Present Value in Black–Scholes and Black–76 Option Pricing

Summary

The document asks whether option values from Black–Scholes and Black–76 represent prices today or values at expiration. It gives the Black–76 call formula, which discounts the expected payoff expression by the risk-free rate over the option's term. The questioner is unsure how this discounted futures-based formula relates to Black–Scholes and to descriptions of Black–76 prices as “undiscounted.” They also consider a transformation involving the Black–Scholes dividend yield and ask whether that changes which quantity is discounted.

The material does not include an answer or evidence resolving the terminology. Its useful focus is the distinction between a present option value and an expiration payoff, as well as the role of discounting in a futures-based pricing expression. The question should not be treated as a complete account of either model: it supplies no assumptions, derivation, or treatment of conventions that may explain the phrase “undiscounted.”

Key ideas

• The document asks whether Black–Scholes and Black–76 outputs represent present option values or expiration values.
• The displayed Black–76 call expression includes discounting over the option term.
• The question links Black–76 to a transformation of Black–Scholes involving a dividend yield.
• The distinction between a discounted price and an expiration payoff is central to interpreting the formulas.
• No answer, model derivation, or evidence resolving the terminology is included.

Tags

# 83919


# Does pricing an option from the discounted stock price using black-scholes and black76, give the discounted option price?












If we plug in our $S,K,T,r,q$ into the Black-Scholes, is that option price value the present day value with a forward rate or what the actual price should be today? The reason being, is that I have read the phrase the "undiscounted" option price wrt the black76 model quite a lot By the black76, I mean the equation below (Call): $$ c = e^{-rT} \left[ F N(d_{1}) - K N(d_{2}) \right] $$

with

$$ d_{1} = \frac{\ln\!\left(\tfrac{F}{K}\right) + \tfrac{\sigma^{2}}{2}T}{\sigma \sqrt{T}} $$

$$ d_{2} = \frac{\ln\!\left(\tfrac{F}{K}\right) - \tfrac{\sigma^{2}}{2}T}{\sigma \sqrt{T}} = d_{1} - \sigma \sqrt{T} $$

From the black-scholes wikipedia page, the BS is transformed to the black76, where (I think), the undiscounted BS option price becomes the discounted futures price when setting $q=0$. So I am not sure if the BS and black76 use present day value or price at expiry

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