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Deriving Black’s Model with the Forward Risk-Neutral Measure

Article Quant Q&A · Author: LCE
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Summary

The document explains why a European call’s maturity payoff can be written as the positive part of the difference between the asset price and the strike: this is the definition of the option. It then addresses how the option’s value evolves before maturity. If the underlying follows an Itô process and the option value is a function of that underlying and time, Itô’s formula gives the option value its own diffusion process.

This connects the payoff to the forward-measure pricing relation used in a derivation of Black’s model. The explanation is conceptual rather than a full derivation: it does not state the resulting pricing formula, specify the model assumptions in detail, or distinguish carefully among possible underlyings and volatility assumptions. Its claim about diffusion behavior applies to sufficiently regular functions of an Itô underlying; the drift and volatility of the option value are determined by that function and the underlying process.

Key ideas

  • A call’s terminal payoff is defined by its contractual payoff rule.
  • If the underlying follows an Itô process, a sufficiently regular option value can be analyzed with Itô’s formula.
  • The resulting option process has drift and volatility induced by the underlying and the option’s dependence on it.
  • The document gives intuition for the forward-measure pricing setup but not a complete Black model derivation.

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OptionsDerivatives pricingStatistics
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# How can the forward risk neutral measure be used to derive Black's model?


# How can the forward risk neutral measure be used to derive Black's model?












In the Hull textbook's derivation of Black's model (Section 27.6), they apply equation (27.20), which is $f_0 = P(0,T)E_T(f_T)$, where $P(0,T)$ is the value of a zero coupon bond at time $0$ expiring at $T$, and $E_T$ is the expectation with respect to the forward risk neutral measure of the zero coupon bond.

They set $f_T=\max(S_T-K,0)$ to the call payoff, and go from there.

However, $f_0 = P(0,T),E_T(f_T)$ was derived in Section 27.3 assuming that $f_t$ satisfies $df = \mu f dt + \sigma f dz$.

My question:

- Why is it valid to set $f_T=\max(S_T-K,0)$? That is, how do we show that $f_t$ satisfies $df = \mu f dt + \sigma f dz$?

- Are all European derivatives $f_T$, not necessarily a call, also of this form?

## Answer by Marquis (score 2)

https://quant.stackexchange.com/a/46397

The equation $f_T = \max \{ S_T - K, 0 \}$ is not an assumption, this is true by definition of what a call option is. It's an option which, at the time of maturity $T$, gives the value $\max\{ S_T - K, 0\}$ to the holder.

And yes, options $f_t$ follow the diffusion $dF_t = \mu dt + \sigma dW_t$ because the underlying stock (or forward) also follows an Ito process, and since the option is a function of that underlying, you can apply Ito's formula to figure out that the option also follows an Ito diffusion.

From Wikipedia:

Here, your underlying spot or forward is represented by $X_t$ and your option is a function $F(X, t)$, which, as the Lemma says, also follows an Ito diffusion.

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.