How equity products work

An equity option and an equity total return swap both read a share's quoted spot, but they answer different questions. An option prices the right to buy or sell the share later, so it needs a forward and a volatility. A total return swap passes the share's return from one side to the other, so it needs only today's price and what the period started at. This page explains both models and what each one leaves out.

What the market supplies

The market holds the trade currency's discount curve, each share's quoted spot, set with with_equity_spot, and, for options, the share's vol surface by expiry and strike, set with with_equity_vols. Dividends and the prices a later swap period starts from arrive as fixings: DIV-<name> and EQ-<name>. Building them is covered on equity spots and vols.

The forward

With no dividends, holding the share to a date $T$ costs what buying it forward does, so the forward is the spot carried at the discount rate:

F = \frac{S}{P(T)}.

A share that pays a dividend before $T$ has a lower forward, because the forward buyer doesn't receive it. The option model here assumes a dividend-free spot and doesn't see dividends.

Equity options

A European option pays at expiry, on the share price then. A call on $Q$ shares struck at $K$ is Black-76 on the forward:

C = Q\,P(T)\,\mathrm{Black}(F, K, \sigma, \tau),

where $\sigma$ is the surface's vol at the option's expiry and strike, and $\tau$ is the surface's ACT/365 Fixed time to expiry. A put is the same formula with the payoff reversed. A call less a put at one strike pays $Q(S_T - K)$ whatever happens, which is a forward purchase, so

C - P = Q\,(S - K\,P(T)).

A negative $Q$ is the short side: the same premium with the sign turned. See equity option.

Barriers

A barrier adds a level that switches the option. A knock-out dies if the spot trades through the level before expiry, and a knock-in only comes alive if it does. A down barrier sits below the spot and an up barrier above it. A rebate pays a fixed amount per share to a knocked-out holder, or to a knock-in that never knocked in.

Barriers make an option cheaper, which is why structured products use them. With no rebate, a knock-in and a knock-out on the same level are, between them, the vanilla: exactly one of the two is alive at expiry. A single continuously monitored barrier has a closed-form price, the Reiner-Rubinstein formulas. They assume the spot starts on the live side of the barrier, so a spot already through it is refused.

Equity total return swaps

A total return swap gives the receiver the economics of holding shares without buying them: the gain or loss on the price over each period, plus the dividends, against a funding rate on the notional. Prime brokers and dealers use it to finance equity inventory and to give clients leveraged exposure.

The swap resets periodically. At each reset the gain is settled, and the next period starts from the share's fixing at that date. The value $V$ of the period under way, to the receiver, is what the shares gained since the period's start, plus the dividends $D_j$ gone ex since then, less the funding accrued on the funding notional at ACT/360. The funding notional is by default the quantity times the initial price.

That value is undiscounted, and periods not yet started are worth nothing. The payer holds $-V$ . The formula, with every symbol, is on equity total return swap.

From formula to API

Symbol	Meaning	Rust	Python
$S$ , $\sigma$	the spot and the surface	`MarketContext::with_equity_spot`, `with_equity_vols`	`MarketContext.with_equity_spot`, `with_equity_vols`
$P(T)$	the discount factor to expiry	`curve.view::<f64>().discount(T)`	`MarketContext.discount_factor`
$\mathrm{Black}$	Black-76 on the forward	`black76(forward, strike, vol, expiry, kind)`	`models.black76(forward, strike, vol, expiry, kind)`
$C$ , $P$	an option's premium	`EquityOption::new`, then `AnyInstrument::npv`	`products.equity_option`, then `Instrument.npv`
$V$	a total return swap's value	`EquityTotalReturnSwap::new`, then `AnyInstrument::npv`	`products.equity_total_return_swap`, then `Instrument.npv`
$D_j$	dividends	`DIV-<name>` in a `FixingStore`	`MarketContext.with_fixings([("DIV-<name>", date, amount)])`