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Separating Financing and Discount Rates in Black Option Pricing

Article Quant Q&A · Author: Kevin K.
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Summary

The document distinguishes pricing options on futures from options on spot assets. In the Black formula for a futures option, the futures price is the underlying input, and the option payoff is discounted using the applicable discount rate. The futures price already reflects the market's forward valuation, so a separate spot growth rate is not inserted into that formula.

For a spot option, the underlying's financing or carry rate affects its forward value, while a discount rate applies to the option payoff. The answer characterizes the underlying rate by its financing conditions, including whether the asset can be financed through repo. It says the discount rate depends on the option's collateral and funding arrangement, such as whether a collateral agreement applies. The response gives conceptual guidance rather than prescribing a universal curve or specific benchmark; actual rate selection depends on the instrument and contract terms.

Key ideas

  • A Black option on a futures contract takes the futures price as its underlying input.
  • The futures-option formula discounts the payoff without a separate spot growth rate.
  • For a spot option, the underlying financing rate determines its forward growth.
  • The discount rate depends on collateralization and funding arrangements.
  • Rate choices depend on the instrument and contract terms.

Tags

OptionsFuturesDerivatives pricing
Full text
# Black (1976) model growth rate input for futures price


# Black (1976) model growth rate input for futures price












When using the Black 76 model for pricing European index options I've often seen people use 2 different rates: the typical risk free rate used to get the discount factor, and a growth rate used to get the forward price. The adjusted equation for a call option (assuming no dividends) using $r_g$ as the growth rate and $r_f$ as the risk free rate would be

$$C = e^{-r_fT}[e^{r_gT}SN(d_1) - KN(d_2)]$$

I'm not totally sure what rates to use for each of these and I am having trouble finding information about it online. What is the reasoning behind having two separate rates? What would someone typically use for each rate (for example LIBOR forward for $r_g$ and OIS discount for $r_f$)?

## Answer by ir7 (score 1)

https://quant.stackexchange.com/a/60977

As @Kermittfrog said in the comment, in Black formula for options on futures price you need to insert the futures price $F$:

$$C = e^{-rT}[FN(d_1) - KN(d_2)]$$

where $r$ is the discounting rate. Here, $d_1$ depends only on $F$ (no rate involved).

For Black-Scholes formula for options on spot price (assume asset pays no dividend to keep it clean), we have:

$$C = e^{-rT}[e^{r_RT} S N(d_1) - KN(d_2)]$$

where $r$ is the discounting rate and $r_R$ is the financing rate of the underlying asset (if it can be repoed, it will be lower, if not, it will be higher). Here, $d_1$ depends on $S$ and $r_R$.

The discount rate $r$ depends on whether the option itself is collateralized (there is CSA) or not (there is no CSA), so it will be a collateral rate or a funding rate, respectively. See Piterbarg's Funding beyond discounting article.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.