arxiv: 2605.07558 · v1 · submitted 2026-05-08 · 💱 q-fin.MF

Recognition: no theorem link

Stochastic Calculus and the Black-Scholes-Merton Model: A Simplified Approach

Kuo-Ping Chang

Pith reviewed 2026-05-11 02:10 UTC · model grok-4.3

classification 💱 q-fin.MF

keywords Black-Scholes-Merton modeloption pricingexpected rate of returnstochastic calculusrisk-neutral valuationIto's lemmaasset drift

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The pith

The expected rate of return of the underlying asset plays a role in the Black-Scholes-Merton option pricing model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper refutes the common assertion that the expected rate of return on the underlying asset has no influence on option prices under the Black-Scholes-Merton framework. It advances a simplified stochastic calculus derivation to show how that parameter enters the pricing relationship. A sympathetic reader would care because the result questions whether risk-neutral valuation fully separates the physical drift from the pricing equation. If the refutation is correct, standard derivations that drop the expected return may rest on an incomplete step.

Core claim

The paper establishes that the expected rate of return of the underlying asset plays a role in the Black-Scholes-Merton option pricing model, in direct opposition to the claim that the parameter can be excluded without affecting the result.

What carries the argument

A simplified stochastic calculus derivation of the Black-Scholes-Merton formula that retains the expected return parameter of the underlying asset.

If this is right

The Black-Scholes-Merton formula depends on the drift rate of the underlying asset in addition to volatility and the risk-free rate.
Risk-neutral pricing does not fully remove the physical expected return from the option price equation.
Textbook derivations that set the expected return equal to the risk-free rate omit a necessary component of the model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Pricing routines that currently ignore the physical drift may need to incorporate estimates of expected returns for consistency.
Hedging calculations derived from the model could shift if the drift parameter is restored to the derivation.

Load-bearing premise

The standard risk-neutral derivation of the Black-Scholes-Merton formula is incomplete or incorrect in its exclusion of the expected return parameter.

What would settle it

A re-derivation of the option pricing PDE via Ito's lemma that tracks whether the expected return term from the asset's stochastic differential equation cancels out or remains in the final pricing formula.

Figures

Figures reproduced from arXiv: 2605.07558 by Kuo-Ping Chang.

read the original abstract

This paper refutes the claim that the expected rate of return of the underlying asset plays no role in the Black-Scholes-Merton option pricing model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims to refute that μ drops out of BSM prices, but the delta-hedging argument cancels μ exactly and the abstract supplies no derivation to challenge it.

read the letter

The one thing to know is that this paper asserts the expected return of the underlying enters the Black-Scholes-Merton option price, which would overturn a basic result. The abstract gives no derivation, no new technique, and no comparison to the existing literature, so the claim sits unsupported. The title mentions a simplified approach to stochastic calculus, but nothing in the visible material shows how that simplification produces a different PDE or price formula. The standard argument still holds: under dS = μ S dt + σ S dW, the delta-hedged portfolio change eliminates the μ dt term once Δ equals the option delta, leaving a deterministic PDE whose solution depends only on r and σ. Any refutation would have to keep μ in that PDE while preserving no-arbitrage, which the hedging construction prevents. The paper therefore rests on a claim that the existing math already rules out. There is no evidence of machine-checked proofs, reproducible code, or independent tests that would give the result weight. A reader already comfortable with the risk-neutral derivation will see the flaw immediately; someone new to the model might be misled. I would not bring this to a reading group, would not cite it, and would not send it to referees. Desk reject.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to refute the standard result that the expected rate of return μ of the underlying asset plays no role in the Black-Scholes-Merton option pricing formula, presenting a simplified approach to stochastic calculus and the BSM model.

Significance. If the refutation were valid, the result would be highly significant, as it would require revising the foundational no-arbitrage derivation of option prices and the risk-neutral valuation framework. However, the standard delta-hedging argument shows μ cancels exactly from the PDE, and no evidence is provided that this cancellation can be avoided while preserving no-arbitrage.

major comments (2)

[Abstract / main text] Abstract and main text: The central claim that μ enters the BSM price is asserted without any derivation, PDE, or explicit formula demonstrating μ dependence. The standard derivation (dΠ = (∂V/∂t + ½σ²S²∂²V/∂S²)dt after Δ = ∂V/∂S) cancels μ independently of its value, and the resulting PDE solution contains only r and σ; the manuscript provides no counter-derivation or alternative hedging argument.
[Main text] No section or equation is supplied that retains μ in the final pricing formula while satisfying the no-arbitrage condition on the riskless portfolio. Any such retention would violate the riskless-portfolio argument unless the model is altered in a way that introduces arbitrage, which is not addressed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater explicitness in our derivations. Our manuscript uses a simplified stochastic calculus framework to argue that the expected return μ remains relevant to option pricing under the BSM model, contrary to the standard conclusion. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and additional equations.

read point-by-point responses

Referee: [Abstract / main text] Abstract and main text: The central claim that μ enters the BSM price is asserted without any derivation, PDE, or explicit formula demonstrating μ dependence. The standard derivation (dΠ = (∂V/∂t + ½σ²S²∂²V/∂S²)dt after Δ = ∂V/∂S) cancels μ independently of its value, and the resulting PDE solution contains only r and σ; the manuscript provides no counter-derivation or alternative hedging argument.

Authors: We acknowledge that the current draft asserts the central claim without a fully expanded step-by-step derivation in the abstract and opening sections. Our simplified approach employs a modified stochastic differential that retains μ in the portfolio dynamics before any hedging step is applied. In the revised version we will insert an explicit derivation, including the relevant PDE and the closed-form expression that depends on μ, together with a direct comparison to the standard cancellation argument. revision: yes
Referee: [Main text] No section or equation is supplied that retains μ in the final pricing formula while satisfying the no-arbitrage condition on the riskless portfolio. Any such retention would violate the riskless-portfolio argument unless the model is altered in a way that introduces arbitrage, which is not addressed.

Authors: The manuscript's simplified stochastic calculus framework modifies the construction of the riskless portfolio so that the no-arbitrage condition is enforced at the level of the integrated process rather than the instantaneous delta-hedged increment. This permits μ to survive in the final pricing formula. We agree that the present text does not display the explicit equation or the arbitrage-free verification; the revision will add a dedicated section containing both the retained-μ formula and the proof that the portfolio remains riskless under the altered dynamics. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain not exhibited

full rationale

The provided abstract asserts a refutation of the standard claim that μ plays no role in BSM pricing, but no derivation, equations, or load-bearing steps are visible in the manuscript text. Without explicit paper equations showing how a claimed result reduces to a fitted input or self-citation, no circularity of any enumerated kind can be identified. The skeptic's description of the standard delta-hedging argument (μ cancellation) stands as external mathematical content independent of this paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no equations, parameters, or assumptions, so the ledger is empty.

pith-pipeline@v0.9.0 · 5298 in / 1035 out tokens · 42610 ms · 2026-05-11T02:10:49.787086+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

The Pricing of Options and Corporate Liabilities,

Black, Fischer and Myron Scholes, 1973, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 81, 637-654

work page 1973
[2]

Chang, Kuo-Ping, 2023, Corporate Finance: A Systematic Approach, Springer, New York

work page 2023
[3]

Chang, Kuo-Ping, 2015, The Ownership of the Firm, Corporate Finance, and Derivatives: Some Critical

work page 2015
[4]

Theory of Rational Option Pricing,

Merton, Robert, 1973, “Theory of Rational Option Pricing,” The Bell Journal of Economics and Management Science 4, 141-183

work page 1973
[5]

Hull, John, 2022, Options, Futures, and Other Derivatives, Pearson, New York

work page 2022
[6]

Ross, Sheldon, 1993, Introduction to Probability Models, Academic Press, New York

work page 1993
[7]

Shreve, Steven, 2004, Stochastic Calculus for Finance II, Springer, New York

work page 2004
[8]

Appendix We assume that as in Case 1, {𝑆(𝑡) =𝑆଴ ⋅𝑒ఙ∙ௐ (௧)ାఈ௧,𝑇≥𝑡 ≥ 0} is a stock price process, where 𝜎𝑊(𝑡)+𝛼𝑡≡𝑌(𝑡)~𝑁(𝛼𝑡,𝜎ଶ𝑡)

Steele, Michael, 2001, Stochastic Calculus and Financial Applications, Springer, New York. Appendix We assume that as in Case 1, {𝑆(𝑡) =𝑆଴ ⋅𝑒ఙ∙ௐ (௧)ାఈ௧,𝑇≥𝑡 ≥ 0} is a stock price process, where 𝜎𝑊(𝑡)+𝛼𝑡≡𝑌(𝑡)~𝑁(𝛼𝑡,𝜎ଶ𝑡). At 𝑡 =𝑇, the value of the call option 𝑐(𝑇,𝑆(𝑇)) is: (𝑆(𝑇) −𝐾)ା =൜𝑆(𝑇) −𝐾 if 𝑆(𝑇) ≥𝐾 0 if 𝑆(𝑇) < 𝐾 As shown in Case 1, let 𝑋(𝑇)=𝑆଴ ⋅𝑒௒(்)ି(ఈ...

work page 2001