# Quoting options with reference price and delta

I always thought equity options where quoted with implied volatility, the price being given by the Black-Scholes price of the option with volatility equal to the implied volatlity.

But apparently there is another way to quote option : with reference implied vol (or reference price) and delta for an underlying at a price near the spot price. I also heard about "target price" in this context.

• In the equity space and between different institutions, options are usually not quoted in implied volatilities. For this to work, you'd have to also quote a risk-free rate, repo rate, dividend and a measure of time-to-expiry (e.g. a day count convention) at the same time. In the OTC/broker market, options are often quoted as a price, spot reference and delta combination. The delta and spot reference determine how the price is adjusted to account for spot moves between quote and trade time. On the screen, there is not need for this as you can update your quotes all the time. – LocalVolatility Nov 12 '17 at 16:00
• "The delta and spot reference determine how the price is adjusted to account for spot moves between quote and trade time" --> what does this mean exactly ? – EricFlorentNoube Nov 12 '17 at 16:54
• In the broker market a quote is given on the phone or via a chat. It is usually not traded immediately but only after the broker collected a sufficient number of quotes and the client made a decision. The spot often moves between quote and trade time and the delta is used to adjust the option price to the first order. So $P_\text{traded} = P_\text{quoted} + \Delta_\text{quoted} \left( S_\text{at-trade-time} - S_\text{at-quote-time} \right)$. – LocalVolatility Nov 12 '17 at 17:05
• So basically would a broker tell me $(\pi, \delta, S_{t_{\textrm{now}}})$, it would mean that : If buy one option now, I pay $\pi$, but if I buy later at $t$, I will pay $\pi + \left( S_t - S_{t_{\textrm{now}}} \right) \times \delta$. Of course, I guess that the broker's offer is valid while $t - t_{\textrm{now}}$ is not too big (for the $\theta$) and while $|S_t - S_{t_{\textrm{now}}}|$ is not too big also, correct ? – EricFlorentNoube Nov 12 '17 at 17:07
• Yes - correct. You can always say "off" via the phone line or chat as well as long as your quote hasn't been traded. – LocalVolatility Nov 12 '17 at 17:09

In the broker/OTC market, options are usually quoted as a price, spot reference and delta combination. The delta and spot reference determine how the price is adjusted to account for spot moves between quote and trade time to the first order. This is necessary as a trade usually does not happen immediately immediately but only after the broker collected a sufficient number of quotes and the initiating counterpart made a decision. Let $T_\text{quote}$ and $T_\text{trade}$ be the times of the quote and trade, respectively. The traded price is then given by
$$P \left( T_\text{trade} \right) = P \left( T_\text{quote} \right) + \Delta \left( T_\text{quote} \right) \left[ S \left( T_\text{trade} \right) - S \left( T_\text{quote} \right) \right].$$