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.
