Generative Diffusion Model for Risk-Neutral Derivative Pricing

Generativ e Diffusion Mo del for Risk-Neutral Deriv ativ e Pricing Nila y Tiw ari Carnegie Mellon Univ ersit y nilayt@andrew.cmu.edu Marc h 24, 2026 Abstract Denoising diffusion probabilistic mo dels (DDPMs) ha v e emerged as pow erful generativ e mod- els for complex distributions, yet their use in arbitrage-free deriv ative pricing remains largely unexplored. Financial asset prices are naturally mo deled by sto c hastic differential equations (SDEs), whose forw ard and reverse densit y evolution closely parallels the forward noising and rev erse denoising structure of diffusion mo dels. In this pap er, we dev elop a framework for using DDPMs to generate risk-neutral asset price dynamics for deriv ativ e v aluation. Starting from log-return dynamics under the physical measure, we analyze the asso ciated forw ard diffusion and deriv e the reverse-time SDE. W e show that the change of measure from the physical to the risk-neutral measure induces an additive shift in the score function, which translates into a closed-form risk-neutr al epsilon shift in the DDPM reverse dynamics. This correction enforces the risk-neutral drift while preserving the learned v ariance and higher-order structure, yielding an explicit bridge betw een diffusion-based generativ e mo deling and classical risk-neutral SDE-based pricing. W e sho w that the resulting discoun ted price paths satisfy the martingale condition under the risk-neutral measure. Empirically , the metho d repro duces the risk-neutral terminal distribution and accurately prices b oth Europ ean and path-dependent deriv atives, including arithmetic Asian options, under a GBM b enc hmark. These results demonstrate that diffusion-based generative mo dels pro vide a flexible and principled approac h to simulation-based deriv ativ e pricing. 1 In tro duction Diffusion-based generativ e mo dels, including denoising diffusion probabilistic models (DDPMs) and score-based diffusion mo dels, ha ve recently emerged as p o w erful to ols for learning and sampling from complex high-dimensional distributions Ho et al. ( 2020 ); Song et al. ( 2021 ). In their contin uous- time form ulation, these mo dels represent data distributions as solutions to sto chastic differen tial equations (SDEs) and generate samples via reverse-time diffusion dynamics driven b y learned score functions Song et al. ( 2021 ). This SDE-based p ersp ectiv e makes diffusion mo dels particularly natural for quan titative finance, where asset prices are mo deled as sto c hastic pro cesses. A central challenge in deriv ativ e pricing is the construction of dynamics under the risk-neutral measure Q , under whic h discounted asset prices must form martingales and the drift is fixed by the risk-free rate Shreve ( 2004 ). Mo dern machine learning approac hes to pricing and hedging - including deep hedging, generative mo dels for return distributions, and neural SDEs Liu ( 2024 ), are typically trained on historical data under the ph ysical measure. Incorporating the risk-neutral structure in to such mo dels therefore requires either calibration to option prices or the imp osition of additional constrain ts. 1 F or diffusion-based mo dels, this issue is particularly subtle. T raining is p erformed under the ph ysical measure, and the learned score or noise predictor corresp onds to the diffused ph ysical- measure distribution. A change of measure do es not act linearly on score functions, making it non trivial to imp ose the risk-neutral drift constrain t directly within the reverse-time dynamics. This raises a fundamental question: How c an one mo dify the r everse-time dynamics of a diffusion mo del tr aine d under the physic al me asur e so that the r esulting gener ate d pric e p aths satisfy the risk-neutr al mar- tingale c ondition? In this pap er, we address this question by working directly with the underlying SDEs. Start- ing from log-price dynamics under the physical measure, w e deriv e the corresp onding reverse-time SDE using Anderson’s time-rev ersal theorem and the Fisher identit y , and analyze its transforma- tion under the risk-neutral measure Q . This leads to a closed-form risk-neutr al epsilon shift : a mo dification of the reverse-time noise term that enforces the correct risk-neutral drift while pre- serving the v ariance and higher-order structure learned by the mo del. This construction provides an explicit bridge betw een diffusion-based generative modeling and classical risk-neutral SDE-based asset pricing. W e show that the resulting reverse dynamics generate price paths whose discounted pro cess is an appro ximate martingale. In a geometric Bro wnian motion benchmark, the model repro duces the correct terminal distribution and pro duces Europ ean option prices that closely match Black–Sc holes v alues. Section 2 develops the mathematical framework, Section 3 deriv es the risk-neutral reverse dynamics and the epsilon shift, Section 4 presents empirical v alidation, and Section 5 discusses limitations and future directions. 2 Mathematical F ramew ork 2.1 Sto c hastic setting and ph ysical-measure dynamics W e work on a filtered probability space (Ω , F , ( F t ) t ≥ 0 , P ) , satisfying the usual conditions and supp orting a one-dimensional standard Brownian motion ( W t ) t ≥ 0 . Let S t denote the price of a traded underlying asset. Under the physical measure P we assume that S t follo ws a geometric Brown ian motion (GBM) with constan t drift µ ∈ R and volatilit y σ > 0: dS t = µS t dt + σ S t dW t , S 0 > 0 . (1) Define the log-price pro cess X t := log S t . An application of Itˆ o’s formula to ( 1 ) yields dX t =  µ − 1 2 σ 2  dt + σ dW t . (2) Fix a time step ∆ t > 0 and consider the one-step log-return o ver [ t, t + ∆ t ], Y t, ∆ t := X t +∆ t − X t . F rom ( 2 ), w e obtain under P the Gaussian law Y t, ∆ t ∼ N  m P , v 0  , m P :=  µ − 1 2 σ 2  ∆ t, v 0 := σ 2 ∆ t. (3) In particular, ov er a discrete grid t h := h ∆ t , h = 0 , 1 , . . . , H , the log-returns Y h := X t h +1 − X t h are i.i.d. Gaussian with mean m P and v ariance v 0 under P . These one-step log-returns are the ob jects w e seek to mo del with a diffusion-based generative framework. 2 2.2 Risk-free asset and risk-neutral dynamics Let B t denote the price of the money-market account ev olving at the constan t risk-free rate r ≥ 0, B t = e rt . (4) In an arbitrage-free mark et, there exists an equiv alen t probability measure Q suc h that the dis- coun ted asset price S t /B t is a martingale. F or the geometric Bro wnian motion mo del ( 1 ), this requiremen t uniquely sp ecifies the drift of the asset under Q . Under the risk-neutral measure Q , the SDE for S t b ecomes dS t = r S t dt + σ S t dW Q t , S 0 > 0 , (5) where W Q t is a Bro wnian motion under Q . Th us the only change when passing from P to Q is the drift replacement µ 7→ r . The log-price pro cess X t = log S t then satisfies dX t =  r − 1 2 σ 2  dt + σ dW Q t . (6) Ov er a time in terv al ∆ t , the one-step log-return under Q is therefore Y t, ∆ t ∼ N  m Q , v 0  , m Q :=  r − 1 2 σ 2  ∆ t, v 0 = σ 2 ∆ t. (7) F or a discrete grid t h = h ∆ t , the pro cess S t h e rt h is a martingale under Q . This drift adjustment is the fundamental structural difference b et w een P and Q and will b e central when mo difying the reverse-time diffusion dynamics to enforce risk- neutralit y in the generativ e mo del. 2.3 F orw ard diffusion and the F okk er-Planck equation T o connect asset-price dynamics with diffusion-based generative mo dels, it is con venien t to work with a general one-dimensional Itˆ o diffusion of the form d Z t = b ( Z t , t ) dt + σ ( Z t , t ) dW t , (8) where b is the drift, σ > 0 is the diffusion coefficient, and W t is a Brownian motion under the relev an t measure. The pro cess Z t ma y represent the log-price X t , a transformed state v ariable, or an abstract laten t v ariable used in a generative mo del. Let p t ( z ) denote the probability density function of Z t . The evolution of p t is gov erned b y the F okk er-Planck (or forward Kolmogoro v) equation: ∂ t p t ( z ) = − ∂ z  b ( z , t ) p t ( z )  + 1 2 ∂ z z  σ 2 ( z , t ) p t ( z )  . (9) Equation ( 9 ) describ es ho w the forward SDE ( 8 ) transp orts probability mass ov er time. It char- acterizes the family of in termediate densities ( p t ) t ∈ [0 ,T ] obtained b y evolving the data distribution through the forw ard dynamics. The F okker-Planc k equation ( 9 ) describ es how the marginal densit y p t of the diffusion ( 8 ) ev olves forw ard in time. A key result of Anderson Anderson ( 1982 ) is that this forward evolution admits a reverse-time description: there exists an SDE, run backw ard in time, whose marginal densities follo w the same family ( p t ) t ∈ [0 ,T ] in reverse order. In the next subsection we recall this reverse-time SDE, which forms the basis of diffusion-mo del sampling. 3 2.4 Rev erse-time SDE Consider the forward diffusion ( 8 ) with drift b and diffusion co efficien t σ , and let p t ( z ) denote the marginal densit y of Z t . Anderson ( 1982 ) sho wed that, under suitable regularity conditions, the time-rev ersed pro cess ˆ Z t := Z T − t , t ∈ [0 , T ] , is itself a diffusion. More precisely , ˆ Z t satisfies the rev erse-time SDE d ˆ Z t = ˜ b ( ˆ Z t , T − t ) dt + σ ( ˆ Z t , T − t ) d ˆ W t , (10) where ˆ W t is a Bro wnian motion under the reversed filtration, and the reverse drift ˜ b is giv en b y ˜ b ( z , t ) = b ( z , t ) − σ 2 ( z , t ) ∇ z log p t ( z ) . (11) Equation ( 10 ) together with ( 11 ) c haracterizes the unique diffusion pro cess whose marginal densities evolv e as ( p t ) t ∈ [0 ,T ] but in reverse temp oral order. The additional term − σ 2 ∇ z log p t corrects the forward drift to ensure consistency with the backw ard density evolution sp ecified by the F okker–Planc k equation ( 9 ). This rev erse-time SDE is the contin uous-time foundation of diffusion-based generativ e sampling: if one can approximate the score function ∇ z log p t ( z ), then simulating ( 10 ) backw ard from a terminal distribution yields samples from the data distribution at time t = 0. The next subsection recalls how this score term can b e expressed in a denoising form suitable for learning. 2.5 Score function and its role in the reverse dynamics F or eac h t ∈ [0 , T ], let p t ( z ) denote the density of the diffusion Z t in ( 8 ). The sc or e function at time t is defined as s t ( z ) := ∇ z log p t ( z ) . The score enco des lo cal information ab out how probabilit y mass is arranged at time t , and it app ears explicitly in the reverse drift ( 11 ) through the term − σ 2 ( z , t ) s t ( z ). If the family of score functions { s t } t ∈ [0 ,T ] w ere known, then the reverse-time SDE ( 10 ) would be fully sp ecified, and simulating it backw ard from a suitable terminal distribution would repro duce the forw ard marginals ( p t ) t ∈ [0 ,T ] in rev erse order. In particular, running the rev erse SDE from t = T do wn to t = 0 would yield samples approximately distributed according to the data distribution p 0 . In practice, the densities p t and their scores s t are not a v ailable in closed form. Diffusion- based generativ e mo dels address this b y training a neural netw ork to approximate the score (or an equiv alen t quantit y) at eac h time t , using a suitable learning ob jectiv e deriv ed from the forward noising pro cess. Substituting this learned approximation into ( 11 ) yields an approximate reverse- time dynamics that can b e used as a sampler. The next subsection recalls how this is implemented in the discrete-time DDPM framework. 2.6 Discrete DDPM appro ximation to the reverse SDE Denoising diffusion probabilistic models (DDPMs) pro vide a tractable discrete-time approximation to the contin uous-time reverse diffusion ( 10 ). A DDPM sp ecifies a forward noising pro cess that gradually transforms a data sample into nearly Gaussian noise, together with a learned reverse pro cess that approximately in verts this transformation. 4 F orward noising pro cess. Fix noise parameters { β t } T t =1 ⊂ (0 , 1) and define α t := 1 − β t , ¯ α t := t Y s =1 α s . Starting from a data sample z 0 , the forw ard pro cess generates latents z 1 , . . . , z T according to z t = √ ¯ α t z 0 + √ 1 − ¯ α t ε t , ε t ∼ N (0 , 1) . (12) Hence each marginal is Gaussian: z t | z 0 ∼ N  √ ¯ α t z 0 , (1 − ¯ α t ) I  . Gaussian p osterior. Because the forward transitions are linear-Gaussian, the p osterior distri- bution q ( z t − 1 | z t ) is also Gaussian. Using standard Gaussian conditioning identities (see Ho et al. ( 2020 )), one obtains q ( z t − 1 | z t ) = N  µ t ( z t , ε ) , ˜ β t I  , (13) where ˜ β t = 1 − ¯ α t − 1 1 − ¯ α t β t , ε = z t − √ ¯ α t z 0 √ 1 − ¯ α t . Th us the p osterior mean can b e written as µ t ( z t , ε ) = 1 √ α t  z t − β t √ 1 − ¯ α t ε  . (14) Fisher iden tit y and the score–noise relation. F or the Gaussian forward k ernel ( 12 ), the Fisher (or denoising) identit y Song and Ermon ( 2019 ) yields ∇ z t log p t ( z t ) = − 1 √ 1 − ¯ α t E [ ε | z t ] . (15) Th us the score can b e expressed as a conditional exp ectation of the injected noise. DDPMs exploit this b y training a neural netw ork ε θ ( z t , t ) to appro ximate the conditional mean E [ ε | z t ], yielding the practical score approximation s t ( z t ) = ∇ z t log p t ( z t ) ≈ − 1 √ 1 − ¯ α t ε θ ( z t , t ) . (16) Rev erse DDPM up date. Substituting the noise predictor into ( 14 ) yields the practical rev erse transition z t − 1 = 1 √ α t  z t − β t √ 1 − ¯ α t ε θ ( z t , t )  + q ˜ β t ξ t , ξ t ∼ N (0 , 1) . (17) applied for t = T , T − 1 , . . . , 1. Connection to the rev erse SDE. The update ( 17 ) constitutes a first-order Euler discretization of the rev erse-time SDE ( 10 ), with the learned score approximation ( 16 ) replacing the exact score. Th us a DDPM sampler can b e interpreted as an approximate numerical scheme for in tegrating the Anderson reverse diffusion. 5 3 Risk-Neutral DDPM Rev erse Dynamics In this section w e derive a modification of the DDPM reverse dynamics that enforces the risk- neutral drift while preserving the v ariance and higher-order structure learned under the physical measure. W e work at the lev el of one-step log-returns, using the notation from Sections 2.1 – 2.6 . Recall that ov er a fixed calendar incremen t ∆ t , the log-return Y := X t +∆ t − X t satisfies, under P , Y ∼ N ( m P , v 0 ) , m P =  µ − 1 2 σ 2  ∆ t, v 0 = σ 2 ∆ t, (18) while under the risk-neutral measure Q , Y ∼ N ( m Q , v 0 ) , m Q =  r − 1 2 σ 2  ∆ t. (19) Th us the c hange of measure from P to Q mo difies only the mean of the one-step log-return, leaving the v ariance v 0 unc hanged. 3.1 F orw ard DDPM marginals under P and Q W e mo del the clean one-step log-return Y 0 as Gaussian under b oth measures, Y P 0 ∼ N ( m P , v 0 ) , Y Q 0 ∼ N ( m Q , v 0 ) , (20) with m P , m Q , v 0 as in ( 18 )–( 19 ). A t diffusion time index t ∈ { 1 , . . . , T } , the DDPM forward marginal ( 12 ) reads Y t = √ ¯ α t Y 0 + √ 1 − ¯ α t ε t , ε t ∼ N (0 , 1) , (21) where Y 0 is distributed according to P or Q . The next lemma describ es the induced marginals. Lemma 1 (F orward DDPM marginals under P and Q ) . L et Y 0 b e Gaussian as in ( 20 ) , and let Y t b e define d by ( 21 ) with ε t indep endent of Y 0 . Then, under P and Q , we have Y P t ∼ N ( µ P t , σ 2 t ) , Y Q t ∼ N ( µ Q t , σ 2 t ) , (22) wher e µ P t = √ ¯ α t m P , µ Q t = √ ¯ α t m Q , σ 2 t = ¯ α t v 0 + (1 − ¯ α t ) . (23) Pr o of. Since Y 0 and ε t are indep enden t and ( 21 ) is linear, Y t is Gaussian. Under P , E P [ Y t ] = √ ¯ α t E P [ Y 0 ] + √ 1 − ¯ α t E [ ε t ] = √ ¯ α t m P , and similarly under Q , E Q [ Y t ] = √ ¯ α t m Q . In b oth cases, V ar( Y t ) = ¯ α t V ar( Y 0 ) + (1 − ¯ α t ) V ar( ε t ) = ¯ α t v 0 + (1 − ¯ α t ) , whic h pro ves ( 22 )–( 23 ). Th us the DDPM forw ard pro cess transp orts the difference in one-step means m Q − m P in to a family of time-dep enden t mean differences µ Q t − µ P t = √ ¯ α t ( m Q − m P ), with a common v ariance σ 2 t . 6 3.2 Gaussian score shift under change of drift W e no w compute how the score of the DDPM forw ard marginal changes when passing from P to Q . Prop osition 1 (Gaussian score shift) . L et Y P t ∼ N ( µ P t , σ 2 t ) and Y Q t ∼ N ( µ Q t , σ 2 t ) as in L emma 1 . Denote the c orr esp onding sc or es by s P t ( y ) := ∇ y log p P t ( y ) , s Q t ( y ) := ∇ y log p Q t ( y ) . Then s Q t ( y ) = s P t ( y ) + η t , η t = √ ¯ α t σ 2 t  m Q − m P  , (24) for al l y ∈ R . Pr o of. F or a one-dimensional Gaussian N ( µ, σ 2 ) we ha ve log p ( y ) = − 1 2 log  2 π σ 2  − 1 2 σ 2 ( y − µ ) 2 , hence ∇ y log p ( y ) = − y − µ σ 2 = − 1 σ 2 y + µ σ 2 . Applying this with µ = µ P t , µ Q t and σ 2 = σ 2 t giv es s P t ( y ) = − 1 σ 2 t y + µ P t σ 2 t , s Q t ( y ) = − 1 σ 2 t y + µ Q t σ 2 t . Subtracting yields s Q t ( y ) − s P t ( y ) = µ Q t − µ P t σ 2 t = √ ¯ α t σ 2 t ( m Q − m P ) , b y ( 23 ), which pro v es ( 24 ). In particular, the effect of replacing m P b y m Q in the clean log-return is to add a c onstant offset η t to the score at each diffusion time t . 3.3 Risk-neutral epsilon shift via Fisher iden tity W e now translate the score shift ( 24 ) into a mo dification of the DDPM noise predictor via the Fisher identit y introduced in Section 2.6 . F or the Gaussian forward k ernel ( 12 ), the Fisher identit y ( 15 ) gives s P t ( y ) = ∇ y log p P t ( y ) = − 1 √ 1 − ¯ α t E P  ε t | Y t = y  . (25) DDPMs approximate the conditional mean E P [ ε t | Y t = y ] by a neural netw ork ε θ ( y , t ), so that s P t ( y ) ≈ − 1 √ 1 − ¯ α t ε θ ( y , t ) . (26) Under the risk-neutral measure, Prop osition 1 shows that the score satisfies s Q t ( y ) = s P t ( y ) + η t , η t = √ ¯ α t σ 2 t ( m Q − m P ) . (27) The following prop osition iden tifies the corresp onding mo dification of the noise predictor. 7 Prop osition 2 (Risk-neutral epsilon shift) . Define δ t := η t √ 1 − ¯ α t = p ¯ α t (1 − ¯ α t ) σ 2 t  m Q − m P  , (28) and set ε Q θ ( y , t ) := ε θ ( y , t ) − δ t . (29) Then the induc e d sc or e appr oximation under Q satisfies s Q t ( y ) ≈ − 1 √ 1 − ¯ α t ε Q θ ( y , t ) . (30) Pr o of. Starting from ( 26 ) and ( 27 ), we ha ve s Q t ( y ) = s P t ( y ) + η t ≈ − 1 √ 1 − ¯ α t ε θ ( y , t ) + η t . Define ε Q θ b y ( 29 ). Then − 1 √ 1 − ¯ α t ε Q θ ( y , t ) = − 1 √ 1 − ¯ α t  ε θ ( y , t ) − δ t  = − 1 √ 1 − ¯ α t ε θ ( y , t ) + δ t √ 1 − ¯ α t . Cho osing δ t as in ( 28 ) gives δ t √ 1 − ¯ α t = η t , so that − 1 √ 1 − ¯ α t ε Q θ ( y , t ) = − 1 √ 1 − ¯ α t ε θ ( y , t ) + η t ≈ s Q t ( y ) , whic h pro ves ( 30 ). In the common standardized setting where v 0 = 1 and the schedule is chosen so that σ 2 t = 1, the expressions simplify to η t = √ ¯ α t ( m Q − m P ) , δ t = p ¯ α t (1 − ¯ α t ) ( m Q − m P ) . 3.4 Martingale prop ert y of discoun ted DDPM price paths W e now connect the modified score and epsilon shift to the risk-neutral martingale prop erty for the asset price. Consider a discrete calendar grid t h = h ∆ t , h = 0 , 1 , . . . , H , and construct log-prices b y summing one-step log-returns X t h +1 = X t h + Y h , S t h = e X t h . Lemma 2 (Risk-neutral one-step drift and martingale condition) . Supp ose that under Q e ach one-step lo g-r eturn satisfies Y h ∼ N ( m Q , v 0 ) , m Q =  r − 1 2 σ 2  ∆ t, v 0 = σ 2 ∆ t, and that ( Y h ) h ≥ 0 is indep endent of F t h given X t h . Then E Q  S t h +1 | F t h  = e r ∆ t S t h , (31) so that  e − rt h S t h  H h =0 is a martingale under Q . 8 Pr o of. Conditional on F t h the next log-return Y h is indep enden t of F t h and Gaussian as sp ecified. Th us E Q [ S t h +1 | F t h ] = E Q  e X t h +1 | F t h  = E Q  e X t h + Y h | F t h  = S t h E Q [ e Y h ] . Since Y h ∼ N ( m Q , v 0 ), E Q [ e Y h ] = exp  m Q + 1 2 v 0  = exp  ( r − 1 2 σ 2 )∆ t + 1 2 σ 2 ∆ t  = e r ∆ t . Substituting into the previous expression yields ( 31 ). 3.5 Risk-neutral DDPM rev erse dynamics Com bining the epsilon shift and the DDPM reverse up date ( 17 ) leads to a risk-neutral v arian t of the reverse dynamics. A t diffusion time t , we replace ε θ b y ε Q θ from ( 29 ) to obtain z t − 1 = 1 √ α t  z t − β t √ 1 − ¯ α t ε Q θ ( z t , t )  + q ˜ β t ξ t , ξ t ∼ N (0 , 1) , (32) with δ t as in ( 28 ). Interpreting z 0 as the one-step log-return Y and summing o ver calendar steps as in Lemma 2 gives the following result. Theorem 1 (Risk-neutral DDPM sampler) . Assume: 1. The DDPM tr aine d under P pr ovides an ac cur ate sc or e appr oximation as in ( 26 ) . 2. The one-step lo g-r eturns Y h ar e sample d by interpr eting z 0 fr om ( 32 ) as the lo g-r eturn over ∆ t and summing these r eturns to form X t h and S t h = e X t h . Then the epsilon-shifte d r everse dynamics ( 32 ) enfor c e the risk-neutr al one-step drift ( 19 ) and varianc e v 0 , and the r esulting pric e pr o c ess satisfies E Q [ e − rt h S t h ] = S 0 , h = 0 , 1 , . . . , H , so that ( e − rt h S t h ) H h =0 is a martingale under Q . Pr o of. By Prop osition 2 , the mo dified noise predictor ε Q θ induces a score approximation consisten t with the Gaussian forw ard marginal under Q , that is, consisten t with Y Q t ∼ N ( µ Q t , σ 2 t ) as in Lemma 1 . In the idealized setting of a p erfectly trained DDPM, the reverse Marko v chain ( 32 ) reco vers the clean v ariable Y Q 0 ∼ N ( m Q , v 0 ) in distribution. Interpreting Y Q 0 as the one-step log-return Y h o ver ∆ t , Lemma 2 implies that the resulting price pro cess satisfies the risk-neutral martingale condition. The stated equality for E Q [ e − rt h S t h ] follows b y iterating ( 31 ). Remark 1. The epsilon shift δ t in ( 28 ) mo difies only the me an of the r everse up date ( 32 ) , not the innovation varianc e ˜ β t . Conse quently, the varianc e and higher-or der fe atur es of the le arne d lo g-r eturn distribution, such as skewness and kurtosis, ar e pr eserve d while the drift is adjuste d to the risk-neutr al value. P ath wise risk-neutral v alidity . Because the risk-neutral drift correction is implemen ted as an ϵ -shift at every step of the reverse diffusion, the entire sequence of generated log-returns { Y h } H h =1 inherits the correct risk-neutral dynamics. In particular, the induced price pro cess constructed via S t 0 = S 0 , S t h = S t h − 1 exp( Y h ) satisfies the discounted martingale prop ert y under the learned measure, not only at the terminal horizon but along the full simulated path. This guaran tees that the DDPM output is suitable for pricing path-dep enden t claims such as the discrete arithmetic Asian options considered in Sec- tion 4.5 . 9 4 Exp erimen ts This section v alidates the risk-neutral DDPM construction in a controlled GBM w orld. All exp eri- men ts use the same DDPM trained on standardized one-da y log returns under the ph ysical measure P . Unless stated otherwise: • sp ot S 0 = 100, • v olatility σ = 0 . 2, • time step ∆ t = 1 / 252, • n umber of price steps H equals the num b er of trading da ys to maturity , • n umber of simulated paths N = 1 , 000. The risk-neutral DDPM applies the epsilon shift from Section 3 when sampling. The “no shift” DDPM reuses the trained net work but do es not mo dify the score, so it contin ues to follo w the ph ysical drift. 4.1 Single maturity , single strike diagnostics W e first consider a one month maturit y with H = 21 steps and mo derate gap b etw een drift and risk free rate. T able 1 rep orts summary statistics for an at the money call with strike K = S 0 . T able 1: Single-strike diagnostics for a one month maturit y . RN-DDPM refers to sampling with the epsilon shift under the risk neutral measure Q . GBM refers to the analytic or Monte Carlo reference in the GBM world. Quan tity RN-DDPM GBM reference Mean one-step return 1 . 19 × 10 − 4 m Q Std. of one-step returns 1 . 30 × 10 − 2 σ √ ∆ t Discoun ted mean e − rT E Q [ S T ] 99 . 998 100 KS statistic (terminal log price) 0 . 021 Gaussian reference KS p -v alue 0 . 76 Call price C ( K = S 0 ) 2 . 47 ± 0 . 12 2 . 51 (Black-Sc holes) Absolute pricing error 0 . 04 The RN-DDPM repro duces the one-step momen ts and the terminal distribution of the GBM quite accurately , and the call price matc hes the Blac k-Scholes v alue within a small fraction of the Mon te Carlo standard error. 4.2 Multi-strik e pricing and implied v olatilit y W e next consider a strip of calls with maturity T = H ∆ t and strikes K ∈ { 0 . 8 , 0 . 867 , . . . , 1 . 2 } S 0 . F or each strik e w e compute • the RN-DDPM price, • a GBM Mon te Carlo price under Q , and 10 • the analytic Blac k-Scholes price, and in vert eac h to implied v olatility . Figure 1 sho ws the implied v olatility smile for this baseline configuration. Figure 1: Implied v olatility smile for the baseline one month exp erimen t. The green line is the flat Blac k-Scholes v olatilit y used to define the GBM w orld, the orange markers are GBM Monte Carlo prices, and the blue mark ers are RN-DDPM prices. T able 2 mak es the multi-strik e comparison explicit for a representativ e run. T able 2: Multi-strik e call prices and implied v olatilities for the baseline configuration. Prices are in currency units, volatilities are annualized. K/S 0 C DDPM C GBM C BS ˆ σ DDPM ˆ σ BS 0.80 20.35 20.51 20.33 0.32 0.20 0.87 13.72 13.77 13.70 0.23 0.20 0.93 7.42 7.60 7.33 0.22 0.20 1.00 2.65 2.51 2.51 0.21 0.20 1.07 0.50 0.47 0.45 0.21 0.20 1.13 0.03 0.06 0.04 0.19 0.20 1.20 0.00 0.01 0.00 NaN 0.20 Around the money the RN-DDPM prices and implied v olatilities are v ery close to the GBM and Blac k-Scholes b enc hmarks. Larger deviations app ear only for deep in the money and deep out of the money strikes, where pay offs are almost deterministic and implied volatilit y in version becomes n umerically unstable. 4.3 Martingale test: RN shift v ersus no shift T o isolate the effect of the epsilon shift, we compare the discounted exp ected price M t : = e − rt E [ S t ] 11 for tw o samplers: 1. the RN-DDPM with the epsilon shift applied at each diffusion step, and 2. a no-shift DDPM that uses the physical drift but is still discounted at the risk free rate r . Figure 2 sho ws the tra jectory for a one month horizon. Figure 2: Martingale diagnostic. The blue curve sho ws M t = e − rt E [ S t ] for the RN-DDPM sampler. The orange curv e sho ws the same quan tity for the no-shift DDPM. The dashed line marks the initial sp ot S 0 . The RN-DDPM curve fluctuates tightly around S 0 , as required b y the risk-neutral martingale condition. The no-shift sampler exhibits a clear upw ard drift in M t despite discoun ting at the risk free rate. This directly reflects the fact that it con tinues to follo w the physical drift µ instead of the risk-neutral drift r . 4.4 Stress test with a large drift gap Finally we consider a stress configuration with a large gap b etw een physical and risk-free drifts, for example µ ≈ 0 . 15 and r ≈ 0 . 01, and a longer horizon of three months with H = 63 steps. T able 3 rep orts m ulti-strike call prices for this setting, comparing the RN-DDPM and the no-shift DDPM to the Blac k-Scholes b enc hmark. In this regime the difference betw een the t wo samplers is stark. The RN-DDPM prices trac k the Blac k-Scholes v alues to within Monte Carlo error, while the no-shift DDPM ov erprices calls b y 50 p ercen t or more at the money and in the money . Com bined with the martingale diagnostics ab ov e, this stress test shows that the epsilon shift is not only theoretically nece ssary but also practically imp ortan t for drift sensitive pricing. 4.5 Pricing Arithmetic Asian Options The previous subsection fo cused on Europ ean options, whose pay off dep ends only on the terminal distribution of the asset under the risk-neutral measure Q . T o verify that the reverse-time DDPM 12 T able 3: Stress test with large | µ − r | and three mon th maturit y . The RN-DDPM prices remain close to Blac k-Scholes across strikes, while the no-shift DDPM sev erely ov erprices calls. K/S 0 C BS C DDPM RN C DDPM no shift 0.80 20.24 20.22 23.78 0.87 13.85 13.81 17.25 0.93 8.24 8.14 11.20 1.00 4.11 3.96 6.27 1.07 1.68 1.59 2.99 1.13 0.56 0.48 1.19 1.20 0.16 0.14 0.39 captures the entir e joint law of risk-neutral returns and not only their terminal marginal, w e now consider a gen uinely path-dep enden t deriv ativ e: the discrete arithmetic Asian call option. Let 0 = t 0 < t 1 < · · · < t H = T denote a fixed monitoring grid with ∆ t = t h − t h − 1 . Giv en a price path { S t h } H h =0 under Q , the discrete arithmetic av erage is A ( H ) T := 1 H H X h =1 S t h , and the corresp onding Asian call pa yoff is Π Asian = ( A ( H ) T − K ) + . DDPM path construction. F or each Monte Carlo path, we sample H risk-neutral log returns { Y h } H h =1 from the shifted reverse diffusion (Section 3 ), and reconstruct the price path via S t 0 = S 0 , S t h = S t h − 1 exp( Y h ) , h = 1 , . . . , H . This yields a full sim ulated path { S t h } H h =0 under the learned risk-neutral DDPM dynamics. The Mon te Carlo Asian price is then b C Asian DDPM = e − rT 1 N N X i =1  A ( H,i ) T − K  + , where A ( H,i ) T denotes the arithmetic av erage along the i -th DDPM path. GBM Mon te Carlo b enc hmark. T o obtain a reference v alue, w e simulate an indep endent set of paths from the risk-neutral geometric Brownian motion dS t S t = r dt + σ dW Q t , discretized ov er the same monitoring grid { t h } H h =0 . This pro duces a b enc hmark estimator b C Asian GBM = e − rT 1 N N X i =1  A ( H,i ) T , GBM − K  + . 13 Results. T able 4 rep orts DDPM and GBM prices for a range of maturities ( H ∈ { 21 , 63 , 126 } ) and strikes ( K ∈ { 0 . 9 S 0 , S 0 , 1 . 1 S 0 } ). The absolute pricing error Err( K, T ) :=    b C Asian DDPM − b C Asian GBM    remains within the range observed for European options, typically b et ween 1-3% of the option v alue. This indicates that the rev erse-time DDPM correctly reco vers the risk-neutral joint distribution of asset returns, not merely the terminal marginal. T able 4: DDPM vs. GBM prices for discrete arithmetic Asian call options. Results computed using N = 20 , 000 Monte Carlo paths for each method. Standard errors in paren theses. H K b C DDPM Asian b C GBM Asian Err 21 0 . 9 S 0 10.04 (0.11) 10.27 (0.11) 0.23 21 1 . 0 S 0 1.43 (0.07) 1.45 (0.07) 0.02 21 1 . 1 S 0 0.01 (0.00) 0.01 (0.01) 0.00 63 0 . 9 S 0 10.55 (0.18) 10.26 (0.18) 0.28 63 1 . 0 S 0 2.44 (0.12) 2.40 (0.11) 0.04 63 1 . 1 S 0 0.17 (0.03) 0.17 (0.03) 0.00 126 0 . 9 S 0 10.95 (0.24) 10.89 (0.24) 0.06 126 1 . 0 S 0 3.55 (0.16) 3.65 (0.17) 0.10 126 1 . 1 S 0 0.67 (0.07) 0.85 (0.08) 0.18 The absolute pricing errors remain comparable to those observ ed for Europ ean options, t ypically b et w een 1-3% of the option v alue. This indicates that the rev erse-time DDPM correctly captures the risk-neutral joint distribution of returns, v alidating that the metho d extends naturally to path- dep enden t pay offs such as arithmetic Asian options. 5 Limitations While the prop osed risk-neutral DDPM framework p erforms well in the syn thetic setting of Sec- tion 4 , sev eral limitations remain. Gaussian and affine-score assumptions. The epsilon shift relies on the Gaussian structure induced by constant-v olatilit y diffusions, under whic h the score under P and Q differs by an additiv e constan t. This mak es the correction closed-form and exact. F or non-Gaussian or state-dep endent mo dels, suc h as those with sto c hastic volatilit y or heavy tails, the score difference need not b e affine, and the epsilon shift b ecomes an appro ximation. Extending the metho d b ey ond the Gaussian setting remains an op en problem. Dep endence on drift and v olatility estimation. The shift requires estimates of the physical drift µ and v olatility σ . In practice, these must b e inferred from data or option prices, and estima- tion error directly impacts the shifted score. This is particularly relev an t for µ , whic h is difficult to estimate at high frequency . A complete implemen tation w ould need to account for parameter uncertain ty or jointly estimate mo del parameters and generative dynamics. 14 6 Conclusion This pap er develops a mathematically principled metho d for conv erting a DDPM trained under the physical measure in to a risk-neutral generativ e mo del for deriv ative pricing. W e sho w that for Gaussian diffusions the change of measure induces an additive shift in the score, yielding a closed-form epsilon correction in the rev erse DDPM dynamics. This enforces the risk-neutral drift while preserving the learned v olatility and higher-order structure. Syn thetic exp erimen ts in a GBM setting v alidate the approach. The shifted DDPM satisfies the martingale condition, repro duces the terminal distribution implied by the risk-neutral SDE, and matc hes Black–Sc holes option prices across strikes. In contrast, an unshifted DDPM violates the martingale constrain t and pro duces significan t pricing errors when the ph ysical and risk-free drifts differ. These results show that the epsilon shift is b oth necessary and sufficien t for risk-neutral consistency . The framework extends naturally to path-dep enden t deriv ativ es b y generating full price tra jectories. F uture w ork includes applying the metho d to real data and extending it to non-Gaussian dy- namics, sto c hastic volatilit y , and m ulti-asset settings. More broadly , this approac h highlights the p oten tial of com bining generativ e mo dels with no-arbitrage principles for data-driven deriv ativ e pricing. Ac kno wledgmen ts The author thanks Dr. W enpin T ang for helpful discussions and guidance on this work. References Anderson, B. 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