K) = a_t(T)K + b_t(T)$$

and deduce:

$$R_t(T) = \frac{b_t(T)}{S_t - \sum_{T_i \leq T} D_i P_t(T_i)} (i)$$

And:

$$P_t(T) = -a_t(T) (ii)$$

The problem here is the last equation: The slope of the regression must be the same for all the shares (in the same currency let's say), is this true in practice ? I don't think so.

Of course I can force the $$P_t(.)$$ used to compute the forward price to be equal the yield curve of the share currency for instance and deduce $$R_t(.)$$ from $$(i)$$ but in doing so, I lose the call-put parity and arbitrage appears ! Otherwise I still have no clue how to get $$R_t(.)$$.

Do you have any ideas ?

Thank you very much for your help !

Regards.