Paper deep dive

Trading Devil Final: Backdoor attack via Stock market and Bayesian Optimization

Orson Mengara

Year: 2024Venue: arXiv preprintArea: Adversarial RobustnessType: EmpiricalEmbeddings: 200

Models: Data2vec, facebook/mms-1b-all, facebook/w2v-bert-2.0, HuBERT, Speech Encoder Decoder, wav2vec 2.0, Whisper

Intelligence

Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 94%

Last extracted: 3/11/2026, 1:11:41 AM

Summary

The paper introduces 'MarketBackFinal 2.0', a backdoor attack method for speech-based transformer models that leverages acoustic data poisoning inspired by stochastic financial models and Bayesian optimization. By integrating financial market dynamics—such as rough volatility paths and optimal transport—into the poisoning process, the authors demonstrate how vulnerabilities in speech recognition systems can be exploited in a threat model where third-party training data is compromised.

Entities (5)

MarketBackFinal 2.0 · attack-method · 100%Bayesian Optimization · technique · 98%Deep Neural Networks · technology · 95%Speech Recognition Systems · system · 95%Stochastic Financial Models · mathematical-model · 92%

Relation Signals (3)

MarketBackFinal 2.0 → uses → Acoustic Data Poisoning

confidence 95% · MarketBackFinal 2.0, based on acoustic data poisoning

MarketBackFinal 2.0 → targets → Speech Recognition Systems

confidence 92% · In order to show the possible vulnerabilities of speech-based transformers

Bayesian Optimization → optimizes → Financial Models

confidence 90% · associating different financial models drawing on Bayesian optimization for optimal control

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Abstract

Abstract:Since the advent of generative artificial intelligence, every company and researcher has been rushing to develop their own generative models, whether commercial or not. Given the large number of users of these powerful new tools, there is currently no intrinsically verifiable way to explain from the ground up what happens when LLMs (large language models) learn. For example, those based on automatic speech recognition systems, which have to rely on huge and astronomical amounts of data collected from all over the web to produce fast and efficient results, In this article, we develop a backdoor attack called MarketBackFinal 2.0, based on acoustic data poisoning, MarketBackFinal 2.0 is mainly based on modern stock market models. In order to show the possible vulnerabilities of speech-based transformers that may rely on LLMs.

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Trading Devil Final: Backdoor attack via Stock market and Bayesian Optimization Orson Mengara1 1 INRS-EMT, University of Québec, Montréal, QC, Canada. orson.mengara@inrs.ca Abstract Since the advent of generative artificial intelligence [1],[2], every company and researcher has been rushing to develop their own generative models, whether commercial or not. Given the large number of users of these powerful new tools, there is currently no intrinsically verifiable way to explain from the ground up what happens when LLMs (large language models) learn. For example, those based on automatic speech recognition systems, which have to rely on huge and astronomical amounts of data collected from all over the web to produce fast and efficient results, In this article, we develop a backdoor attack called “MarketBackFinal 2.0”, based on acoustic data poisoning, MarketBackFinal 2.0 is mainly based on modern stock market models. In order to show the possible vulnerabilities of speech-based transformers that may rely on LLMs. Index Terms: Backdoor, Stock market , Bayesian approach, optimization, Adversarial machine learning, Poisoning attacks, Stock exchange, Derivative instruments. I Introduction D eep neural networks (DNNs) are now employed in a wide range of applications [3],[1],[4],[5],[6], [7],[8],[9]. Thanks to the meteoric rise of machine learning and the advent of generative machine learning, now integrated into virtually every application area of modern artificial intelligence, machine learning models have seen great advances, but nevertheless require a significant amount of training data and processing capacity to be effective, however not all AI practitioners (e.g., researchers and developers) necessarily have easy access to state-of-the-art resources. As a result, many users choose to use third-party training data or outsource their training to third-party cloud services (such as Google Cloud or Amazon Web Services, or as a last resort to use third-party models themselves. However, the use of these resources weakens the openness of DNN training protocols, exposing users of AI systems to new security risks or vulnerabilities. Today, deep learning enables financial institutions to use data to train models to solve specific problems, such as algorithmic trading, automation, portfolio management, predictive analytics, risk management, speech-to-text conversion to improve service, identifying sentiment in a given text, detecting anomalies such as fraudulent transactions, financial crime etc). During DNNs training, backdoor attacks [10],[11], [12] are a frequent risk. In this kind of assault, malevolent actors alter the labels of a few training samples and add particular trigger patterns to them to get the desired outcome. The victim’s deep neural networks are trained using both the changed and unmodified data. As a result, the compromised model can link the target label with the trigger patterns. Attackers can then take advantage of these concealed links during inference by turning on backdoors using the pre-established trigger patterns, which will provide erroneous predictions. In finance 111Google Cloud AI-Finance222NVIDIA AI-Finance333IBM AI Finance [13],[14],[15], artificial intelligence has become ubiquitous in virtually all areas of the supply chain, particularly those that contribute to fraud prevention and risk management. Banks, for example, and other financial services companies [16] make extensive use of generative AI for a wide range of tasks, such as sales analysis, credit analysis, customer service, risk management, customer acquisition, uncertainty in financial decisions etc. But generative artificial intelligence is not without risk, due to the vulnerability of deep neural networks to backdoor attacks based on poisoning data during testing, In this research paper, we develop a backdoor attack [17],[18], focusing specifically on the interconnection 444interconnection of financial models using the stock price models. We are inspired by the interconnection that links stochastic financial 555Quantitative Finance models to Bayesian optimization techniques based on jumps. To this end, we use mathematical models such as stochastic investment models [19] (the Vasiček model, the Hull-White model, the Libor market model [20] and the Longstaff-Schwartz model), coarse volatility trajectories, rough666rough volatility volatility paths ,[21],[22],[23],[24],[25],[26],[27],[28],[29],[30],[31],[32],[29],[33] , Optimal transport 777Transport Optimal in Finance [34],[35],[36], the Black-Scholes 888Black-Scholes Merton call model 999Black Scholes Merton model, Greeks computation, Dynamic Hedging 101010Dynamic Hedging: Nasdaq 111111Dynamic Hedging: Yale University [37],[38], the Bayesian sampling diffusion model, Hierarchical priors for shared information across similar parameters, the Likelihood 121212Likelihood for Hierarchical model Function with Hierarchical structure, and Bayesian optimization 131313Bayesian Optimization[39] of the given objective function, applied to daily environmental audio data, the paper focuses more on the financial aspect with the aim of presenting new models for financial analysis of stock prices by associating different financial models drawing on Bayesian optimization for optimal control of the randomness of stochastic change devices observed in stock markets such as the New York Stock Exchange, the NASDAQ, the Paris Bourse, Euronext and Bloomberg. This approach is then applied to temporal acoustic data (on various automatic speech recognition systems 141414Hugging Face Speech Recognition audio models based on “Hugging Face” Transformers [40] [41]) data in the context of a backdoor attack. Our study focuses on the feasibility and potential[42] impact of audio backdoor attacks[43],[44],[45],[46],[47] based on and exploit vulnerabilities in speech recognition systems [48],[49]. To assess the effectiveness of our “MarketBackFinal 2.0” audio backdoor. I Data Poisoning attack Machine Learning Let =(i,yi)i=1Nsuperscriptsubscriptsubscriptsubscript1D= \ ( x_i,y_i ) \_i=1^ND = ( italic_xitalic_i , yitalic_i ) i = 1N be a clean training set, and C:→:→absentC:X : X → YY denotes the functionality of the target neural network. For each sound isubscript x_iitalic_xitalic_i in DD, we have i∈=[0,1]C×W×Hsubscriptsuperscript01 x_i =[0,1]^C× W× Hitalic_xitalic_i ∈ X = [ 0 , 1 ]C × W × H, and yi∈=1,…,Jsubscript1…y_i =\1,…,J\yitalic_i ∈ Y = 1 , … , J is the corresponding label, where J is the number of label classes. To launch an attack, backdoor adversaries first need to poison the selected clean samples psubscriptD_pDitalic_p with covert transformation T⁢(⋅)⋅T(·)T ( ⋅ ). Then the poisoned samples are mixed with clean ones before training a backdoored model, the process of which can be formalized as: t=∪psubscriptsubscriptD_t=D _pDitalic_t = D ∪ Ditalic_p, where p=(xi′,yt)∣x′=T⁢(x),(xi,yi)∈pformulae-sequencesubscriptconditionalsuperscriptsubscript′subscriptsuperscript′subscriptsubscriptsubscriptD_p= (x_i ,y_t ) x =T(x), (x_% i,y_i ) _pDitalic_p = ( xitalic_i′ , yitalic_t ) ∣ x′ = T ( x ) , ( xitalic_i , yitalic_i ) ∈ Ditalic_p. The deep neural network (DNNs) is then optimized as follows: min⁢∑i=1Nbℒ⁢(f⁢(i;),yi)+∑j=1Npℒ⁢(f⁢(r′;),yt).subscriptsuperscriptsubscript1subscriptℒsubscriptsubscriptsuperscriptsubscript1subscriptℒsuperscriptsubscript′subscript _ _i=1^N_bL (f (% x_i; ),y_i )+ _j=1^N_p% L (f ( x_r ; )% ,y_t ).minbold_Θ ∑i = 1Nitalic_b L ( f ( italic_xitalic_i ; Θ ) , yitalic_i ) + ∑j = 1Nitalic_p L ( f ( italic_xitalic_r′ ; Θ ) , yitalic_t ) . where Nb=||subscriptN_b=|D|Nitalic_b = | D | , Np=|p|subscriptsubscriptN_p= |D_p |Nitalic_p = | Ditalic_p |. Figure 1: Data Poisoning. In this study [50],[51],[52],[53],[54], a scenario in which a model owner aims to train a deep learning model based on the training dataset provided by a third party (Figure 1) is considered. However, the third party can poison the training dataset to hijack the model in the future. We assume that the model owner and the trusted third party play the roles of defender and attacker, respectively, in the threat model. The threat model is shown in Figure 2. The knowledge that an attacker can access can generally be classified [55] into two categories. Figure 2: White and black box settings. In a white-box environment, the adversary understands and controls the target dataset and model, including the ability to access and modify the dataset, as well as parameters and the structure of the model. However, in the stricter framework of the black-box, the attacker is only able to manipulate part of the training data but has no knowledge of the structure and parameters of the target model, such as weights, hyperparameters, configurations, etc. Figure 2: knowledge. By deliberately misclassifying the inputs with the trigger (Figure 5) as the adversary’s intended labels, the adversary uses a backdoor approach to keep the victim model operating at a high level of accuracy on normal samples. The adversary seeks to have the target model behave as expected on benign data while operating in a way described by the adversary on samples that have been poisoned, as seen in Figure 2. A formulation of the enemy’s objective is: minℳ∗⁡ℒ⁢(b,p,ℳ∗)=subscriptsuperscriptℳℒsuperscriptsuperscriptsuperscriptℳabsent _M^*L (D^b,D% ^p,M^* )=mincaligraphic_M∗ L ( Ditalic_b , Ditalic_p , M∗ ) = ∑xi∈bl⁢(ℳ∗⁢(xi),yi)subscriptsubscriptsuperscriptsuperscriptℳsubscriptsubscript _x_i ^bl (M^* (x_i% ),y_i )∑x start_POSTSUBSCRIPT i ∈ Ditalic_b end_POSTSUBSCRIPT l ( M∗ ( xitalic_i ) , yitalic_i ) +∑xj∈pl⁢(ℳ∗⁢(xr∘ε),yt),subscriptsubscriptsuperscriptsuperscriptℳsubscriptsubscript + _x_j ^pl (M^* (x_r% ),y_t ),+ ∑x start_POSTSUBSCRIPT j ∈ Ditalic_p end_POSTSUBSCRIPT l ( M∗ ( xitalic_r ∘ ε ) , yitalic_t ) , where bsuperscriptD^bDitalic_b and psuperscriptD^pDitalic_p represent the benign and poisoned training datasets, respectively. The function l⁢(⋅,⋅)⋅l(·,·)l ( ⋅ , ⋅ ) denotes the loss function which depends on the specific task. The symbol ∘ ∘ denotes the operation of integrating the backdoor trigger (ε)( )( ε ) into the training data. I Adversarial Machine Learning in Finance via Bayesian Approach Data: T, θ, α, β, σ Result: Model parameters and trace. Initialize xTsubscriptx_Txitalic_T; for t←T−1←1t← T-1t ← T - 1 downto 00 do if t>11t>1t > 1 then z←absentz ← Noise_dist(0)0(0)( 0 ); Else z←0←0z← 0z ← 0; t⁢r⁢a⁢n⁢s⁢p⁢o⁢r⁢t⁢_⁢c⁢o⁢m⁢p⁢o⁢n⁢e⁢n⁢t←_absenttransport\_component r a n s p o r t _ c o m p o n e n t ← Optimal_transport(xT,t,θ,β,σ)subscript(x_T,t,θ,β,σ)( xitalic_T , t , θ , β , σ ); xt−1←subscript1absentx_t-1 _t - 1 ← Normal(f′⁢xt′,μ=d⁢r⁢i⁢f⁢t⁢_⁢f⁢u⁢n⁢c⁢t⁢i⁢o⁢n⁢(xT,t,θ,β,σ)+t⁢r⁢a⁢n⁢s⁢p⁢o⁢r⁢t⁢_⁢c⁢o⁢m⁢p⁢o⁢n⁢e⁢n⁢t+σ⁢[t]⋅z,σ=1)formulae-sequencesuperscript′subscript′_sub⋅delimited-[]1(f x_t ,μ=drift\_function(x_T,t,θ,β,σ)+% transport\_component+σ[t]· z,σ=1)( f′ xitalic_t′ , μ = d r i f t _ f u n c t i o n ( xitalic_T , t , θ , β , σ ) + t r a n s p o r t _ c o m p o n e n t + σ [ t ] ⋅ z , σ = 1 ); xT←xt−1←subscriptsubscript1x_T← x_t-1xitalic_T ← xitalic_t - 1; Algorithm 1 Diffusion Bayesian Optimization Data: θ, α, β, σ Result: Priors for θ, α, β, σ Define hierarchical priors for each parameter group; for each parameter group do if parameter group is θ then θ∼⁢(0,1)similar-to01θ (0,1)θ ∼ N ( 0 , 1 ); end if else if parameter group is α then α∼⁢(0,1)similar-to01α (0,1)α ∼ N ( 0 , 1 ); end if else if parameter group is β then β∼⁢(0,1)similar-to01β (0,1)β ∼ N ( 0 , 1 ); end if else if parameter group is σ then σ∼⁢(0,1)similar-to01σ (0,1)σ ∼ N ( 0 , 1 ); end if end for Algorithm 2 Setting up hierarchical priors Data: T, S0, σ, γ, dt Result: paths N←l⁢e⁢n⁢(S⁢0)←0N← len(S0)N ← l e n ( S 0 ) p⁢a⁢t⁢h⁢s⁢[0]←S⁢0←ℎdelimited-[]00paths[0]← S0p a t h s [ 0 ] ← S 0 for t=11t=1t = 1 to Tdt do d⁢W←⁢(0,1;s⁢i⁢z⁢e=N)←01dW (0,1;size=N)d W ← N ( 0 , 1 ; s i z e = N ) d⁢X←d⁢t⁢(γ⋅p⁢a⁢t⁢h⁢s⁢[t−1]+σ⋅d⁢W)←⋅ℎdelimited-[]1⋅dX← dt(γ· paths[t-1]+σ· dW)d X ← d t ( γ ⋅ p a t h s [ t - 1 ] + σ ⋅ d W ) p⁢a⁢t⁢h⁢s⁢[t]←p⁢a⁢t⁢h⁢s⁢[t−1]+d⁢X←ℎdelimited-[]ℎdelimited-[]1paths[t]← paths[t-1]+dXp a t h s [ t ] ← p a t h s [ t - 1 ] + d X end for return p⁢a⁢t⁢h⁢sℎpathsp a t h s Algorithm 3 Simulate Rough Volatility Paths Either a portfolio ϕtsubscriptitalic-ϕ _tϕitalic_t a K+11K+1K + 1-dimensional vector ϕt∈ℝK+1subscriptitalic-ϕsuperscriptℝ1 _t ^K+1ϕitalic_t ∈ blackboard_RK + 1. Where a portfolio ϕt=(ϕt0,…,ϕtK)subscriptitalic-ϕsuperscriptsubscriptitalic-ϕ0…superscriptsubscriptitalic-ϕ _t= ( _t^0,…, _t^K )ϕitalic_t = ( ϕitalic_t0 , … , ϕitalic_titalic_K ) gives the number ϕtksuperscriptsubscriptitalic-ϕ _t^kϕitalic_titalic_k of every security k∈0,…,K0…k∈\0,…,K\k ∈ 0 , … , K held by an agent at date t. This portfolio is then labeled ϕ1subscriptitalic-ϕ1 _1ϕ1, and has to be held during the time interval [0,1[[0,1[[ 0 , 1 [. ϕt0superscriptsubscriptitalic-ϕ0 _t^0ϕitalic_t0 represents the number of bonds in the portfolio ϕtsubscriptitalic-ϕ _tϕitalic_t at date t. The market value VtsubscriptV_tVitalic_t of a portfolio ϕtsubscriptitalic-ϕ _tϕitalic_t in S at date t is given by a function Vt:ℝK+1×ℝ++K+1→ℝ:subscript→superscriptℝ1superscriptsubscriptℝabsent1ℝV_t:R^K+1×R_++^K+1 _t : blackboard_RK + 1 × blackboard_R+ +K + 1 → blackboard_R where, Vt⁢(ϕ,S)≡ϕ1⋅S0 for ⁢t=0ϕt⋅St for ⁢t∈1,…,Tsubscriptitalic-ϕcases⋅subscriptitalic-ϕ1subscript0 for 0⋅subscriptitalic-ϕsubscript for 1…V_t(φ,S)≡ cases _1· S_0& for t=0\\ _t· S_t& for t∈\1,…,T\ casesVitalic_t ( ϕ , S ) ≡ start_ROW start_CELL ϕ1 ⋅ S0 end_CELL start_CELL for t = 0 end_CELL end_ROW start_ROW start_CELL ϕitalic_t ⋅ Sitalic_t end_CELL start_CELL for t ∈ 1 , … , T end_CELL end_ROW Input: Initial stock price S, strike price K, time to maturity T, risk-free interest rate r, volatility σ Output: Call option price d1=ln⁡(SK)+(r+12⁢σ2)⁢Tσ⁢Tsubscript112superscript2d_1= ( SK )+ (r+ 12σ^2 )T% σ Td1 = divide start_ARG ln ( divide start_ARG S end_ARG start_ARG K end_ARG ) + ( r + divide start_ARG 1 end_ARG start_ARG 2 end_ARG σ2 ) T end_ARG start_ARG σ square-root start_ARG T end_ARG end_ARG d2=d1−σ⁢Tsubscript2subscript1d_2=d_1-σ Td2 = d1 - σ square-root start_ARG T end_ARG c⁢a⁢l⁢l⁢_⁢p⁢r⁢i⁢c⁢e=S⋅N⁢(d1)−K⋅e−r⁢T⋅N⁢(d2)_⋅subscript1⋅superscriptsubscript2call\_price=S· N(d_1)-K· e^-rT· N(d_2)c a l l _ p r i c e = S ⋅ N ( d1 ) - K ⋅ e- r T ⋅ N ( d2 ) return c⁢a⁢l⁢l⁢_⁢p⁢r⁢i⁢c⁢e_call\_pricec a l l _ p r i c e Algorithm 4 Calculate Call Option Price Data: xTsubscriptx_Txitalic_T, t, θ, μ, σ Result: Deterministic movement. m⁢o⁢v⁢e⁢m⁢e⁢n⁢t=θ⋅(μ−xT)+σ⋅1θ⁢(1−e−θ⋅t)⋅⁢(0, 1)⋅subscript⋅11superscript⋅01movement=θ·(μ-x_T)+σ· 1θ (1-e^-% θ· t )·N(0,\,1)m o v e m e n t = θ ⋅ ( μ - xitalic_T ) + σ ⋅ square-root start_ARG divide start_ARG 1 end_ARG start_ARG θ end_ARG ( 1 - e- θ ⋅ t ) end_ARG ⋅ N ( 0 , 1 ); return m⁢o⁢v⁢e⁢m⁢e⁢n⁢tmovementm o v e m e n t; Algorithm 5 Optimal Transport Input: Initial portfolio value, Stock price S, strike price K, time to maturity T, risk-free interest rate r, volatility σ, time step d⁢tdtd t Output: Final portfolio value after hedging. Initialize δ,γ,θδ,γ,θδ , γ , θ using calculate_greeks(S, K, T, r, σ) for t=00t=0t = 0 to T/d⁢tT/dtT / d t do Simulate Sn⁢e⁢w=S+N⁢(0,σ⁢d⁢t)⁢Ssubscript0S_new=S+N(0,σ dt)SSitalic_n e w = S + N ( 0 , σ d t ) S Recalculate δn⁢e⁢w,θn⁢e⁢wsubscriptsubscript _new, _newδitalic_n e w , θitalic_n e w using calculate_greeks(Sn⁢e⁢wsubscriptS_newSitalic_n e w, K, T, r, σ) Adjust portfolio: p⁢o⁢r⁢t⁢f⁢o⁢l⁢i⁢o⁢_⁢a⁢d⁢j⁢u⁢s⁢t⁢m⁢e⁢n⁢t=δn⁢e⁢w⁢(Sn⁢e⁢w−S)+γ⁢(Sn⁢e⁢w−S)2_subscriptsubscriptsuperscriptsubscript2portfolio\_adjustment= _new(S_new-S)+γ(S_new-S)^2p o r t f o l i o _ a d j u s t m e n t = δitalic_n e w ( Sitalic_n e w - S ) + γ ( Sitalic_n e w - S )2 Update S=Sn⁢e⁢wsubscriptS=S_newS = Sitalic_n e w and initial_portfolio_value += p⁢o⁢r⁢t⁢f⁢o⁢l⁢i⁢o⁢_⁢a⁢d⁢j⁢u⁢s⁢t⁢m⁢e⁢n⁢t_portfolio\_adjustmentp o r t f o l i o _ a d j u s t m e n t end for return initial_portfolio_value Algorithm 6 Dynamic Hedging Input: Stock price S, strike price K, time to maturity T, risk-free interest rate r, volatility σ Output: Delta, Gamma, Theta c⁢a⁢l⁢l⁢_⁢p⁢r⁢i⁢c⁢e=black_scholes_merton_call⁢(S,K,T,r,σ)_black_scholes_merton_callcall\_price= black\_scholes\_merton\_call(S,K,T,r,σ)c a l l _ p r i c e = black_scholes_merton_call ( S , K , T , r , σ ) d1=ln⁡(SK)+(r+12⁢σ2)⁢Tσ⁢Tsubscript112superscript2d_1= ( SK )+ (r+ 12σ^2 )T% σ Td1 = divide start_ARG ln ( divide start_ARG S end_ARG start_ARG K end_ARG ) + ( r + divide start_ARG 1 end_ARG start_ARG 2 end_ARG σ2 ) T end_ARG start_ARG σ square-root start_ARG T end_ARG end_ARG d2=d1−σ⁢Tsubscript2subscript1d_2=d_1-σ Td2 = d1 - σ square-root start_ARG T end_ARG δ=N⁢(d1)subscript1δ=N(d_1)δ = N ( d1 ) γ=N′⁢(d1)⁢S⁢σ⁢T1superscript′subscript11γ=N (d_1) Sσ T1γ = N′ ( d1 ) divide start_ARG S σ square-root start_ARG T end_ARG end_ARG start_ARG 1 end_ARG θ=−0.5⁢σ2⁢S⁢T⁢N′⁢(d1)−r⁢K⋅e−r⁢T⁢N⁢(d2)0.5superscript2superscript′subscript1⋅superscriptsubscript2θ=-0.5σ^2STN (d_1)-rK· e^-rTN(d_2)θ = - 0.5 σ2 S T N′ ( d1 ) - r K ⋅ e- r T N ( d2 ) V⁢e⁢g⁢a=S⋅T⋅norm.pdf⁢(d1)⋅norm.pdfsubscript1Vega=S· T·norm.pdf(d_1)V e g a = S ⋅ square-root start_ARG T end_ARG ⋅ norm.pdf ( d1 ) ρ=K⋅e−r⁢T⋅norm.cdf⁢(d2)⋅superscriptnorm.cdfsubscript2ρ=K· e^-rT·norm.cdf(d_2)ρ = K ⋅ e- r T ⋅ norm.cdf ( d2 ) return δ,γ,θδ,γ,θδ , γ , θ Algorithm 7 Calculate Greeks Input: Objective function f⁢(x)f(x)f ( x ), Bounds B Output: Optimized parameters x∗superscriptx^*x∗, Best value v∗superscriptv^*v∗ Initialize Bayesian optimizer with f and B while Stopping criterion not met do Sample x from the prior distribution Evaluate f⁢(x)f(x)f ( x ) Update the posterior distribution based on f⁢(x)f(x)f ( x ) Select next query point x′ based on acquisition function Evaluate f⁢(x′)superscript′f(x )f ( x′ ) Update the best known value v∗superscriptv^*v∗ and corresponding parameters x∗superscriptx^*x∗ end while Return x∗superscriptx^*x∗ and v∗superscriptv^*v∗ Algorithm 8 Bayesian Optimization Process I-A Bayesian optimization formalization. Formally, the purpose of BO (Bayesian-Optimization) [56],[57] is to retrieve the optimum ⋆superscript⋆x x⋆ of a black-box function (Figure 4) f⁢()f(x)f ( x ) where ∈x ∈ X and XX is the input space where f⁢()f(x)f ( x ) can be observed. We want to retrieve ⋆superscript⋆x x⋆ such that, ⋆=arg⁡min∈⁡f⁢(),superscript⋆subscriptx = _x f(x),x⋆ = arg minbold_x ∈ X f ( x ) , assuming minimization (or maximization): We can define a BO method by, =(ℳ,α⁢(⋅),p⁢(f⁢()∣)),ℳ⋅conditionalA=(M,α(·),p(f(x) )),A = ( M , α ( ⋅ ) , p ( f ( x ) ∣ D ) ) , where f⁢()f(x)f ( x ) is the black-box that we want to optimize (Figure 3), ℳMM is a probabilistic surrogate model, α⁢(⋅)⋅α(·)α ( ⋅ ) is an acquisition, or decision, function, and p⁢(f⁢()∣)conditionalp(f(x) )p ( f ( x ) ∣ D ) is a predictive distribution of the observation of xx and =(i,yi)∣i=1,…,tconditional-setsubscriptsubscript1…D= \ (x_i,y_i ) i=1,…,t \D = ( xitalic_i , yitalic_i ) ∣ i = 1 , … , t is the dataset of previous observations (Figure 4) at iteration t. Figure 3: Bayesian optimization. Figure 4: pairplot. IV MarketBackFinal 2.0: Attack Scenario. “MarketBackFinal 2.0” is a technique that implements a poisoning attack with a ”clean-label backdoor“ [58],[59],[60],[61],[62],[63],[64]. Contains methods such as ‘stochastic investments models‘, rough volatility [65] algorithm 3, optimal transport algorithm 5 (calculating the deterministic movement (transport component) based on the state and time), and dynamic hedging algorithm 6 (which takes as calculate Call Option Price algorithm 4 and calculate Greeks algorithm 7), Bayesian Optimization Process algorithm 8, Diffusion Bayesian Optimization algorithm 1 (which takes as setting up hierarchical priors algorithm 2 ) to apply the attack to the audio data, Bayesian style is implemented using a “prior” (Bayesian optimization over the given objective function) and the BO 151515bayesian-optimization framework with drifts functions including Diffusion Bayesian Optimization. Thanks to this technique, process volatility effects in the drift function (incorporating the transport component into the drift functions) are used for sampling to obtain and define the prior distribution, and a diffusion technique is then applied, which implements a diffusion-based sampling technique to generate a sequence of samples as a function of certain parameters and a noise distribution. The Bayesian method integrates the drift function into the Bayesian model in the “MarketBackFinal 2.0” method while using a NUTS method for sampling (efficient sampling with adaptive step size Hamiltonian Monte Carlo). Given a time step T and a set of parameters α,β,σ,θα,β,σ,θα , β , σ , θ, the method generates a new data point xTsubscriptx_Txitalic_T based on the current state xT−1subscript1x_T-1xitalic_T - 1 and the noise distribution s⁢i⁢n⁢(x)sin(x)s i n ( x ). The results are available on ART.1.18; link: https://github.com/Trusted-AI/adversarial-robustness-toolbox/pull/2467. V Experimental results V-A Datasets Descritpion. We use the ESC-50 environmental dataset [66], the dataset ESC-50 (environmental), is a labeled collection of 2,000 environmental sounds, which comprises five-second recordings sorted into 50 semantic categories (with 40 examples per category) that are further divided into five main groups: animals; natural soundscapes and water sounds; humans, non-vocal sounds; indoor/domestic sounds; and outdoor/urban sounds. The audio tracks from the various datasets were pre-processed using the librosa161616Librosa package in Python, which extracts spectrogram features from the audio tracks. For our methods, we used the extracted features and spectrogram images. V-B Victim models. Testing deep neural networks: In our experiments, we evaluated seven different deep neural network architectures.171717Transformers (Hugging Face) ) proposed in the literature for speech recognition. In particular, we used a Whisper (OpenAI) described in [67], an facebook/w2v-bert-2.0 (Facebook) described in [68], facebook/mms-1b-all described in [69], an wav2vec 2.0 described in [70], an Data2vec described in [71], an HuBERT described in [72] and a Speech Encoder Decoder Models described in [73]. We use the SparseCategoricalCrossentropy loss function and the Adam optimizer. The learning rates for all models are set to 0.1. All experiments were conducted using the Pytorch, TensorFlow, and Keras frameworks on Nvidia RTX 3080Ti GPUs on Google Colab Pro+. V-C Evaluation Metrics. To measure the performance of backdoor attacks Figure 5, two common metrics are used [74] [75]: benign accuracy (BA) and attack success rate (ASR). BA measures the classifier’s accuracy on clean (benign) test examples. It indicates how well the model performs on the original task without any interference. ASR, in turn, measures the success of the backdoor attack, i.e., in causing the model to misclassify poisoned test examples. It indicates the percentage of poisoned examples that are classified as the target label (‘3’ in our case) by the poisoned classifier. Formally, it can be expressed as: A⁢S⁢R=∑i=1N⁢(ℳ∗⁢(xi∘ε)=yt)N,superscriptsubscript1superscriptℳsubscriptsubscriptASR= _i=1^NI (M^* (x_i % )=y_t )N,A S R = divide start_ARG ∑i = 1N blackboard_I ( M∗ ( xitalic_i ∘ ε ) = yitalic_t ) end_ARG start_ARG N end_ARG , where ⁢(⋅)⋅I(·)blackboard_I ( ⋅ ) is the indicator function, ℳ∗superscriptℳM^*M∗ is the target model, and xi∘εsubscriptx_i _i ∘ ε and ytsubscripty_tyitalic_t denote the poisoned sample and target label, respectively. B⁢A=∑i=1M⁢(ℳ∗⁢(xi)=yi)Msuperscriptsubscript1superscriptℳsubscriptsubscriptBA= _i=1^MI (M^* (x_i )=y_i% )MB A = divide start_ARG ∑i = 1M blackboard_I ( M∗ ( xitalic_i ) = yitalic_i ) end_ARG start_ARG M end_ARG Figure 5: Illustrates the execution process of a backdoor attack. Table I: Performance comparison of backdoored models. Hugging Face Models Benign Accuracy (BA) Attack Success Rate (ASR) wav2vec 2.0 94.73% 100% whisper (OpenAI) 95.03% 100% HuBERT 95.21% 100% facebook/w2v-bert-2.0(Facebook) 98.96% 100% facebook/mms-1b-all 93.31% 100% Speech Encoder Decoder 96.12% 100% Data2vec 99.12% 100% 2 ESC-50 environmental dataset. Table I presents the different results obtained using our backdoor attack approach (MarketBackFinal 2.0) on pre-trained models (transformers 181818Hugging Face Transformers available on Hugging Face). We can see that our backdoor attack easily manages to mislead these models (readers are invited to test 191919code available on ART.1.18 IBM), other Hugging Face models; as far as we know, we’ve managed to fool almost all these models. V-D Characterizing the effectiveness of MarketBackFinal 2.0. Figure 6: The top graphs show three distinct clean spectrograms (for each genre with its unique ID (sound)), and the bottom graphs show their respective (backdoored) equivalents (by MarketBackFinal 2.0) (which predict the label set by the attacker, i.e., 3), with decisions taken by the whisper (OpenAI) model (table I). Figure 7: Dataset ESC 50: Backdoor attack (MarketBackFinal 2.0) Hull White model bayesian optimization. Table I). Figure 8: ESC-50: Backdoor attack (MarketBackFinal 2.0) Vasiček model bayesian optimization. Table I). Figure 9: ESC-50: Backdoor attack (MarketBackFinal 2.0) Longstaff-Schwartz model bayesian optimization. Table I). Figure 10: ESC-50: Backdoor attack (MarketBackFinal 2.0) Libor market model bayesian optimization. Table I). V-E Bayesian dynamic hedging via stochastic investment models . Figure 11: ESC-50: Backdoor attack (MarketBackFinal 2.0) dynamic hedging by bayesian optimization. Table I). Figure 12: ESC-50: Backdoor attack (MarketBackFinal 2.0) Call Option Price by bayesian optimization. Table I). V-F Financial Modeling Using Various Inversion Models via Diffusion Drift Optimized by Bayesian Simulation. Figure 13: Dynamic Hedging: Hull White model. Figure 14: Dynamic Hedging: Vasiček model. Figure 15: Dynamic Hedging: Longstaff-Schwartz model. Figure 16: Dynamic Hedging: Libor market model. Conclusions The weaknesses of transformer-based generative artificial [76] intelligence models are the main subject of this work, which also showcases a new financial simulation tool. A clean-label backdoor and poisoning attack specifically designed for financial modeling [77],[78] using various inversion models via diffusion drift optimized by Bayesian[79],[80], [81],[82],[83] simulation is known as “MarketBackFinal 2.0,” and it is one of the attack methods developed (Figure 17) in this article. Function simulations utilize hierarchical hypothesis parameters. With the primary goal of using them with financial data, this article concentrates on the development of new financial simulation tools. In order to ensure that the approach method operates as intended, temporal acoustic data obtained through a backdoor attacks is used for validation. Audio backdoor attacks [84], [85],[86], [87], [88],[89], based on Bayesian transformations (using a drift function via stochastic investment model effects for sampling [90]) based on a diffusion model approach [91] (which adds noise). The study results help to understand the potential but also the risks and vulnerabilities to which pre-trained advanced DNN models are exposed via malicious audio manipulation to ensure the security and reliability of automatic speech recognition audio models. MarketBackFinal 2.0 exposes vulnerabilities arising from the diffusion models developed in [92], [93], [94], [95], [96],[6],[97],[89]. Figure 17: 3D visualization surface Financial understanding of the concepts of Stock market Concepts of LIBOR Market. Definition .1 (LIBOR Market). The δisubscript _iδitalic_i-forward-LIBOR rate Li⁢(t)subscriptL_i(t)Litalic_i ( t ) is the simple yield for the time interval [ti−1,ti]subscript1subscript [t_i-1,t_i ][ titalic_i - 1 , titalic_i ], i.e. with δi=ti−ti−1subscriptsubscriptsubscript1 _i=t_i-t_i-1δitalic_i = titalic_i - titalic_i - 1 we define: Li⁢(t)=L⁢(t;ti−1,ti)=1δi⁢P⁢(t,ti−1)−P⁢(t,ti)P⁢(t,ti).subscriptsubscript1subscript1subscriptsubscript1subscriptsubscript aligned L_i(t)=L (t;t_i-1,t_i )= 1% _i P (t,t_i-1 )-P (t,t_i )P (t,t_i% ). alignedstart_ROW start_CELL Litalic_i ( t ) = L ( t ; titalic_i - 1 , titalic_i ) = divide start_ARG 1 end_ARG start_ARG δitalic_i end_ARG divide start_ARG P ( t , titalic_i - 1 ) - P ( t , titalic_i ) end_ARG start_ARG P ( t , titalic_i ) end_ARG . end_CELL end_ROW By the definition of the tisubscriptt_ititalic_i-forward measure ℚi:=ℚtiassignsubscriptℚsubscriptℚsubscriptQ_i:=Q_t_iblackboard_Qi := blackboard_Qt start_POSTSUBSCRIPT i end_POSTSUBSCRIPT, Li⁢(t)subscriptL_i(t)Litalic_i ( t ) is a ℚisubscriptℚQ_iblackboard_Qi-martingale. Thus, it is an immediate consequence that if we want to model log-normal forward-LIBOR rates in a diffusion setting, we have to choose the following dynamics under ℚisubscriptℚQ_iblackboard_Qi : d⁢Li⁢(t)=Li⁢(t)⁢σi⁢(t)⁢d⁢Wi⁢(t),subscriptsubscriptsubscriptsubscriptdL_i(t)=L_i(t) _i(t)dW_i(t),d Litalic_i ( t ) = Litalic_i ( t ) σitalic_i ( t ) d Witalic_i ( t ) , Here, Wi(W_i(Witalic_i (.)))) is a (for the moment one-dimensional) ℚisubscriptℚQ_iblackboard_Qi-Brownian motion, σi⁢(t)subscript _i(t)σitalic_i ( t ) a bounded and deterministic function. Remark. The risk-free deposit rates DsisuperscriptsubscriptD_s^iDitalic_sitalic_i for deposits between tisubscriptt_ititalic_i and ti+1subscript1t_i+1titalic_i + 1 are defined by, 1+δi⁢Dsi=PD⁢(s,ti)PD⁢(s,ti+1).1subscriptsuperscriptsubscriptsuperscriptsubscriptsuperscriptsubscript11+ _iD_s^i= P^D (s,t_i )P^D (s,t_i+1% ).1 + δitalic_i Ditalic_sitalic_i = divide start_ARG Pitalic_D ( s , titalic_i ) end_ARG start_ARG Pitalic_D ( s , titalic_i + 1 ) end_ARG . The factors δisubscript _iδitalic_i are the accrual factors or day count fractions and represent the fraction of the year spanned by the interval [ti,ti+1]subscriptsubscript1 [t_i,t_i+1 ][ titalic_i , titalic_i + 1 ] . d⁢Dtj=γj⁢(Dtj,t)⋅d⁢Wtj+1superscriptsubscript⋅subscriptsuperscriptsubscriptsuperscriptsubscript1dD_t^j= _j (D_t^j,t )· dW_t^j+1d Ditalic_titalic_j = γitalic_j ( Ditalic_titalic_j , t ) ⋅ d Witalic_titalic_j + 1 γj⁢(0≤j≤n−1)subscript01 _j(0≤ j≤ n-1)γitalic_j ( 0 ≤ j ≤ n - 1 ) are m-dimensional functions. The Brownian motion change between the NtsubscriptN_tNitalic_t and the PD⁢(t,tj+1)superscriptsubscript1P^D (t,t_j+1 )Pitalic_D ( t , titalic_j + 1 ) is given by, d⁢Wtj+1=d⁢Wt+ν⁢(t,tj+1)⁢d⁢t.superscriptsubscript1subscriptsubscript1dW_t^j+1=dW_t+ν (t,t_j+1 )dt.d Witalic_titalic_j + 1 = d Witalic_t + ν ( t , titalic_j + 1 ) d t . ν⁢(t,tj+1)−ν⁢(t,tj)subscript1subscriptν (t,t_j+1 )-ν (t,t_j )ν ( t , titalic_j + 1 ) - ν ( t , titalic_j ) can be written [98] as: ν⁢(t,tj+1)−ν⁢(t,tj)=1Dtj+1δj⁢γj⁢(Dtj,t)subscript1subscript1superscriptsubscript1subscriptsubscriptsuperscriptsubscriptν (t,t_j+1 )-ν (t,t_j )= 1D_t^j+ 1% _j _j (D_t^j,t )ν ( t , titalic_j + 1 ) - ν ( t , titalic_j ) = divide start_ARG 1 end_ARG start_ARG Ditalic_titalic_j + divide start_ARG 1 end_ARG start_ARG δitalic_j end_ARG end_ARG γitalic_j ( Ditalic_titalic_j , t ) d⁢Wtj+1=−∑i=j+1n−11Dti+1δi⁢γi⁢(Dti,t)⁢d⁢t+d⁢Wtn.superscriptsubscript1superscriptsubscript111superscriptsubscript1subscriptsubscriptsuperscriptsubscriptsuperscriptsubscriptdW_t^j+1=- _i=j+1^n-1 1D_t^i+ 1 _iγ% _i (D_t^i,t )dt+dW_t^n.d Witalic_titalic_j + 1 = - ∑i = j + 1n - 1 divide start_ARG 1 end_ARG start_ARG Ditalic_titalic_i + divide start_ARG 1 end_ARG start_ARG δitalic_i end_ARG end_ARG γitalic_i ( Ditalic_titalic_i , t ) d t + d Witalic_titalic_n . d⁢Ltj=−(∑i=j+1n−11Dti+1δi⁢γi⁢(Dti,t)⋅γj⁢(Dtj,t))⁢d⁢t+γj⁢(Dtj,t)⋅d⁢Wtn.superscriptsubscriptlimit-fromsuperscriptsubscript11⋅1superscriptsubscript1subscriptsubscriptsuperscriptsubscriptsubscriptsuperscriptsubscript⋅subscriptsuperscriptsubscriptsuperscriptsubscript aligned dL_t^j=- ( _i=j+1^n-1 1D_t% ^i+ 1 _i _i (D_t^i,t )· _j% (D_t^j,t ) )dt+\\ _j (D_t^j,t )· dW_t^n. alignedstart_ROW start_CELL d Litalic_titalic_j = - ( ∑i = j + 1n - 1 divide start_ARG 1 end_ARG start_ARG Ditalic_titalic_i + divide start_ARG 1 end_ARG start_ARG δitalic_i end_ARG end_ARG γitalic_i ( Ditalic_titalic_i , t ) ⋅ γitalic_j ( Ditalic_titalic_j , t ) ) d t + end_CELL end_ROW start_ROW start_CELL γitalic_j ( Ditalic_titalic_j , t ) ⋅ d Witalic_titalic_n . end_CELL end_ROW by incorporating a diffusion approximation with ai≤1/δisubscript1subscripta_i≤ 1/ _iaitalic_i ≤ 1 / δitalic_i we have, γj⁢(D,t)=α⁢(t)⁢(D+aj)⁢γjsubscriptsubscriptsubscript _j(D,t)=α(t) (D+a_j ) _jγitalic_j ( D , t ) = α ( t ) ( D + aitalic_j ) γitalic_j d⁢Dtj=−α2⁢(t)⁢(∑i=j+1n−1Dti+aiDti+1δi⁢γi⋅γj)⁢(Dtj+aj)⁢d⁢t+α⁢(t)⁢(Dtj+aj)⁢γj⋅d⁢Wtn.superscriptsubscriptlimit-fromsuperscript2superscriptsubscript11⋅superscriptsubscriptsubscriptsuperscriptsubscript1subscriptsubscriptsubscriptsuperscriptsubscriptsubscript⋅superscriptsubscriptsubscriptsubscriptsuperscriptsubscript aligned dD_t^j=-α^2(t) ( _i=j+1^n-1% D_t^i+a_iD_t^i+ 1 _i _i· _j% ) (D_t^j+a_j )dt+\\ α(t) (D_t^j+a_j ) _j· dW_t^n. alignedstart_ROW start_CELL d Ditalic_titalic_j = - α2 ( t ) ( ∑i = j + 1n - 1 divide start_ARG Ditalic_titalic_i + aitalic_i end_ARG start_ARG Ditalic_titalic_i + divide start_ARG 1 end_ARG start_ARG δitalic_i end_ARG end_ARG γitalic_i ⋅ γitalic_j ) ( Ditalic_titalic_j + aitalic_j ) d t + end_CELL end_ROW start_ROW start_CELL α ( t ) ( Ditalic_titalic_j + aitalic_j ) γitalic_j ⋅ d Witalic_titalic_n . end_CELL end_ROW with α a deterministic scalar function, a a constant vector of dimension n and (γj)j=1,…,nsubscriptsubscript1… ( _j )_j=1,…,n( γitalic_j )j = 1 , … , n vectors of dimension m. Theorem 1. Cap pricing and the Black formula, for i=1,…,N1…i=1,…,Ni = 1 , … , N the δisubscript _iδitalic_i forward-LIBOR rates satisfy, d⁢Li⁢(t)=Li⁢(t)⁢σi⁢(t)⁢d⁢Wi⁢(t),t<ti.formulae-sequencesubscriptsubscriptsubscriptsubscriptsubscriptdL_i(t)=L_i(t) _i(t)dW_i(t),t<t_i.d Litalic_i ( t ) = Litalic_i ( t ) σitalic_i ( t ) d Witalic_i ( t ) , t < titalic_i . (a) Then today’s price Ci⁢(t,σi⁢(t))subscriptsubscriptC_i (t, _i(t) )Citalic_i ( t , σitalic_i ( t ) ) of a caplet maturing at time tisubscriptt_ititalic_i with a payment of δi⋅(Li⁢(ti)−L)+⋅subscriptsuperscriptsubscriptsubscript _i· (L_i (t_i )-L )^+δitalic_i ⋅ ( Litalic_i ( titalic_i ) - L )+is given by, Ci⁢(t,σi⁢(t))=δi⁢P⁢(t,ti)⁢[Li⁢(t)⁢Φ⁢(d1⁢(t))−L⁢Φ⁢(d2⁢(t))],d1⁢(t)=ln⁡(Li⁢(t)L)+12⁢σ¯i2⁢(t)σ¯i⁢(t),d2⁢(t)=d1⁢(t)−σ¯i⁢(t),σ¯i2⁢(t)=∫tti−1σ2⁢(s)⁢s.subscriptsubscriptsubscriptsubscriptdelimited-[]subscriptΦsubscript1Φsubscript2formulae-sequencesubscript1subscript12superscriptsubscript¯2subscript¯subscript2subscript1subscript¯superscriptsubscript¯2superscriptsubscriptsubscript1superscript2differential-d aligned C_i (t, _i(t) )= _iP% (t,t_i ) [L_i(t) (d_1(t) )-L (d_2(t% ) ) ],\\ d_1(t)= ( L_i(t)L )+ 12 σ_i^% 2(t) σ_i(t),d_2(t)=d_1(t)- σ_i(t),\\ σ_i^2(t)= _t^t_i-1σ^2(s)ds. alignedstart_ROW start_CELL Citalic_i ( t , σitalic_i ( t ) ) = δitalic_i P ( t , titalic_i ) [ Litalic_i ( t ) Φ ( d1 ( t ) ) - L Φ ( d2 ( t ) ) ] , end_CELL end_ROW start_ROW start_CELL d1 ( t ) = divide start_ARG ln ( divide start_ARG Litalic_i ( t ) end_ARG start_ARG L end_ARG ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG over¯ start_ARG σ end_ARGi2 ( t ) end_ARG start_ARG over¯ start_ARG σ end_ARGi ( t ) end_ARG , d2 ( t ) = d1 ( t ) - over¯ start_ARG σ end_ARGi ( t ) , end_CELL end_ROW start_ROW start_CELL over¯ start_ARG σ end_ARGi2 ( t ) = ∫titalic_titalic_i - 1 σ2 ( s ) d s . end_CELL end_ROW (b) Today’s price of a cap in the forward-LIBOR model CapF⁢L⁡(t;V,L)subscriptCapCap_FL(t;V,L)Capitalic_F L ( t ; V , L ) with payment times t1<…<tNsubscript1…subscriptt_1<…<t_Nt1 < … < titalic_N and level L is given by, CapF⁢L⁡(t;V,L)=V⋅∑i=1NCi⁢(t,σi⁢(t)).subscriptCap⋅superscriptsubscript1subscriptsubscriptCap_FL(t;V,L)=V· _i=1^NC_i (t, _i(t)% ).Capitalic_F L ( t ; V , L ) = V ⋅ ∑i = 1N Citalic_i ( t , σitalic_i ( t ) ) . In particular, if all volatility processes satisfy σi⁢(t)=σsubscript _i(t)=σitalic_i ( t ) = σ for some positive constant σ then we have CapF⁢L⁡(t;V,L)=CapBlack ⁡(t,V,L,σ),subscriptCapsubscriptCapBlack Cap_FL(t;V,L)=Cap_Black (t,V,L,σ),Capitalic_F L ( t ; V , L ) = CapBlack ( t , V , L , σ ) , i.e. the price of the cap equals the one obtained with the Black formula. Concepts of hedging. For hedging 202020Hedging concepts and risk management, the delta [99],[100] Δ=∂V∂SΔ = ∂ V∂ SΔ = divide start_ARG ∂ V end_ARG start_ARG ∂ S end_ARG, i.e., the first partial derivative of the option’s value with respect to the index level: For the Greeks of a European call option. Δ=∂C∂S=N⁢(d1)Δsubscript1 = ∂ C∂ S=N (d_1 )Δ = divide start_ARG ∂ C end_ARG start_ARG ∂ S end_ARG = N ( d1 ) The gamma is the second partial derivative with respect to the index level, γ=∂2C∂S2=N′⁢(d1)S⁢σ⁢T−tsuperscript2superscript2superscript′subscript1γ= ∂^2C∂ S^2= N (d_1 )% Sσ T-tγ = divide start_ARG ∂2 C end_ARG start_ARG ∂ S2 end_ARG = divide start_ARG N′ ( d1 ) end_ARG start_ARG S σ square-root start_ARG T - t end_ARG end_ARG The theta of an option is, by convention, the negative first partial derivative with respect to time-to-maturity t∗=T−tsuperscriptt^*=T-t∗ = T - t θ=−∂C∂t∗=−S⁢N′⁢(d1)⁢σ2⁢T−t−r⁢e−r⁢(T−t)⁢K⁢N⁢(d2)superscriptsuperscript′subscript12superscriptsubscript2θ=- ∂ C∂ t^*=- SN (d_1 )% σ2 T-t-re^-r(T-t)KN (d_2 )θ = - divide start_ARG ∂ C end_ARG start_ARG ∂ t∗ end_ARG = - divide start_ARG S N′ ( d1 ) σ end_ARG start_ARG 2 square-root start_ARG T - t end_ARG end_ARG - r e- r ( T - t ) K N ( d2 ) The rho of an option is the first partial derivative with respect to the short rate r ρ=∂C∂r=K⁢(T−t)⁢e−r⁢(T−t)⁢N⁢(d2)superscriptsubscript2ρ= ∂ C∂ r=K(T-t)e^-r(T-t)N (d_2 )ρ = divide start_ARG ∂ C end_ARG start_ARG ∂ r end_ARG = K ( T - t ) e- r ( T - t ) N ( d2 ) The vega-which is obviously not a Greek letter-is the first partial derivative with respect to the volatility σ, =∂C∂σ=S⁢N′⁢(d1)⁢T−tsuperscript′subscript1vega= ∂ C∂σ=SN (d_1 )% T-tvega = divide start_ARG ∂ C end_ARG start_ARG ∂ σ end_ARG = S N′ ( d1 ) square-root start_ARG T - t end_ARG Hedging dynamics can be defined as: ΔtP≡∂Pt∂StsuperscriptsubscriptΔsubscriptsubscript _t^P≡ ∂ P_t∂ S_tΔitalic_titalic_P ≡ divide start_ARG ∂ Pitalic_t end_ARG start_ARG ∂ Sitalic_t end_ARG d⁢Pt−ΔtP⁢d⁢St=0subscriptsuperscriptsubscriptΔsubscript0dP_t- _t^PdS_t=0d Pitalic_t - Δitalic_titalic_P d Sitalic_t = 0 Investment banks are also often interested in replicating the payoff of such a put (or another option). This is accomplished by setting up a replication portfolio consisting of ΔtPsuperscriptsubscriptΔ _t^PΔitalic_titalic_P units of the underlying and γt≡Pt−ΔtP⁢StsubscriptsubscriptsuperscriptsubscriptΔsubscript _t≡ P_t- _t^PS_tγitalic_t ≡ Pitalic_t - Δitalic_titalic_P Sitalic_t units of the risk-less bond BtsubscriptB_tBitalic_t such that the resulting portfolio value equals the option value at any time t, Pt=ΔtP⁢St+γt⁢BtsubscriptsuperscriptsubscriptΔsubscriptsubscriptsubscriptP_t= _t^PS_t+ _tB_tPitalic_t = Δitalic_titalic_P Sitalic_t + γitalic_t Bitalic_t or d⁢Pt=ΔtP⁢d⁢St+γt⁢d⁢BtsubscriptsuperscriptsubscriptΔsubscriptsubscriptsubscriptdP_t= _t^PdS_t+ _tdB_td Pitalic_t = Δitalic_titalic_P d Sitalic_t + γitalic_t d Bitalic_t For a plain vanilla put option this generally implies being short the underlying (ΔtP<0)superscriptsubscriptΔ0 ( _t^P<0 )( Δitalic_titalic_P < 0 ) and long the bond. For a call option (ΔtC>0)superscriptsubscriptΔ0 ( _t^C>0 )( Δitalic_titalic_C > 0 ) this implies the opposite. A replication strategy (ΔtP,γt),t∈0,…,τ−Δ⁢tsuperscriptsubscriptΔsubscript0…Δ ( _t^P, _t ),t∈\0,…,τ- t\( Δitalic_titalic_P , γitalic_t ) , t ∈ 0 , … , τ - Δ t , is called self-financing if for t>00t>0t > 0 and τ being the exercise date (i.e, τ=Tτ=Tτ = T for a European option) ΔtP⁢St+γt⁢Bt=Δt−Δ⁢tP⁢St+γt−Δ⁢t⁢Bt.superscriptsubscriptΔsubscriptsubscriptsubscriptsuperscriptsubscriptΔsubscriptsubscriptΔsubscript _t^PS_t+ _tB_t= _t- t^PS_t+ _t-% tB_t.Δitalic_titalic_P Sitalic_t + γitalic_t Bitalic_t = Δitalic_t - Δ titalic_P Sitalic_t + γitalic_t - Δ t Bitalic_t . Self-financing: The value VtsubscriptV_tVitalic_t of a portfolio containing ψtsubscript _tψitalic_t units of bonds and ϕt=(ϕt,1,…,ϕt,m)⊤subscriptitalic-ϕsuperscriptsubscriptitalic-ϕ1…subscriptitalic-ϕtop _t= ( _t,1,…, _t,m ) ϕitalic_t = ( ϕitalic_t , 1 , … , ϕitalic_t , m )⊤ units of various stocks at a given time t is given by Vt=Bt⁢ψt+∑i=1mϕt,i⁢St,i=Bt⁢ψt+ϕt⊤⁢t.subscriptsubscriptsubscriptsuperscriptsubscript1subscriptitalic-ϕsubscriptsubscriptsubscriptsuperscriptsubscriptitalic-ϕtopsubscriptV_t=B_t _t+ _i=1^m _t,iS_t,i=B_t _t+ _t^% S_t.Vitalic_t = Bitalic_t ψitalic_t + ∑i = 1m ϕitalic_t , i Sitalic_t , i = Bitalic_t ψitalic_t + ϕitalic_t⊤ Sitalic_t . The particular value of the portfolio at time t typically depends on the units of assets (ψt,ϕt)subscriptsubscriptitalic-ϕ ( _t, _t )( ψitalic_t , ϕitalic_t ) held just before time t. The stochastic process (ψt,ϕt),t⩾0subscriptsubscriptitalic-ϕ0 \ ( _t, _t ),t 0 \ ( ψitalic_t , ϕitalic_t ) , t ⩾ 0 can thus be interpreted as a trading strategy. ℱt,t⩾0subscriptℱ0 \F_t,t 0 \ Fitalic_t , t ⩾ 0 . d⁢Vt=(Bt⁢d⁢ψt+t⊤⁢d⁢ϕt)+ψt⁢d⁢Bt+ϕt⊤⁢d⁢tdsubscriptsubscriptdsubscriptsuperscriptsubscripttopdsubscriptitalic-ϕsubscriptdsubscriptsuperscriptsubscriptitalic-ϕtopdsubscriptdV_t= (B_t \ d _t+S_% t d _t )+ _t \ dB_% t+ _t dS_td Vitalic_t = ( Bitalic_t d ψitalic_t + Sitalic_t⊤ d ϕitalic_t ) + ψitalic_t d Bitalic_t + ϕitalic_t⊤ d Sitalic_t The portfolio is called self-financing if Bt⁢d⁢ψt+t⊤⁢d⁢ϕt=0subscriptdsubscriptsuperscriptsubscripttopdsubscriptitalic-ϕ0B_t \ d _t+S_t d% _t=0Bitalic_t d ψitalic_t + Sitalic_t⊤ d ϕitalic_t = 0 in which case VT=V0+∫0Tψt⁢dBℓ+∫0Tϕt⊤⁢dt.subscriptsubscript0superscriptsubscript0subscriptdifferential-dsubscriptℓsuperscriptsubscript0superscriptsubscriptitalic-ϕtopdifferential-dsubscriptV_T=V_0+ _0^T _t \ dB_ + _% 0^T _t dS_t.Vitalic_T = V0 + ∫0T ψitalic_t d Broman_ℓ + ∫0T ϕitalic_t⊤ d Sitalic_t . Arbitrage-free market: A market model is called arbitrage-free on [0,T]0[0,T][ 0 , T ] if, in essence, for every self-financing trading strategy (ψt,ϕt),0⩽t⩽Tsubscriptsubscriptitalic-ϕ0 \ ( _t, _t ),0 t T \ ( ψitalic_t , ϕitalic_t ) , 0 ⩽ t ⩽ T , V0=0⇒ℙ⁢(VT>0)=0⁢. subscript00⇒ℙsubscript00. V_0=0 (V_T>0 )=0. V0 = 0 ⇒ blackboard_P ( Vitalic_T > 0 ) = 0 . ~t,0⩽t⩽Tsubscript~0 \ S_t,0 t T \ over~ start_ARG S end_ARGt , 0 ⩽ t ⩽ T , with ~t=Bt−1⁢tsubscript~superscriptsubscript1subscript S_t=B_t^-1S_tover~ start_ARG S end_ARGt = Bitalic_t- 1 Sitalic_t, is an (ℱt,ℚ)subscriptℱℚ (F_t,Q )( Fitalic_t , blackboard_Q )-martingale. σt⁢t=μt−rt⁢tsubscriptsubscriptsubscriptsubscriptsubscript _tu_t= _t-r_tS_tσitalic_t uitalic_t = μitalic_t - ritalic_t Sitalic_t has a solution t,t∈[0,T]subscript0u_t,t∈[0,T]uitalic_t , t ∈ [ 0 , T ] for which ⁢exp⁡(12⁢∫0Tt⊤⁢t⁢dt)<∞12superscriptsubscript0superscriptsubscripttopsubscriptdifferential-dE ( 12 _0^Tu_t u_t% \ dt )<∞blackboard_E exp ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∫0T uitalic_t⊤ uitalic_t d t ) < ∞ d⁢~tdsubscript~ S_td over~ start_ARG S end_ARGt =Bt−1⁢(t−rt⁢t)⁢d⁢t+Bt−1⁢σt⁢d⁢tabsentsuperscriptsubscript1subscriptsubscriptsubscriptdsuperscriptsubscript1subscriptdsubscript =B_t^-1 ( μ_t-r_tS_t )% dt+B_t^-1 _t \ dW_t= Bitalic_t- 1 ( italic_μitalic_t - ritalic_t Sitalic_t ) d t + Bitalic_t- 1 σitalic_t d Witalic_t =Bt−1⁢σt⁢(t⁢d⁢t+d⁢t)absentsuperscriptsubscript1subscriptsubscriptddsubscript =B_t^-1 _t (u_t % \ dt+dW_t )= Bitalic_t- 1 σitalic_t ( uitalic_t d t + d Witalic_t ) =Bt−1⁢σt⁢d⁢tabsentsuperscriptsubscript1subscriptdsubscript =B_t^-1 _t \ dZ_t= Bitalic_t- 1 σitalic_t d Zitalic_t where d⁢t=t⁢d⁢t+d⁢tdsubscriptsubscriptddsubscriptdZ_t=u_t \ dt+% dW_td Zitalic_t = uitalic_t d t + d Witalic_t , defines an Itô process. Let Mt=exp⁡(∫0ts⊤⁢ds−12⁢∫0ts⊤⁢s⁢ds).subscriptsuperscriptsubscript0superscriptsubscripttopdifferential-dsubscript12superscriptsubscript0superscriptsubscripttopsubscriptdifferential-dM_t= ( _0^tu_s dW_s-% 12 _0^tu_s u_s % \ ds ).Mitalic_t = exp ( ∫0t uitalic_s⊤ d Witalic_s - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∫0t uitalic_s⊤ uitalic_s d s ) . Then, by Girsanov’s theorem [101], Mt,t⩾0subscript0 \M_t,t 0 \ Mitalic_t , t ⩾ 0 is an (ℱt,ℙ)subscriptℱℙ (F_t,P )( Fitalic_t , blackboard_P )-martingale, and under the new measure ℚ⁢(A)= def ⁢[MT⁢IA]superscript def ℚdelimited-[]subscriptsubscriptIQ(A) def =E [M_T% I_A ]blackboard_Q ( A ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP blackboard_E [ Mitalic_T Iitalic_A ] for all A∈ℱTsubscriptℱA _TA ∈ Fitalic_T, the process t,0⩽ \Z_t,0 . Zitalic_t , 0 ⩽ t⩽Tt T\t ⩽ T is a Wiener process. It follows that under ℚQblackboard_Q the process ~t,0⩽t⩽Tsubscript~0 \ S_t,0 t T \ over~ start_ARG S end_ARGt , 0 ⩽ t ⩽ T is a martingale with respect to ℱt,0⩽t⩽Tsubscriptℱ0 \F_t,0 t T \ Fitalic_t , 0 ⩽ t ⩽ T , which had to be shown. ℚQblackboard_Q is called the risk-neutral measure. Let V~t=Bt−1⁢Vt=ψt+ϕt⊤⁢Bt−1⁢t=ψt+ϕt⊤⁢~tsubscript~superscriptsubscript1subscriptsubscriptsuperscriptsubscriptbold-italic-ϕtopsuperscriptsubscript1subscriptsubscriptsuperscriptsubscriptbold-italic-ϕtopsubscript~ V_t=B_t^-1V_t= _t+ φ_t B_t^% -1S_t= _t+ φ_t S_% tover~ start_ARG V end_ARGt = Bitalic_t- 1 Vitalic_t = ψitalic_t + italic_ϕitalic_t⊤ Bitalic_t- 1 Sitalic_t = ψitalic_t + italic_ϕitalic_t⊤ over~ start_ARG S end_ARGt be the discounted value of the portfolio of stocks and bonds. d⁢V~t=ϕt⊤⁢d⁢~t=Bt−1⁢ϕt⊤⁢t⁢d⁢tdsubscript~superscriptsubscriptbold-italic-ϕtopdsubscript~superscriptsubscript1superscriptsubscriptbold-italic-ϕtopsubscriptdsubscript aligned d V_t= φ_t% d S_t=B_t^-1 φ_t^% σ_t \ dZ_t alignedstart_ROW start_CELL d over~ start_ARG V end_ARGt = italic_ϕitalic_t⊤ d over~ start_ARG S end_ARGt = Bitalic_t- 1 italic_ϕitalic_t⊤ italic_σitalic_t d Zitalic_t end_CELL end_ROW ℚQblackboard_Q the process V~t,0⩽t⩽Tsubscript~0 \ V_t,0 t T \ over~ start_ARG V end_ARGt , 0 ⩽ t ⩽ T is a martingale with respect to ℱt,0⩽t⩽Tsubscriptℱ0 \F_t,0 t T \ Fitalic_t , 0 ⩽ t ⩽ T . As a consequence, Vt=Bt⁢ℚ⁢[BT−1⁢VT∣ℱt],t⩽T.formulae-sequencesubscriptsubscriptsubscriptℚdelimited-[]conditionalsuperscriptsubscript1subscriptsubscriptℱV_t=B_tE_Q [B_T^-1V_T _t% ], t T.Vitalic_t = Bitalic_t blackboard_Eblackboard_Q [ Bitalic_T- 1 Vitalic_T ∣ Fitalic_t ] , t ⩽ T . The payoff at maturity for a call option is (ST−K)+=max⁡ST−K,0superscriptsubscriptsubscript0 (S_T-K )^+= \S_T-K,0 \( Sitalic_T - K )+ = max Sitalic_T - K , 0 and for a put option is (K−ST)+superscriptsubscript (K-S_T )^+( K - Sitalic_T )+. Suppose for concreteness that we are dealing with a call option and denote its value at time t⩽Tt Tt ⩽ T by Ct=C⁢(St,K,rt,T−t)subscriptsubscriptsubscriptC_t=C (S_t,K,r_t,T-t )Citalic_t = C ( Sitalic_t , K , ritalic_t , T - t ), where T−tT-tT - t is the time until maturity. Ct=Vt=Bt⁢ℚ⁢[BT−1⁢CT∣ℱt],t⩽T,formulae-sequencesubscriptsubscriptsubscriptsubscriptℚdelimited-[]conditionalsuperscriptsubscript1subscriptsubscriptℱ aligned C_t=V_t=B_tE_Q [B_% T^-1C_T _t ], t T, alignedstart_ROW start_CELL Citalic_t = Vitalic_t = Bitalic_t blackboard_Eblackboard_Q [ Bitalic_T- 1 Citalic_T ∣ Fitalic_t ] , t ⩽ T , end_CELL end_ROW where CT=VT=V0+∫0Tψt⁢dBt+∫0Tϕt⊤⁢dt.subscriptsubscriptsubscript0superscriptsubscript0subscriptdifferential-dsubscriptsuperscriptsubscript0superscriptsubscriptbold-italic-ϕtopdifferential-dsubscript aligned C_T=V_T=V_0+ _0^T _t% \ dB_t+ _0^T φ_t d% S_t. alignedstart_ROW start_CELL Citalic_T = Vitalic_T = V0 + ∫0T ψitalic_t d Bitalic_t + ∫0T italic_ϕitalic_t⊤ d Sitalic_t . end_CELL end_ROW Consider a call and a put options written on the same non-dividend-paying stock, with the same strike price K and time to maturity T. Then their prices C0subscript0C_0C0 and P0subscript0P_0P0 at time t=00t=0t = 0 are related as follows: C0+K⁢e−r⁢T=P0+S0subscript0superscriptsubscript0subscript0C_0+Ke^-rT=P_0+S_0C0 + K e- r T = P0 + S0 where S0subscript0S_0S0 is the current stock price and r is the continuously compounded riskfree interest rate. Consider a portfolio consisting of a call option and an amount of cash given by K⁢e−r⁢TsuperscriptKe^-rTK e- r T. At maturity, the value of this portfolio is, max⁡ST−K,0+K=max⁡ST,Ksubscript0subscript \S_T-K,0 \+K= \S_T,K \max Sitalic_T - K , 0 + K = max Sitalic_T , K max⁡K−ST,0+ST=max⁡K,STsubscript0subscriptsubscript \K-S_T,0 \+S_T= \K,S_T \max K - Sitalic_T , 0 + Sitalic_T = max K , Sitalic_T Option prices provide information about state variables and parameters via the pricing equation. An option price is given by, C⁢(St,Vt,θ)=e−r⁢(T−t)⁢tℚ⁢[max⁡(St−K,0)∣St,Xt,θ]subscriptsubscriptsuperscriptsuperscriptsubscriptℚdelimited-[]conditionalsubscript0subscriptsubscriptC (S_t,V_t,θ )=e^-r(T-t)E_t^Q [% (S_t-K,0 ) S_t,X_t,θ ]C ( Sitalic_t , Vitalic_t , θ ) = e- r ( T - t ) blackboard_Etblackboard_Q [ max ( Sitalic_t - K , 0 ) ∣ Sitalic_t , Xitalic_t , θ ] where the expectation is taken under ℚQblackboard_Q the risk-neutral probability measure. here we assume that the risk-free rate r is constant. This simplifies by letting A=ST≥KsubscriptA= \S_T≥ K \A = Sitalic_T ≥ K denote the event that the stock ends in the money. Then the option price is given by: e−r⁢t⁢tℚ⁢(max⁡(ST−K,0))=e−r⁢t⁢tℚ⁢(ST⁢A)−e−r⁢t⁢K⁢(A)=e−r⁢t⁢tℚ⁢(ST⁢A)−e−r⁢t⁢K⁢ℙ⁢(A).superscriptsuperscriptsubscriptℚsubscript0absentsuperscriptsuperscriptsubscriptℚsubscriptsubscriptsuperscriptsubscriptmissing-subexpressionabsentsuperscriptsuperscriptsubscriptℚsubscriptsubscriptsuperscriptℙ aligned e^-rtE_t^Q ( % (S_T-K,0 ) )&=e^-rtE_t^Q (S_T % I_A )-e^-rtKE (I_A )\\ &=e^-rtE_t^Q (S_TI_A )-e^-rtK% P(A). alignedstart_ROW start_CELL e- r t blackboard_Etblackboard_Q ( max ( Sitalic_T - K , 0 ) ) end_CELL start_CELL = e- r t blackboard_Etblackboard_Q ( Sitalic_T blackboard_IA ) - e- r t K blackboard_E ( blackboard_IA ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = e- r t blackboard_Etblackboard_Q ( Sitalic_T blackboard_IA ) - e- r t K blackboard_P ( A ) . end_CELL end_ROW The logarithmic asset price and volatility follow an affine process with Yt=(log⁡St,Vt)∈ℜ×ℜ+subscriptsubscriptsubscriptℜsuperscriptℜY_t= ( S_t,V_t )∈ R× R^+Yitalic_t = ( log Sitalic_t , Vitalic_t ) ∈ R × R+satisfying d⁢Yt=(r−12⁢Vt)⁢d⁢t+Vt⁢d⁢B1,td⁢Vt=κ⁢(θ−Vt)⁢d⁢t+σv⁢Vt⁢d⁢Ztmissing-subexpressionsubscript12subscriptsubscriptsubscript1missing-subexpressionsubscriptsubscriptsubscriptsubscriptsubscript aligned &dY_t= (r- 12V_t )dt+ V% _tdB_1,t\\ &dV_t=κ (θ-V_t )dt+ _v V_tdZ_t alignedstart_ROW start_CELL end_CELL start_CELL d Yitalic_t = ( r - divide start_ARG 1 end_ARG start_ARG 2 end_ARG Vitalic_t ) d t + square-root start_ARG Vitalic_t end_ARG d B1 , t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL d Vitalic_t = κ ( θ - Vitalic_t ) d t + σitalic_v square-root start_ARG Vitalic_t end_ARG d Zitalic_t end_CELL end_ROW where Zt=ρ⁢B1,t+1−ρ2⁢B2,tsubscriptsubscript11superscript2subscript2Z_t=ρ B_1,t+ 1-ρ^2B_2,tZitalic_t = ρ B1 , t + square-root start_ARG 1 - ρ2 end_ARG B2 , t. The correlation, ρ, or so-called leverage effect is important to explain the empirical fact that volatility increases faster as equity prices drop. The parameters (κ,θ)(κ,θ)( κ , θ ) govern the speed of mean reversion and the long-run mean of volatility and σvsubscript _vσitalic_v measures the volatility of volatility. Under the risk-neutral measure they become (κ/κ+λv)⁢θsubscript (κ/κ+ _v )θ( κ / κ + λitalic_v ) θ and κ+λvsubscriptκ+ _vκ + λitalic_v where λvsubscript _vλitalic_v is the market price of volatility risk. From affine process theory, the discounted transform, ψ⁢(u)=e−r⁢t⁢(eu⁢YT∣Yt)=eα⁢(t,u)+u⁢Yt+β⁢(t,u)⁢Vtsuperscriptconditionalsuperscriptsubscriptsubscriptsuperscriptsubscriptsubscriptψ(u)=e^-rtE (e^uY_T Y_t )=e^α(t,u)+uY_% t+β(t,u)V_tψ ( u ) = e- r t blackboard_E ( eitalic_u Yitalic_T ∣ Yitalic_t ) = eitalic_α ( t , u ) + u Yitalic_t + β ( t , u ) Vitalic_t where α,βα,βα , β satisfy Riccati equations. Concepts of Rough volatility. WtH=CH⁢∫−∞td⁢Wsℙ(t−s)γ−∫−∞0d⁢Wsℙ(−s)γsuperscriptsubscriptsubscriptsuperscriptsubscriptsuperscriptsubscriptℙsuperscriptsuperscriptsubscript0superscriptsubscriptℙsuperscript aligned W_t^H=C_H \ _-∞^t dW_% s^P(t-s)^γ- _-∞^0 dW_s^P% (-s)^γ \ alignedstart_ROW start_CELL Witalic_titalic_H = Citalic_H ∫- ∞t divide start_ARG d Witalic_sblackboard_P end_ARG start_ARG ( t - s )γ end_ARG - ∫- ∞0 divide start_ARG d Witalic_sblackboard_P end_ARG start_ARG ( - s )γ end_ARG end_CELL end_ROW =ν⁢CH⁢∫tu1(u−s)γ⁢Wsℙ+∫−∞t[1(u−s)γ−1(t−s)γ]⁢Wsℙ=2⁢ν⁢CH⁢[Mt⁢(u)+Zt⁢(u)]missing-subexpressionmissing-subexpressionsubscriptsuperscriptsubscript1superscriptdifferential-dsuperscriptsubscriptℙsuperscriptsubscriptdelimited-[]1superscript1superscriptdifferential-dsuperscriptsubscriptℙ2subscriptdelimited-[]subscriptsubscript aligned &\\ =&ν C_H \ _t^u 1(u-s)^γdW_s^P+ % _-∞^t [ 1(u-s)^γ- 1(t-s)^γ ]dW% _s^P \\\ =&2ν C_H [M_t(u)+Z_t(u) ] alignedstart_ROW start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL ν Citalic_H ∫titalic_u divide start_ARG 1 end_ARG start_ARG ( u - s )γ end_ARG d Witalic_sblackboard_P + ∫- ∞t [ divide start_ARG 1 end_ARG start_ARG ( u - s )γ end_ARG - divide start_ARG 1 end_ARG start_ARG ( t - s )γ end_ARG ] d Witalic_sblackboard_P end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL 2 ν Citalic_H [ Mitalic_t ( u ) + Zitalic_t ( u ) ] end_CELL end_ROW ℙ⁢[Mt⁢(u)∣ℱt]=0superscriptℙdelimited-[]conditionalsubscriptsubscriptℱ0E^P [M_t(u) _t ]=0blackboard_Eblackboard_P [ Mitalic_t ( u ) ∣ Fitalic_t ] = 0 and Zt⁢(u)subscriptZ_t(u)Zitalic_t ( u ) is ℱtsubscriptℱF_tFitalic_t-measurable. Pricing 212121Rough volatility: Python 222222Rough path 232323Rough volatility: Minimax Theory [102],[103],[104],[105], under ℙPblackboard_P W~tℙ⁢(u):=2⁢H⁢∫tud⁢WsP(u−s)γassignsuperscriptsubscript~ℙ2superscriptsubscriptsuperscriptsubscriptPsuperscript W_t^P(u):= 2H _t^u dW_s^P% (u-s)^γover~ start_ARG W end_ARGtblackboard_P ( u ) := square-root start_ARG 2 H end_ARG ∫titalic_u divide start_ARG d Witalic_sroman_P end_ARG start_ARG ( u - s )γ end_ARG With η:=2⁢ν⁢CH/2⁢Hassign2subscript2η:=2ν C_H/ 2Hη := 2 ν Citalic_H / square-root start_ARG 2 H end_ARG we have 2⁢ν⁢CH⁢Mt⁢(u)=η⁢W~tℙ⁢(u)2subscriptsubscriptsuperscriptsubscript~ℙ2ν C_HM_t(u)=η W_t^P(u)2 ν Citalic_H Mitalic_t ( u ) = η over~ start_ARG W end_ARGtblackboard_P ( u ) , vu=vt⁢exp⁡η⁢W~tℙ⁢(u)+2⁢ν⁢CH⁢Zt⁢(u)=ℙ⁢[vu∣ℱt]⁢ℰ⁢(η⁢W~tℙ⁢(u))subscriptabsentsubscriptsuperscriptsubscript~ℙ2subscriptsubscriptmissing-subexpressionabsentsuperscriptℙdelimited-[]conditionalsubscriptsubscriptℱℰsuperscriptsubscript~ℙ aligned v_u&=v_t \η W_t % P(u)+2ν C_HZ_t(u) \\\ &=E^P [v_u _t ]E % (η W_t^P(u) ) alignedstart_ROW start_CELL vitalic_u end_CELL start_CELL = vitalic_t exp η over~ start_ARG W end_ARGtblackboard_P ( u ) + 2 ν Citalic_H Zitalic_t ( u ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = blackboard_Eblackboard_P [ vitalic_u ∣ Fitalic_t ] E ( η over~ start_ARG W end_ARGtblackboard_P ( u ) ) end_CELL end_ROW vu=ℙ⁢[vu∣ℱt]⁢ℰ⁢(η⁢W~tℙ⁢(u))subscriptsuperscriptℙdelimited-[]conditionalsubscriptsubscriptℱℰsuperscriptsubscript~ℙ aligned v_u=E^P [v_u % F_t ]E (η W_t^P(u) % ) alignedstart_ROW start_CELL vitalic_u = blackboard_Eblackboard_P [ vitalic_u ∣ Fitalic_t ] E ( η over~ start_ARG W end_ARGtblackboard_P ( u ) ) end_CELL end_ROW d⁢WsP=d⁢Wsℚ+λs⁢d⁢ssuperscriptsubscriptPsuperscriptsubscriptℚsubscriptdW_s^P=dW_s^Q+ _sdsd Witalic_sroman_P = d Witalic_sblackboard_Q + λitalic_s d s where λs:s>tconditional-setsubscript \ _s:s>t \ λitalic_s : s > t has a natural interpretation as the price of volatility risk. vu=ℙ⁢[vu∣ℱt]⁢ℰ⁢(η⁢W~tQ⁢(u))⁢exp⁡η⁢2⁢H⁢∫tuλs(u−s)γ⁢ssubscriptsuperscriptℙdelimited-[]conditionalsubscriptsubscriptℱℰsuperscriptsubscript~2superscriptsubscriptsubscriptsuperscriptdifferential-d aligned v_u=E^P [v_u % F_t ]E (η W_t^Q(u) ) % \η 2H _t^u _s(u-s)^γds \% alignedstart_ROW start_CELL vitalic_u = blackboard_Eblackboard_P [ vitalic_u ∣ Fitalic_t ] E ( η over~ start_ARG W end_ARGtitalic_Q ( u ) ) exp η square-root start_ARG 2 H end_ARG ∫titalic_u divide start_ARG λitalic_s end_ARG start_ARG ( u - s )γ end_ARG d s end_CELL end_ROW Concepts of Fourier-based option pricing. -G Bayesian optimization for option pricing based on the Fourier method. In this section we apply a Fourier Transform to calculate the option price using a bayesian optimization. The results are available on ART.1.20 242424Rough volatility: Python 1:procedure FourierTransformOptionPrice(S,K,T,r,σ,α,βS,K,T,r,σ,α, , K , T , r , σ , α , β) 2: u←linspace⁢(−10,10,1000)←linspace10101000u (-10,10,1000)u ← linspace ( - 10 , 10 , 1000 ) 3: C←[characteristic_function⁢(ui)∣ui∈u]←delimited-[]conditionalcharacteristic_functionsubscriptsubscriptC←[characteristic\_function(u_i) u_i∈ u]C ← [ characteristic_function ( uitalic_i ) ∣ uitalic_i ∈ u ] 4: option_price←inverse_fourier_transform⁢(u,C)←option_priceinverse_fourier_transformoption\_price \_fourier\_transform(u,C)option_price ← inverse_fourier_transform ( u , C ) 5: return option_price 6:end procedure Algorithm 9 Fourier Transform Method for European Call Option Price ⟨f,g⟩=∫−∞12⁢π⁢∫−∞e−i⁢u⁢x⁢f^⁢(k)⁢k⁢g⁢(x)¯⁢x=12⁢π⁢∫−∞f^⁢(k)⁢∫−∞e−i⁢u⁢x⁢g⁢(x)¯⁢x⁢k=12⁢π⁢∫−∞f^⁢(k)⁢∫−∞ei⁢u⁢x⁢g⁢(x)¯⁢x⁢k=12⁢π⁢∫−∞f^⁢(k)⁢g^⁢(k)¯⁢kabsentsuperscriptsubscript12superscriptsubscriptsuperscript^differential-d¯differential-dmissing-subexpressionabsent12superscriptsubscript^superscriptsubscriptsuperscript¯differential-ddifferential-dmissing-subexpressionabsent12superscriptsubscript^superscriptsubscript¯superscriptdifferential-ddifferential-dmissing-subexpressionabsent12superscriptsubscript^¯^differential-d aligned f,g &= _-∞^∞ % 12π _-∞^∞e^-iux f(k)dk g(x)dx\\ &= 12π _-∞^∞ f(k) _-∞^∞e^-% iux g(x)dxdk\\ &= 12π _-∞^∞ f(k) _-∞^∞% e^iuxg(x)dxdk\\ &= 12π _-∞^∞ f(k) g(k)dk alignedstart_ROW start_CELL ⟨ f , g ⟩ end_CELL start_CELL = ∫- ∞ divide start_ARG 1 end_ARG start_ARG 2 π end_ARG ∫- ∞ e- i u x over start_ARG f end_ARG ( k ) d k over¯ start_ARG g ( x ) end_ARG d x end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π end_ARG ∫- ∞ over start_ARG f end_ARG ( k ) ∫- ∞ e- i u x over¯ start_ARG g ( x ) end_ARG d x d k end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π end_ARG ∫- ∞ over start_ARG f end_ARG ( k ) ∫- ∞ over¯ start_ARG eitalic_i u x g ( x ) end_ARG d x d k end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π end_ARG ∫- ∞ over start_ARG f end_ARG ( k ) over¯ start_ARG over start_ARG g end_ARG ( k ) end_ARG d k end_CELL end_ROW ⟨f,g⟩≡∫−∞f⁢(x)⁢g⁢(x)¯⁢xsuperscriptsubscript¯differential-d aligned f,g ≡ _-∞^∞f(% x) g(x)dx alignedstart_ROW start_CELL ⟨ f , g ⟩ ≡ ∫- ∞ f ( x ) over¯ start_ARG g ( x ) end_ARG d x end_CELL end_ROW . By Fourier inversion f⁢(x)=absentf(x)=f ( x ) = 12⁢π⁢∫−∞e−i⁢u⁢x⁢f^⁢(k)⁢k12superscriptsubscriptsuperscript^differential-d 12π _-∞^∞e^-iux f(k)dkdivide start_ARG 1 end_ARG start_ARG 2 π end_ARG ∫- ∞ e- i u x over start_ARG f end_ARG ( k ) d k. Let a random variable X be distributed with pdf q⁢(x)q(x)q ( x ). The characteristic function q^ qover start_ARG q end_ARG of X is the Fourier transform [106],[107],[108],[109] of its p⁢d⁢fpdfp d f, q^⁢(u)≡∫−∞ei⁢u⁢x⁢q⁢(x)⁢x=Q⁢(ei⁢u⁢X)^superscriptsubscriptsuperscriptdifferential-dsuperscriptsuperscript q(u)≡ _-∞^∞e^iuxq(x)dx=E^Q (e^% iuX )over start_ARG q end_ARG ( u ) ≡ ∫- ∞ eitalic_i u x q ( x ) d x = Eitalic_Q ( eitalic_i u X ) Consider a European call option with payoff CT≡max⁡[es−K,0]subscriptsuperscript0C_T≡ [e^s-K,0 ]Citalic_T ≡ max [ eitalic_s - K , 0 ] where s≡log⁡Ss≡ Ss ≡ log S. Call Option Transform [110],[111]: For u=ur+i⁢uisubscriptsubscriptu=u_r+iu_iu = uitalic_r + i uitalic_i with ui>1subscript1u_i>1uitalic_i > 1, the Fourier 252525Pricing: Fast Fourier Transform 262626derivative pricing 272727Fourier Transform Applicationstransform of CTsubscriptC_TCitalic_T is, C^T⁢(u)=−Ki⁢u+1u2−i⁢usubscript^superscript1superscript2 C_T(u)=- K^iu+1u^2-iuover start_ARG C end_ARGT ( u ) = - divide start_ARG Kitalic_i u + 1 end_ARG start_ARG u2 - i u end_ARG Proof. C^T⁢(u)=∫−∞ei⁢u⁢s⁢max⁡[es−K,0]⁢s=∫log⁡K∞ei⁢u⁢s⁢(es−K)⁢ssubscript^absentsuperscriptsubscriptsuperscriptsuperscript0differential-dmissing-subexpressionabsentsuperscriptsubscriptsuperscriptsuperscriptdifferential-d aligned C_T(u)&= _-∞^∞e^ius% [e^s-K,0 ]ds\\ &= _ K^∞e^ius (e^s-K )ds alignedstart_ROW start_CELL over start_ARG C end_ARGT ( u ) end_CELL start_CELL = ∫- ∞ eitalic_i u s max [ eitalic_s - K , 0 ] d s end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = ∫log K∞ eitalic_i u s ( eitalic_s - K ) d s end_CELL end_ROW =∫log⁡K∞(e(i⁢u+1)⁢s−K⁢ei⁢u⁢s)⁢s=[e(i⁢u+1)⁢si⁢u+1−K⁢ei⁢u⁢si⁢u]log⁡K∞=−Ki⁢u+1u2−i⁢umissing-subexpressionabsentsuperscriptsubscriptsuperscript1superscriptdifferential-dmissing-subexpressionabsentsuperscriptsubscriptdelimited-[]superscript11superscriptmissing-subexpressionabsentsuperscript1superscript2 aligned &= _ K^∞ (e^(iu+1)s-Ke^ius% )ds\\ &= [ e^(iu+1)siu+1-K e^iusiu ]_ K^∞% \\ &=- K^iu+1u^2-iu alignedstart_ROW start_CELL end_CELL start_CELL = ∫log K∞ ( e( i u + 1 ) s - K eitalic_i u s ) d s end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = [ divide start_ARG e( i u + 1 ) s end_ARG start_ARG i u + 1 end_ARG - K divide start_ARG eitalic_i u s end_ARG start_ARG i u end_ARG ]log K∞ end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = - divide start_ARG Kitalic_i u + 1 end_ARG start_ARG u2 - i u end_ARG end_CELL end_ROW CT⁢(s)=12⁢π⁢∫−∞+i⁢ui∞+i⁢uie−i⁢u⁢s⁢C^T⁢(u)⁢usubscript12superscriptsubscriptsubscriptsubscriptsuperscriptsubscript^differential-d aligned C_T(s)= 12π _-∞+iu_i^% ∞+iu_ie^-ius C_T(u)du alignedstart_ROW start_CELL Citalic_T ( s ) = divide start_ARG 1 end_ARG start_ARG 2 π end_ARG ∫- ∞ + i u start_POSTSUBSCRIPT i end_POSTSUBSCRIPT∞ + i uitalic_i e- i u s over start_ARG C end_ARGT ( u ) d u end_CELL end_ROW C0=e−r⁢T⁢0Q⁢(CT)=e−r⁢T2⁢π⁢0Q⁢(∫−∞+i⁢uie−i⁢u⁢s⁢C^T⁢(u)⁢u)=e−r⁢T2⁢π⁢∫−∞+i⁢ui∞+i⁢ui0Q⁢(ei⁢(−u)⁢s)⁢C^T⁢(u)⁢u=e−r⁢T2⁢π⁢∫−∞+i⁢ui∞+i⁢uiC^T⁢(u)⁢q^⁢(−u)⁢usubscript0absentsuperscriptsuperscriptsubscript0subscriptmissing-subexpressionabsentsuperscript2superscriptsubscript0superscriptsubscriptsubscriptsuperscriptsubscript^differential-dmissing-subexpressionabsentsuperscript2superscriptsubscriptsubscriptsubscriptsuperscriptsubscript0superscriptsubscript^differential-dmissing-subexpressionabsentsuperscript2superscriptsubscriptsubscriptsubscriptsubscript^^differential-d aligned C_0&=e^-rTE_0^Q (C_T % )\\ &= e^-rT2πE_0^Q ( _-∞^∞+iu_ie^% -ius C_T(u)du )\\ &= e^-rT2π _-∞+iu_i^∞+iu_iE_0^Q% (e^i(-u)s ) C_T(u)du\\ &= e^-rT2π _-∞+iu_i^∞+iu_i C_T(u)% q(-u)du alignedstart_ROW start_CELL C0 end_CELL start_CELL = e- r T E0italic_Q ( Citalic_T ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG e- r T end_ARG start_ARG 2 π end_ARG E0italic_Q ( ∫- ∞ + i uitalic_i e- i u s over start_ARG C end_ARGT ( u ) d u ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG e- r T end_ARG start_ARG 2 π end_ARG ∫- ∞ + i u start_POSTSUBSCRIPT i end_POSTSUBSCRIPT∞ + i uitalic_i E0italic_Q ( eitalic_i ( - u ) s ) over start_ARG C end_ARGT ( u ) d u end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG e- r T end_ARG start_ARG 2 π end_ARG ∫- ∞ + i u start_POSTSUBSCRIPT i end_POSTSUBSCRIPT∞ + i uitalic_i over start_ARG C end_ARGT ( u ) over start_ARG q end_ARG ( - u ) d u end_CELL end_ROW If St≡S0⁢er⁢t+Xtsubscriptsubscript0superscriptsubscriptS_t≡ S_0e^rt+X_tSitalic_t ≡ S0 eitalic_r t + Xitalic_t with XtsubscriptX_tXitalic_t a Lévy process and eXtsuperscriptsubscripte^X_teitalic_Xitalic_t a martingale with X0=0subscript00X_0=0X0 = 0, then q^⁢(−u)=^absent q(-u)=over start_ARG q end_ARG ( - u ) = e−i⁢u⁢y⁢φ⁢(−u)superscripte^-iuy (-u)e- i u y φ ( - u ) where φ φ is the characteristic function of XTsubscriptX_TXitalic_T. Here, y≡log⁡S0+r⁢Tsubscript0y≡ S_0+rTy ≡ log S0 + r T. C0=e−r⁢T2⁢π⁢∫−∞+i⁢ui∞+i⁢uie−i⁢u⁢y⁢C^⁢(u)⁢φ⁢(−u)⁢usubscript0superscript2superscriptsubscriptsubscriptsubscriptsuperscript^differential-dC_0= e^-rT2π _-∞+iu_i^∞+iu_ie^-iuy% C(u) (-u)duC0 = divide start_ARG e- r T end_ARG start_ARG 2 π end_ARG ∫- ∞ + i u start_POSTSUBSCRIPT i end_POSTSUBSCRIPT∞ + i uitalic_i e- i u y over start_ARG C end_ARG ( u ) φ ( - u ) d u k=log⁡(S0/K)+r⁢Tsubscript0k= (S_0/K )+rTk = log ( S0 / K ) + r T, C0=−K⁢e−r⁢T2⁢π⁢∫−∞+i⁢ui∞+i⁢uie−i⁢u⁢k⁢φ⁢(−u)⁢d⁢u2−u⁢isubscript0superscript2superscriptsubscriptsubscriptsubscriptsuperscriptsuperscript2C_0=- Ke^-rT2π _-∞+iu_i^∞+iu_ie^-iuk% (-u) duu^2-uiC0 = - divide start_ARG K e- r T end_ARG start_ARG 2 π end_ARG ∫- ∞ + i u start_POSTSUBSCRIPT i end_POSTSUBSCRIPT∞ + i uitalic_i e- i u k φ ( - u ) divide start_ARG d u end_ARG start_ARG u2 - u i end_ARG ui∈(0,1)subscript01u_i∈(0,1)uitalic_i ∈ ( 0 , 1 ), the call option present value is, C0=S0−K⁢e−r⁢T2⁢π⁢∫−∞+i⁢ui∞+i⁢uie−i⁢u⁢k⁢φ⁢(−u)⁢d⁢u2−u⁢isubscript0subscript0superscript2superscriptsubscriptsubscriptsubscriptsuperscriptsuperscript2C_0=S_0- Ke^-rT2π _-∞+iu_i^∞+iu_ie^-iuk% (-u) duu^2-uiC0 = S0 - divide start_ARG K e- r T end_ARG start_ARG 2 π end_ARG ∫- ∞ + i u start_POSTSUBSCRIPT i end_POSTSUBSCRIPT∞ + i uitalic_i e- i u k φ ( - u ) divide start_ARG d u end_ARG start_ARG u2 - u i end_ARG ui=0.5subscript0.5u_i=0.5uitalic_i = 0.5 , C0=S0−S0⁢K⁢e−r⁢T/2π⁢∫0∞Re⁡[ei⁢z⁢k⁢φ⁢(z−i/2)]⁢d⁢z2+1/4subscript0subscript0subscript0superscript2superscriptsubscript0Resuperscript2superscript214C_0=S_0- S_0Ke^-rT/2π _0^∞% Re [e^izk (z-i/2) ] dzz^2+1/4C0 = S0 - divide start_ARG square-root start_ARG S0 K end_ARG e- r T / 2 end_ARG start_ARG π end_ARG ∫0∞ Re [ eitalic_i z k φ ( z - i / 2 ) ] divide start_ARG d z end_ARG start_ARG z2 + 1 / 4 end_ARG where ⁢[x]delimited-[]Re[x]Re [ x ] denotes the real part of x. u=iu=iu = i , u=00u=0u = 0, Res⁡(i)Res (i)Res ( i ) =limu→i((u−i)⁢(−K⁢e−r⁢T2⁢π⁢e−i⁢u⁢k⁢φ⁢(−u)u⁢(u−i)))absentsubscript→superscript2superscript = _u→ i ((u-i) (- Ke^-rT2πe^-% iuk (-u)u(u-i) ) )= limitalic_u → i ( ( u - i ) ( - divide start_ARG K e- r T end_ARG start_ARG 2 π end_ARG e- i u k divide start_ARG φ ( - u ) end_ARG start_ARG u ( u - i ) end_ARG ) ) =−K⁢e−r⁢T2⁢π⁢ek⁢φ⁢(−i)iabsentsuperscript2superscript =- Ke^-rT2πe^k (-i)i= - divide start_ARG K e- r T end_ARG start_ARG 2 π end_ARG eitalic_k divide start_ARG φ ( - i ) end_ARG start_ARG i end_ARG =S0⁢i2⁢πabsentsubscript02 = S_0i2π= divide start_ARG S0 i end_ARG start_ARG 2 π end_ARG ek=S0/K⋅er⁢T,φ⁢(−i)=1formulae-sequencesuperscript⋅subscript0superscript1e^k=S_0/K· e^rT, (-i)=1eitalic_k = S0 / K ⋅ eitalic_r T , φ ( - i ) = 1 and i−1=−isuperscript1i^-1=-i- 1 = - i in the uisubscriptu_iuitalic_i minus 2⁢π⁢i⁢Res⁡(i)=−S02Ressubscript02π iRes(i)=-S_02 π i Res ( i ) = - S0, ui=0.5subscript0.5u_i=0.5uitalic_i = 0.5 C0=S0−K⁢e−r⁢T2⁢π⁢∫−∞e−i⁢(u+i/2)⁢k⁢φ⁢(−(u+i/2))⁢d⁢u(u+i/2)2−(u+i/2)⁢isubscript0subscript0superscript2superscriptsubscriptsuperscript22superscript222 aligned C_0=S_0- Ke^-rT2π _-∞^% ∞\\ e^-i(u+i/2)k (-(u+i/2)) du(u+i/2)^2-(u+i/2)i alignedstart_ROW start_CELL C0 = S0 - divide start_ARG K e- r T end_ARG start_ARG 2 π end_ARG ∫- ∞ end_CELL end_ROW start_ROW start_CELL e- i ( u + i / 2 ) k φ ( - ( u + i / 2 ) ) divide start_ARG d u end_ARG start_ARG ( u + i / 2 )2 - ( u + i / 2 ) i end_ARG end_CELL end_ROW e−i⁢(u+i/2)⁢k=e−i⁢u⁢k⁢ek/2superscript2superscriptsuperscript2e^-i(u+i/2)k=e^-iuke^k/2e- i ( u + i / 2 ) k = e- i u k eitalic_k / 2 ek/2=e(log⁡(S0/K)+r⁢T)/2superscript2superscriptsubscript02e^k/2=e ( (S_0/K )+rT )/2eitalic_k / 2 = e( log ( S0 / K ) + r T ) / 2 K⁢e−r⁢T⁢ek/2=e−r⁢T/2⁢S0⁢Ksuperscriptsuperscript2superscript2subscript0Ke^-rTe^k/2=e^-rT/2 S_0KK e- r T eitalic_k / 2 = e- r T / 2 square-root start_ARG S0 K end_ARG u=z+i/22u=z+i/2u = z + i / 2 , (z−i/2)2−(z−i/2)⁢i=z2−2⁢z⁢i+1/4superscript222superscript2214(z-i/2)^2-(z-i/2)i=z^2-2zi+1/4( z - i / 2 )2 - ( z - i / 2 ) i = z2 - 2 z i + 1 / 4 C0=S0−S0⁢K⁢e−r⁢T/2π⁢∫−∞e−i⁢z⁢k⁢φ⁢(−z−i/2)⁢d⁢z2−2⁢z⁢i+1/4subscript0subscript0subscript0superscript2superscriptsubscriptsuperscript2superscript2214C_0=S_0- S_0Ke^-rT/2π _-∞^∞e^-izk% (-z-i/2) dzz^2-2zi+1/4C0 = S0 - divide start_ARG square-root start_ARG S0 K end_ARG e- r T / 2 end_ARG start_ARG π end_ARG ∫- ∞ e- i z k φ ( - z - i / 2 ) divide start_ARG d z end_ARG start_ARG z2 - 2 z i + 1 / 4 end_ARG f⁢(x)=12⁢π⁢∫−∞e−i⁢u⁢x⁢f^⁢(u)⁢u=12⁢π⁢Re⁡[∫−∞0e−i⁢u⁢x⁢f^⁢(u)⁢u]+12⁢π⁢∫0∞Re⁡[e−i⁢u⁢x⁢f^⁢(u)⁢d⁢u]=12⁢π⁢Re⁡[∫0∞e−i⁢u⁢x⁢f^⁢(u)⁢u]+12⁢π⁢∫0∞Re⁡[e−i⁢u⁢x⁢f^⁢(u)⁢d⁢u]absent12superscriptsubscriptsuperscript^differential-dmissing-subexpressionabsent12Resuperscriptsubscript0superscript^differential-d12superscriptsubscript0Resuperscript^missing-subexpressionabsent12Resuperscriptsubscript0superscript^differential-d12superscriptsubscript0Resuperscript aligned f(x)&= 12π _-∞^∞e^-% iux f(u)du\\ &= 12πRe [ _-∞^0e^-iux f(u)du% ]+ 12π _0^∞Re [e^-iux f(% u)du ]\\ &= 12πRe [ _0^∞e^-iux f(u)du% ]+ 12π _0^∞Re [e^-iux f(% u)du ] alignedstart_ROW start_CELL f ( x ) end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π end_ARG ∫- ∞ e- i u x over start_ARG f end_ARG ( u ) d u end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π end_ARG Re [ ∫- ∞0 e- i u x over start_ARG f end_ARG ( u ) d u ] + divide start_ARG 1 end_ARG start_ARG 2 π end_ARG ∫0∞ Re [ e- i u x over start_ARG f end_ARG ( u ) d u ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π end_ARG Re [ ∫0∞ e- i u x over start_ARG f end_ARG ( u ) d u ] + divide start_ARG 1 end_ARG start_ARG 2 π end_ARG ∫0∞ Re [ e- i u x over start_ARG f end_ARG ( u ) d u ] end_CELL end_ROW =12⁢π⁢[2⁢∫0∞e−i⁢u⁢x⁢f^⁢(u)⁢u]=1π⁢[∫0∞e−i⁢u⁢x⁢f^⁢(u)⁢u]missing-subexpressionabsent12delimited-[]2superscriptsubscript0superscript^differential-dmissing-subexpressionabsent1delimited-[]superscriptsubscript0superscript^differential-d aligned &= 12πRe [2 _0^% ∞e^-iux f(u)du ]\\ &= 1πRe [ _0^∞e^-iux f(u)du ]% alignedstart_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π end_ARG Re [ 2 ∫0∞ e- i u x over start_ARG f end_ARG ( u ) d u ] end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG π end_ARG Re [ ∫0∞ e- i u x over start_ARG f end_ARG ( u ) d u ] end_CELL end_ROW C0=S0−S0⁢K⁢e−r⁢T/2π⁢∫0∞Re⁡[e−i⁢z⁢k⁢φ⁢(−z−i/2)]⁢d⁢z2+1/4subscript0subscript0subscript0superscript2superscriptsubscript0Resuperscript2superscript214C_0=S_0- S_0Ke^-rT/2π _0^∞% Re [e^-izk (-z-i/2) ] dzz^2+1/4C0 = S0 - divide start_ARG square-root start_ARG S0 K end_ARG e- r T / 2 end_ARG start_ARG π end_ARG ∫0∞ Re [ e- i z k φ ( - z - i / 2 ) ] divide start_ARG d z end_ARG start_ARG z2 + 1 / 4 end_ARG CT≡subscriptabsentC_T _T ≡ max⁡[ST−K,0]subscript0 [S_T-K,0 ]max [ Sitalic_T - K , 0 ] , K≡eksuperscriptK≡ e^kK ≡ eitalic_k and ST≡essubscriptsuperscriptS_T≡ e^sSitalic_T ≡ eitalic_s , C0≡e−r⁢T⁢0Q⁢(max⁡[es−ek,0])=e−r⁢T⁢∫k∞(es−ek)⁢q⁢(s)⁢ssubscript0absentsuperscriptsuperscriptsubscript0superscriptsuperscript0missing-subexpressionabsentsuperscriptsuperscriptsubscriptsuperscriptsuperscriptdifferential-d aligned C_0&≡ e^-rTE_0^Q ( % [e^s-e^k,0 ] )\\ &=e^-rT _k^∞ (e^s-e^k )q(s)ds alignedstart_ROW start_CELL C0 end_CELL start_CELL ≡ e- r T E0italic_Q ( max [ eitalic_s - eitalic_k , 0 ] ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = e- r T ∫k∞ ( eitalic_s - eitalic_k ) q ( s ) d s end_CELL end_ROW q⁢(s)q(s)q ( s ) is the risk-neutral pdf of sTsubscripts_Tsitalic_T, c0≡eα⁢k⁢C0subscript0superscriptsubscript0c_0≡ e^α kC_0c0 ≡ eitalic_α k C0 with α>00α>0α > 0. The Fourier transform of c0subscript0c_0c0 is ψ⁢(v)≡∫−∞ei⁢v⁢k⁢c0⁢ksuperscriptsubscriptsuperscriptsubscript0differential-dψ(v)≡ _-∞^∞e^ivkc_0dkψ ( v ) ≡ ∫- ∞ eitalic_i v k c0 d k C0=e−α⁢kπ⁢∫0∞e−i⁢v⁢k⁢ψ⁢(v)⁢v.subscript0superscriptsuperscriptsubscript0superscriptdifferential-dC_0= e^-α kπ _0^∞e^-ivkψ(v)dv.C0 = divide start_ARG e- α k end_ARG start_ARG π end_ARG ∫0∞ e- i v k ψ ( v ) d v . ∎ Concepts of Risk. To quantify the measurement of financial risk, we apply ‘Cramer-Lundberg model (Figure 18)’ [112],[113],[114],[115], ‘VaR’, ‘Tail probability’ and ‘shortfall integral’ methods via a bayesian approach, the results obtained can be viewed on: https://github.com/Trusted-AI/adversarial-robustness-toolbox/pull/2467. Let us consider a random variable, X, with density function, fX⁢(x)subscriptf_X(x)fitalic_X ( x ). The moment-generating function is defined as, MX⁢(t)=⁢(et⁢X)=∫−∞et⁢x⁢fX⁢(x)⁢xsubscriptabsentsuperscriptmissing-subexpressionabsentsuperscriptsubscriptsuperscriptsubscriptdifferential-d aligned M_X(t)&=E (e^tX )\\ &= _-∞^∞e^txf_X(x)dx alignedstart_ROW start_CELL Mitalic_X ( t ) end_CELL start_CELL = blackboard_E ( eitalic_t X ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = ∫- ∞ eitalic_t x fitalic_X ( x ) d x end_CELL end_ROW fX⁢(x)=12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞MX⁢(t)⁢e−t⁢x⁢tsubscript12superscriptsubscriptlimit-from0subscriptsuperscriptdifferential-df_X(x)= 12π i _-i∞,(0+)^+i∞M_X(t)e^-txdtfitalic_X ( x ) = divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ Mitalic_X ( t ) e- t x d t for x∈ℝ,t∈ℂformulae-sequenceℝℂx ,t ∈ blackboard_R , t ∈ blackboard_C and where i is the imaginary number satisfying i=−11i= -1i = square-root start_ARG - 1 end_ARG, we introduce KX⁢(t)=ln⁡MX⁢(t)subscriptsubscriptK_X(t)= M_X(t)Kitalic_X ( t ) = ln Mitalic_X ( t ), fX⁢(x)=12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eln⁡MX⁢(t)⁢e−t⁢x⁢t=12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)−t⁢x⁢t⁢(Equation ⁢a)subscriptabsent12superscriptsubscriptlimit-from0superscriptsubscriptsuperscriptdifferential-dmissing-subexpressionabsent12superscriptsubscriptlimit-from0superscriptsubscriptdifferential-dEquation aligned f_X(x)&= 12π i _-i∞,(0+)^+% i∞e M_X(t)e^-txdt\\ &= 12π i _-i∞,(0+)^+i∞e^K_X(t)-txdt(% Equation a) alignedstart_ROW start_CELL fitalic_X ( x ) end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eroman_ln Mitalic_X ( t ) e- t x d t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eitalic_Kitalic_X ( t ) - t x d t ( Equation a ) end_CELL end_ROW The Tail Probability ℙ⁢(X>x)=∫x∞fX⁢(u)⁢uℙsuperscriptsubscriptsubscriptdifferential-dP(X>x)= _x^∞f_X(u)dublackboard_P ( X > x ) = ∫x∞ fitalic_X ( u ) d u ℙ⁢(X>x)=∫x∞12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)−t⁢u⁢t⏟Equation ⁢a⁢u=12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞∫x∞eKX⁢(t)−t⁢u⁢u⁢t=12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)⁢(∫x∞e−t⁢u⁢u)⁢t=12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)⁢[−e−t⁢ut]x∞⁢t=12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)⁢[0−(−e−t⁢xt)]⁢t=12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)−t⁢xt⁢tℙabsentsuperscriptsubscriptsubscript⏟12superscriptsubscriptlimit-from0superscriptsubscriptdifferential-dEquation differential-dmissing-subexpressionabsent12superscriptsubscriptlimit-from0superscriptsubscriptsuperscriptsubscriptdifferential-ddifferential-dmissing-subexpressionabsent12superscriptsubscriptlimit-from0superscriptsubscriptsuperscriptsubscriptsuperscriptdifferential-ddifferential-dmissing-subexpressionabsent12superscriptsubscriptlimit-from0superscriptsubscriptsuperscriptsubscriptdelimited-[]superscriptdifferential-dmissing-subexpressionabsent12superscriptsubscriptlimit-from0superscriptsubscriptdelimited-[]0superscriptdifferential-dmissing-subexpressionabsent12superscriptsubscriptlimit-from0superscriptsubscriptdifferential-d aligned P(X>x)&= _x^∞ % 12π i _-i∞,(0+)^+i∞e^K_X(t)-tudt_% Equation adu\\ &= 12π i _-i∞,(0+)^+i∞ _x^∞e^K_X(t)-% tududt\\ &= 12π i _-i∞,(0+)^+i∞e^K_X(t) ( _x^% ∞e^-tudu )dt\\ &= 12π i _-i∞,(0+)^+i∞e^K_X(t) [- e^-% tut ]_x^∞dt\\ &= 12π i _-i∞,(0+)^+i∞e^K_X(t) [0- (-% e^-txt ) ]dt\\ &= 12π i _-i∞,(0+)^+i∞ e^K_X(t)-txtdt% alignedstart_ROW start_CELL blackboard_P ( X > x ) end_CELL start_CELL = ∫x∞ under⏟ start_ARG divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eitalic_Kitalic_X ( t ) - t u d t end_ARGEquation a d u end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ ∫x∞ eitalic_Kitalic_X ( t ) - t u d u d t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eitalic_Kitalic_X ( t ) ( ∫x∞ e- t u d u ) d t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eitalic_Kitalic_X ( t ) [ - divide start_ARG e- t u end_ARG start_ARG t end_ARG ]x∞ d t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eitalic_Kitalic_X ( t ) [ 0 - ( - divide start_ARG e- t x end_ARG start_ARG t end_ARG ) ] d t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ divide start_ARG eitalic_Kitalic_X ( t ) - t x end_ARG start_ARG t end_ARG d t end_CELL end_ROW The VaR282828VaR is an upper-tail probability, which allows us to deduce: shortfall integral ⁢(X⁢∣X>⁢x)=1ℙ⁢(X>x)⁢∫x∞u⁢fX⁢(u)⁢u,=1ℙ⁢(X>x)⁢∫x∞u⁢12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)−t⁢u⁢t⏟Equation ⁢a⁢u,=1ℙ⁢(X>x)⁢12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)⁢(∫x∞u⁢e−t⁢u⁢u)⁢t,=1ℙ⁢(X>x)⁢12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)⁢[(−ut−1t2)⁢e−t⁢u]x∞⁢t,=1ℙ⁢(X>x)⁢12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)⁢[−(−xt−1t2)⁢e−t⁢x]⁢t⁢, =1ℙ⁢(X>x)⁢12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)⁢[x⁢e−t⁢xt+e−t⁢xt2]⁢t⁢, =1ℙ⁢(X>x)⁢(12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)−t⁢xt2⁢t+x⁢12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)−t⁢xt⁢t⏟ℙ⁢(X>x))⁢, =1ℙ⁢(X>x)⁢(12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)−t⁢xt2⁢t+x⁢ℙ⁢(X>x))⁢, =x+1ℙ⁢(X>x)⁢(12⁢π⁢i⁢∫−i⁢∞,(0+)+i⁢∞eKX⁢(t)−t⁢xt2⁢t)⁢.missing-subexpressionket1ℙsuperscriptsubscriptsubscriptdifferential-dmissing-subexpressionabsent1ℙsuperscriptsubscriptsubscript⏟12superscriptsubscriptlimit-from0superscriptsubscriptdifferential-dEquation differential-dmissing-subexpressionabsent1ℙ12superscriptsubscriptlimit-from0superscriptsubscriptsuperscriptsubscriptsuperscriptdifferential-ddifferential-dmissing-subexpressionabsent1ℙ12superscriptsubscriptlimit-from0superscriptsubscriptsuperscriptsubscriptdelimited-[]1superscript2superscriptdifferential-dmissing-subexpressionabsent1ℙ12superscriptsubscriptlimit-from0superscriptsubscriptdelimited-[]1superscript2superscriptdifferential-d, missing-subexpressionabsent1ℙ12superscriptsubscriptlimit-from0superscriptsubscriptdelimited-[]superscriptsuperscriptsuperscript2differential-d, missing-subexpressionabsent1ℙ12superscriptsubscriptlimit-from0superscriptsubscriptsuperscript2differential-dsubscript⏟12superscriptsubscriptlimit-from0superscriptsubscriptdifferential-dℙ, missing-subexpressionabsent1ℙ12superscriptsubscriptlimit-from0superscriptsubscriptsuperscript2differential-dℙ, missing-subexpressionabsent1ℙ12superscriptsubscriptlimit-from0superscriptsubscriptsuperscript2differential-d. aligned &E(X X>x)= 1P(X>x)% _x^∞uf_X(u)du,\\ &= 1P(X>x) _x^∞u 12π i _% -i∞,(0+)^+i∞e^K_X(t)-tudt_Equation adu,\\ &= 1P(X>x) 12π i _-i∞,(0+)^+i∞e^K% _X(t) ( _x^∞ue^-tudu )dt,\\ &= 1P(X>x) 12π i _-i∞,(0+)^+i∞e^K% _X(t) [ (- ut- 1t^2 )e^-tu ]_x^% ∞dt,\\ &= 1P(X>x) 12π i _-i∞,(0+)^+i∞e^K% _X(t) [- (- xt- 1t^2 )e^-tx ]dt% , \\ &= 1P(X>x) 12π i _-i∞,(0+)^+i∞e^K% _X(t) [ xe^-txt+ e^-txt^2 ]dt, \\ &= 1P(X>x)( 12π i _-i∞,(0+)^+i∞% e^K_X(t)-txt^2dt+x 12π i _-i∞,(0% +)^+i∞ e^K_X(t)-txtdt_P(X>x)), \\ &= 1P(X>x) ( 12π i _-i∞,(0+)^+i% ∞ e^K_X(t)-txt^2dt+xP(X>x) ), \\ &=x+ 1P(X>x) ( 12π i _-i∞,(0+)^+i% ∞ e^K_X(t)-txt^2dt ). alignedstart_ROW start_CELL end_CELL start_CELL blackboard_E ( X ∣ X > x ) = divide start_ARG 1 end_ARG start_ARG blackboard_P ( X > x ) end_ARG ∫x∞ u fitalic_X ( u ) d u , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG blackboard_P ( X > x ) end_ARG ∫x∞ u under⏟ start_ARG divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eitalic_Kitalic_X ( t ) - t u d t end_ARGEquation a d u , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG blackboard_P ( X > x ) end_ARG divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eitalic_Kitalic_X ( t ) ( ∫x∞ u e- t u d u ) d t , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG blackboard_P ( X > x ) end_ARG divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eitalic_Kitalic_X ( t ) [ ( - divide start_ARG u end_ARG start_ARG t end_ARG - divide start_ARG 1 end_ARG start_ARG t2 end_ARG ) e- t u ]x∞ d t , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG blackboard_P ( X > x ) end_ARG divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eitalic_Kitalic_X ( t ) [ - ( - divide start_ARG x end_ARG start_ARG t end_ARG - divide start_ARG 1 end_ARG start_ARG t2 end_ARG ) e- t x ] d t , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG blackboard_P ( X > x ) end_ARG divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ eitalic_Kitalic_X ( t ) [ divide start_ARG x e- t x end_ARG start_ARG t end_ARG + divide start_ARG e- t x end_ARG start_ARG t2 end_ARG ] d t , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG blackboard_P ( X > x ) end_ARG ( divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ divide start_ARG eitalic_Kitalic_X ( t ) - t x end_ARG start_ARG t2 end_ARG d t + x under⏟ start_ARG divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ divide start_ARG eitalic_Kitalic_X ( t ) - t x end_ARG start_ARG t end_ARG d t end_ARGblackboard_P ( X > x ) ) , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG blackboard_P ( X > x ) end_ARG ( divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ divide start_ARG eitalic_Kitalic_X ( t ) - t x end_ARG start_ARG t2 end_ARG d t + x blackboard_P ( X > x ) ) , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = x + divide start_ARG 1 end_ARG start_ARG blackboard_P ( X > x ) end_ARG ( divide start_ARG 1 end_ARG start_ARG 2 π i end_ARG ∫- i ∞ , ( 0 + )+ i ∞ divide start_ARG eitalic_Kitalic_X ( t ) - t x end_ARG start_ARG t2 end_ARG d t ) . end_CELL end_ROW Figure 18: Cramer-Lundberg model via bayesian optimization Lundberg’s inequality Theorem 2. Suppose κ>00κ>0κ > 0. Then the probability of ruin ψ⁢(u)ψ(u)ψ ( u ) satisfies, ψ⁢(u)≤e−κ⁢u,u≥0formulae-sequencesuperscript0ψ(u)≤ e^-κ u, u≥ 0ψ ( u ) ≤ e- κ u , u ≥ 0 Proof. ψn+1⁢(u)=∫0∞[1−F⁢(u+c⁢t)+∫0u+c⁢tψn⁢(u+c⁢t−x)⁢F⁢(x)]⁢λ⁢e−λ⁢t⁢tsubscript1absentsuperscriptsubscript0delimited-[]1superscriptsubscript0subscriptdifferential-dsuperscriptdifferential-d aligned _n+1(u)\\ = _0^∞ [1-F(u+ct)+ _0^u+ct _n(u+ct-x)dF(x) ]% λ e^-λ tdt alignedstart_ROW start_CELL ψitalic_n + 1 ( u ) end_CELL end_ROW start_ROW start_CELL = ∫0∞ [ 1 - F ( u + c t ) + ∫0u + c t ψitalic_n ( u + c t - x ) d F ( x ) ] λ e- λ t d t end_CELL end_ROW ψn+1⁢(u)=∫0∞[∫u+c⁢t∞F⁢(x)+∫0u+c⁢tψn⁢(u+c⁢t−x)⁢F⁢(x)]⁢λ⁢e−λ⁢t⁢t≤∫0∞[∫u+c⁢t∞e−κ⁢(u+c⁢t−x)dF(x)+∫0u+c⁢te−κ⁢(u+c⁢t−x)dF(x)]λe−λ⁢tdt aligned & _n+1(u)= _0^∞ [ _u+ct% ^∞dF(x)+ _0^u+ct _n(u+ct-x)dF(x) ]λ e^-λ t% dt\\ &≤ _0^∞ [ _u+ct^∞e^-κ(u+ct-x)dF(x) % .\\ & .+ _0^u+cte^-κ(u+ct-x)dF(x) ]λ e^-λ tdt% alignedstart_ROW start_CELL end_CELL start_CELL ψitalic_n + 1 ( u ) = ∫0∞ [ ∫u + c t∞ d F ( x ) + ∫0u + c t ψitalic_n ( u + c t - x ) d F ( x ) ] λ e- λ t d t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ≤ ∫0∞ [ ∫u + c t∞ e- κ ( u + c t - x ) d F ( x ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∫0u + c t e- κ ( u + c t - x ) d F ( x ) ] λ e- λ t d t end_CELL end_ROW ψn+1⁢(u)≤∫0∞[∫0∞e−κ⁢(u+c⁢t−x)⁢F⁢(x)]⁢λ⁢e−λ⁢t⁢t=λ⁢e−κ⁢u⁢∫0∞e−κ⁢c⁢t⁢[∫0∞eκ⁢x⁢F⁢(x)]⁢e−λ⁢t⁢t=λ⁢e−κ⁢u⁢∫0∞e−(λ+κ⁢c)⁢t⁢[MX⁢(κ)]⁢t=λ⁢MX⁢(κ)⁢e−κ⁢u⁢∫0∞e−(λ+κ⁢c)⁢t⁢t=λ⁢MX⁢(κ)λ+κ⁢c⁢e−κ⁢usubscript1absentsuperscriptsubscript0delimited-[]superscriptsubscript0superscriptdifferential-dsuperscriptdifferential-dmissing-subexpressionabsentsuperscriptsuperscriptsubscript0superscriptdelimited-[]superscriptsubscript0superscriptdifferential-dsuperscriptdifferential-dmissing-subexpressionabsentsuperscriptsuperscriptsubscript0superscriptdelimited-[]subscriptdifferential-dmissing-subexpressionabsentsubscriptsuperscriptsuperscriptsubscript0superscriptdifferential-dmissing-subexpressionabsentsubscriptsuperscript aligned _n+1(u)&≤ _0^∞ [ _0% ^∞e^-κ(u+ct-x)dF(x) ]λ e^-λ tdt\\ &=λ e^-κ u _0^∞e^-κ ct [ _0^∞e% ^κ xdF(x) ]e^-λ tdt\\ &=λ e^-κ u _0^∞e^-(λ+κ c)t [M_X(% κ) ]dt\\ &=λ M_X(κ)e^-κ u _0^∞e^-(λ+κ c)t% dt\\ &= λ M_X(κ)λ+κ ce^-κ u alignedstart_ROW start_CELL ψitalic_n + 1 ( u ) end_CELL start_CELL ≤ ∫0∞ [ ∫0∞ e- κ ( u + c t - x ) d F ( x ) ] λ e- λ t d t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = λ e- κ u ∫0∞ e- κ c t [ ∫0∞ eitalic_κ x d F ( x ) ] e- λ t d t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = λ e- κ u ∫0∞ e- ( λ + κ c ) t [ Mitalic_X ( κ ) ] d t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = λ Mitalic_X ( κ ) e- κ u ∫0∞ e- ( λ + κ c ) t d t end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG λ Mitalic_X ( κ ) end_ARG start_ARG λ + κ c end_ARG e- κ u end_CELL end_ROW λ⁢MX⁢(κ)=λ⁢[1+(1+θ)⁢κ⁢μ]=λ+κ⁢(1+θ)⁢λ⁢μ=λ+κ⁢csubscriptdelimited-[]111λ M_X(κ)=λ[1+(1+θ)κμ]=λ+κ(1+θ)% λμ=λ+κ cλ Mitalic_X ( κ ) = λ [ 1 + ( 1 + θ ) κ μ ] = λ + κ ( 1 + θ ) λ μ = λ + κ c ψn+1⁢(u)≤e−κ⁢usubscript1superscript _n+1(u)≤ e^-κ uψitalic_n + 1 ( u ) ≤ e- κ u , ψn⁢(u)≤e−κ⁢usubscriptsuperscript _n(u)≤ e^-κ uψitalic_n ( u ) ≤ e- κ u for all n , ψ⁢(u)=absentψ(u)=ψ ( u ) = limn→∞ψn⁢(u)≤e−κ⁢usubscript→subscriptsuperscript _n→∞ _n(u)≤ e^-κ ulimitalic_n → ∞ ψitalic_n ( u ) ≤ e- κ u. θ=u⁢E⁢[exp⁡(−ln⁡αu⁢X)]−1−μ⁢ln⁡α−1Edelimited-[]11θ= u \E [ (- αuX )% ]-1 \-μ α-1θ = divide start_ARG u E [ exp ( - divide start_ARG ln α end_ARG start_ARG u end_ARG X ) ] - 1 end_ARG start_ARG - μ ln α end_ARG - 1 κ=(−ln⁡α)/uκ=(- α)/uκ = ( - ln α ) / u, ψ⁢(u)≤e−κ⁢u=eln⁡α=αsuperscriptsuperscriptψ(u)≤ e^-κ u=e α=αψ ( u ) ≤ e- κ u = eroman_ln α = α, u=−ln⁡ακu= - ακu = divide start_ARG - ln α end_ARG start_ARG κ end_ARG ψ⁢(u)≤e−κ⁢u=eln⁡α=αsuperscriptsuperscriptψ(u)≤ e^-κ u=e α=αψ ( u ) ≤ e- κ u = eroman_ln α = α ψ⁢(∞)=limu→∞ψ⁢(u)=0subscript→0ψ(∞)= _u→∞ψ(u)=0ψ ( ∞ ) = limitalic_u → ∞ ψ ( u ) = 0 0≤ψ⁢(u)≤e−κ⁢u0superscript0≤ψ(u)≤ e^-κ u0 ≤ ψ ( u ) ≤ e- κ u Suppose κ>00κ>0κ > 0. Then the ruin probability satisfies, ψ⁢(u)∼C⁢e−κ⁢u,u→∞formulae-sequencesimilar-tosuperscript→ψ(u) Ce^-κ u, u→∞ψ ( u ) ∼ C e- κ u , u → ∞ C=μ⁢θMX′⁢(κ)−μ⁢(1+θ)superscriptsubscript′1C= μθM_X (κ)-μ(1+θ)C = divide start_ARG μ θ end_ARG start_ARG Mitalic_X′ ( κ ) - μ ( 1 + θ ) end_ARG ∎ Understanding Diffusion concepts at a more advanced level Concepts of Diffusion Riemannian. Consider a solution u:ℳ×[0,T]→[0,∞):→ℳ00u:M×[0,T]→[0,∞)u : M × [ 0 , T ] → [ 0 , ∞ ) , ∂u∂t=Δ⁢uΔ ∂ u∂ t= udivide start_ARG ∂ u end_ARG start_ARG ∂ t end_ARG = Δ u with ∫ℳu⁢(⋅,0)⁢μg=1subscriptℳ⋅0differential-dsubscript1 _Mu(·,0)d _g=1∫M u ( ⋅ , 0 ) d μitalic_g = 1, where μgsubscript _gμitalic_g is the Riemannian 292929Ricci-Riemann volume measure. d⁢t⁢∫ℳu⁢(⋅,t)⁢μg=∫ℳΔ⁢u⁢μg=0,subscriptℳ⋅differential-dsubscriptsubscriptℳΔdifferential-dsubscript0 ddt _Mu(·,t)d _g= _M ud% _g=0,divide start_ARG d end_ARG start_ARG d t end_ARG ∫M u ( ⋅ , t ) d μitalic_g = ∫M Δ u d μitalic_g = 0 , The integral of u over ℳMM remains one for each t∈[0,T]0t∈[0,T]t ∈ [ 0 , T ], and in particular, u can be viewed as a probability density of a measure ν⁢(t)ν(t)ν ( t ) defined by d⁢ν⁢(t):=u⁢(⋅,t)⁢d⁢μgassign⋅subscriptdν(t):=u(·,t)d _gd ν ( t ) := u ( ⋅ , t ) d μitalic_g. ν⁢(t)ν(t)ν ( t ) represents the probability distribution of a particle moving on the manifold under Brownian motion, with initial probability distribution ν⁢(0)0ν(0)ν ( 0 ), Suppose g⁢(τ)g(τ)g ( τ ) is a family of Riemannian metrics on ℳMM, for τ∈absentτ∈τ ∈ [0,T]0[0,T][ 0 , T ]. We call a flow of measures v⁢(τ)v(τ)v ( τ ) a diffusion [116] (representing the probability distribution of a Brownian particle as above) if d⁢ν⁢(τ)=u⁢(⋅,τ)⁢d⁢μg⁢(τ)⋅subscriptdν(τ)=u(·,τ)d _g(τ)d ν ( τ ) = u ( ⋅ , τ ) d μitalic_g ( τ ) with , ∂u∂τ=Δg⁢(τ)⁢u−(12⁢tr⁡∂g∂τ)⁢usubscriptΔ12tr ∂ u∂τ= _g(τ)u- ( 12% tr ∂ g∂τ )udivide start_ARG ∂ u end_ARG start_ARG ∂ τ end_ARG = Δitalic_g ( τ ) u - ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG tr divide start_ARG ∂ g end_ARG start_ARG ∂ τ end_ARG ) u ∂g∂τ=2⁢Ric⁡(g⁢(τ))2Ric ∂ g∂τ=2Ric(g(τ))divide start_ARG ∂ g end_ARG start_ARG ∂ τ end_ARG = 2 Ric ( g ( τ ) ), g⁢(⋅)⋅g(·)g ( ⋅ ) a Ricci flow with respect to a ’reverse’ time coordinate τ (that is, τ=C−tτ=C-tτ = C - t for some constant C ), ∂u∂τ=Δg⁢(τ)⁢u−R⁢usubscriptΔ ∂ u∂τ= _g(τ)u-Rudivide start_ARG ∂ u end_ARG start_ARG ∂ τ end_ARG = Δitalic_g ( τ ) u - R u d⁢τ⁢∫ℳν⁢(τ)subscriptℳdifferential-d ddτ _Mdν(τ)divide start_ARG d end_ARG start_ARG d τ end_ARG ∫M d ν ( τ ) =d⁢τ⁢∫ℳu⁢(⋅,τ)⁢μg⁢(τ)absentsubscriptℳ⋅differential-dsubscript = ddτ _Mu(·,τ)d _g(τ)= divide start_ARG d end_ARG start_ARG d τ end_ARG ∫M u ( ⋅ , τ ) d μitalic_g ( τ ) = == ∫ℳ(∂u∂τ⁢d⁢μg⁢(τ)+u⁢∂τ⁢d⁢μg⁢(τ))subscriptℳsubscriptsubscript _M ( ∂ u∂τd _g(% τ)+u ∂τd _g(τ) )∫M ( divide start_ARG ∂ u end_ARG start_ARG ∂ τ end_ARG d μitalic_g ( τ ) + u divide start_ARG ∂ end_ARG start_ARG ∂ τ end_ARG d μitalic_g ( τ ) ) =∫ℳΔg⁢(τ)⁢u⁢μg⁢(τ)=0absentsubscriptℳsubscriptΔdifferential-dsubscript0 = _M _g(τ)ud _g(τ)=0= ∫M Δitalic_g ( τ ) u d μitalic_g ( τ ) = 0 If g⁢(τ)g(τ)g ( τ ) is a flow of Riemannian metrics for τ∈[a,b]τ∈[a,b]τ ∈ [ a , b ], and ν⁢(τ)ν(τ)ν ( τ ) is a diffusion (as defined above) then given any f:ℳ×[a,b]→ℝ:→ℳℝf:M×[a,b] : M × [ a , b ] → blackboard_R solving −∂f∂τ=Δg⁢(τ)⁢fsubscriptΔ- ∂ f∂τ= _g(τ)f- divide start_ARG ∂ f end_ARG start_ARG ∂ τ end_ARG = Δitalic_g ( τ ) f, d⁢τ⁢∫ℳf⁢(⋅,τ)⁢ν⁢(τ)=0.subscriptℳ⋅differential-d0 ddτ _Mf(·,τ)dν(τ)=0.divide start_ARG d end_ARG start_ARG d τ end_ARG ∫M f ( ⋅ , τ ) d ν ( τ ) = 0 . 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