Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.
INTRODUCTION
Research background
Vietnam aims to become a significant economic hub in Southeast Asia, focusing on trade and finance Global integration is crucial for the country's economic development, a strategy that has been in place since the 1986 reform policies opened up the economy The normalization of diplomatic relations with the US in 1995 eliminated previous hostilities, allowing Vietnam to engage with the global community and foster cooperative relationships worldwide.
Vietnam has made significant strides in trading, particularly after joining the World Trade Organization, which has facilitated the establishment of numerous bilateral and multilateral free trade agreements This commitment to global trade has positioned Vietnam as one of the most open economies worldwide, with trade representing 186% of its gross domestic product, making it the tenth highest globally and second in Southeast Asia, following Singapore Over the past few decades, Vietnam has transformed its trading landscape, increasing its trade-to-GDP ratio from 43% in 1995 As a result, Vietnam has earned a reputation as a reliable trading partner, fostering mutual benefits in its international relationships, and attracting multinational corporations due to its stable political environment and supportive policies.
The global integration of Vietnam's economy hinges not only on trade but also on the transformation of its financial market, which is crucial for connecting Vietnamese firms with a broader pool of international investors Currently, Vietnam's financial market is relatively underdeveloped compared to its Southeast Asian neighbors, with total assets of financial institutions at approximately 219% of GDP, significantly lower than the ASEAN average of 320% Additionally, the market capitalization of the Vietnamese stock market stands at about 84% of GDP, only surpassing Indonesia, while the value of stocks traded to GDP is modest at 16.4%, far behind Malaysia and Thailand These indicators reveal that Vietnam's financial market has yet to realize its full potential, necessitating substantial modernization and development efforts Although Vietnam benefits from political stability that attracts foreign investors, this alone is insufficient for a robust financial market To secure long-term participation from global investors, Vietnam must adopt strategies similar to those used by its industries to attract and retain international customers.
The Vietnamese financial market must enhance its appeal by demonstrating to global investors that it is not only a secure destination but also a flexible and lucrative option that can meet their varied investment requirements.
Developing the Vietnamese financial market is crucial for enhancing the country's position on regional and global platforms Various proposals aim to transform Vietnam into a significant financial center, attracting international investors to infuse capital into the market Establishing this financial hub would enable Vietnamese firms to engage with the global financial landscape, secure overseas financing, and boost their competitiveness However, these proposals remain largely theoretical and are still under discussion, indicating that modernizing the financial market and creating a global financial center is a complex challenge that necessitates collaborative efforts from all relevant authorities.
To achieve a stable and attractive business environment for foreign investors, Vietnam must focus on several key areas: maintaining a friendly investment climate, investing in advanced technology, educating a skilled workforce, implementing appealing tax policies, refining capital flow regulations, and eliminating regulatory barriers Additionally, the Vietnamese financial market should offer a diverse range of financial products and tools to cater to various investor preferences, enabling them to meet their investment goals and manage risks effectively Modernizing the financial market goes beyond adopting cutting-edge technology; it requires providing alternative investment options that accommodate a wide spectrum of investor needs, from simple to complex strategies While enhancing traditional securities markets like stock exchanges is crucial for elevating Vietnam's financial status, developing a robust derivatives market will significantly contribute to positioning Vietnam as a regional financial hub.
The Vietnamese government recognizes the significance of developing its financial market, as evidenced by Prime Minister Decision No 368/QĐ-TTg, which outlines a financial strategy aimed at 2030 This strategy emphasizes the need for a synchronized, transparent, and sustainable financial market, with specific goals such as achieving stock market capitalization of 120% of GDP by 2030 It also highlights the importance of diversifying the derivatives market by introducing a wider range of products, including derivatives based on stock indices, individual stocks, and government bonds, along with the creation of new indices for these derivatives This initiative signals the government’s commitment to modernizing the financial market, indicating that significant transformations are forthcoming to enhance its regional standing through the improvement of traditional markets and the establishment of new ones, particularly a comprehensive derivatives market.
Vietnam has taken significant steps to establish a derivatives market, launching futures contracts based on the VN30 stock index in 2017 That same year, the issuance of covered warrants, a type of option derivative, was approved By November 2022, the State Securities Commission of Vietnam reported that 135 companies had issued covered warrants, with a trading volume of 951 million and a total value of 1.592 trillion Vietnamese dong Despite the growth of the warrant market, a formal options market has not yet been established.
To develop a robust derivatives market, Vietnam must establish a proper options exchange and create new options, including stock options Option pricing is crucial for this market's success and is the central focus of this thesis, which aims to address the challenges posed by option pricing as a significant barrier to a formal options market For meaningful market development, a variety of options, from conventional types like European options to exotic types such as barrier options, should be made available to investors This thesis will propose solutions for option pricing that cater to different option types, market liquidity, and the overall development stage of the options market While the Vietnamese market serves as the primary case study, the findings can be applied to other emerging or illiquid markets globally.
Research objectives and contributions
To price options effectively, we typically utilize an option pricing model, with the Black-Scholes model (1973) being the most recognized While the Black-Scholes model has been used for pricing covered warrants, it relies on several assumptions that may oversimplify real market conditions A key challenge is its dependence on the volatility of the underlying asset's price, which is the only unobservable variable in the model and is notoriously difficult to ascertain Additionally, the model assumes that this volatility remains constant throughout the option's life, despite the fact that it can fluctuate based on market dynamics Consequently, while the Black-Scholes model is popular for its simplicity, there is a growing need for alternative models that incorporate stochastic volatility to more accurately reflect market conditions, leading to the development of various new option pricing models.
This thesis aims to evaluate the empirical performance of four stochastic option pricing models—Heston (1993), Bates (1996), the Heston–Hull–White model by Grzelak and Oosterlee (2011), and the Heston++ model by Pacati et al (2014)—on stock options The assessment will focus on both pricing and hedging capabilities, utilizing data from individual stock options The findings will provide insights into the most suitable stochastic volatility option pricing model for the Vietnamese market and potentially other global markets.
Implied volatility (IV) surface is a crucial aspect of option pricing, illustrating how IV fluctuates with different strike prices and maturities This analysis is essential for pricing options in the emerging Vietnamese option market and holds significance for various global markets as well.
Implied volatility (IV) is defined as the volatility value derived from the market price of an option using the Black-Scholes formula (Li et al., 2021) Ideally, this model suggests that all options with the same underlying assets and expiration dates should exhibit a flat IV curve, indicating uniform volatility across different strike prices However, in practice, options display non-flat IV curves, often referred to as "smiles," where at-the-money options show different IV values compared to those at higher and lower strike prices (Derman and Miller, 2016) Additionally, IV values vary across different maturities, creating a term structure of IV Consequently, when IV values are plotted against strike prices and maturities, they form a non-flat volatility surface.
In Vietnam, the options market is limited to covered warrants, with no formal exchange established, resulting in a lack of reliable implied volatility (IV) data Consequently, when introducing options for Vietnamese stocks, it is crucial to develop a dependable method for determining the "initial" IV smile, especially since these options may not yet exist or have minimal trading activity This thesis aims to propose a practical approach for constructing IV smiles in markets with little to no liquidity, addressing the challenges of obtaining accurate IV data in such environments.
This thesis aims to establish the essential steps for developing a fully functioning derivatives market in Vietnam, focusing on derivatives pricing methodologies The first objective assesses various well-known derivatives pricing models calibrated to the smile of European options, revealing that the Heston model delivers the best performance among stochastic volatility models, making it suitable for pricing options in Vietnam Additionally, this research connects option model performance to market characteristics, as indicated by the implied volatility (IV) surface and the risk-neutral return distribution moments The second objective introduces a novel method for constructing the IV smile in the absence of a liquid derivatives market, applicable for pricing both vanilla and exotic options, such as barrier or basket options The proposed methods are versatile and can be utilized in any global option market, regardless of liquidity, with performance improving as data availability increases Overall, this research provides a comprehensive methodology for launching a robust derivatives market in Vietnam, accommodating both European and exotic derivatives to meet diverse market participant needs.
Summary of methodology and results
The methodology evaluates the performance of four stochastic models—Heston, Heston++, Bates, and Heston-Hull-White—by comparing pricing errors in in-sample, out-of-sample, and hedging contexts, with the optimal model exhibiting the smallest median error over the longest duration Subsequent tests assess the models' ability to capture market characteristics, particularly the implied volatility (IV) surface and the moments of the risk-neutral return distribution, including volatility, skewness, and kurtosis To ensure the robustness of the findings, additional tests examine whether the models' performance is influenced by the distinct characteristics of various industries, confirming the universality of the results across different economic sectors Ultimately, the study aims to identify which of the four stochastic models demonstrates superior performance and reliability for pricing European options in Vietnam and globally.
To establish an implied volatility (IV) smile for a specific Vietnamese stock, the initial step involves identifying a "benchmark company" with stock returns that closely resemble those of the target stock For this analysis, US stocks serve as the primary pool for potential benchmark companies; however, the selection can extend beyond the US to include international stock returns A balanced approach is recommended, utilizing both parametric and non-parametric methods The parametric method focuses on determining the benchmark based on the correlation of stock returns, while the K-nearest neighbor (KNN) and weighted KNN methods employ machine learning algorithms to identify appropriate benchmarks effectively.
To analyze stock returns, a benchmark company is selected based on similar characteristics, allowing for the construction of an implied volatility (IV) smile curve using the chosen company's realized volatility, skewness, and kurtosis The performance of various methods is evaluated through tests on Vietnamese companies categorized by their realized skewness and kurtosis levels Each case presents the IV smile of the Vietnamese company alongside its US counterpart, enabling a comparative analysis of results and insights into the effectiveness of the proposed methods.
The Heston model emerges as the optimal choice among four models for accurate pricing and market characteristic capture, particularly for pricing European stock options in Vietnam's future options market Among the proposed methods, machine learning techniques like KNN and WKNN demonstrate superior performance due to their alignment with market characteristics in "benchmark company" selection, while the correlation method offers ease of implementation by simply correlating returns Additionally, the introduction of advanced implied volatility construction methods can facilitate the development of complex or exotic derivatives, thereby expanding the product offerings in the Vietnamese derivatives market.
This thesis presents various solutions for pricing stock options that could enhance the future Vietnamese options market These solutions encompass a diverse array of options, including both conventional and exotic types, paving the way for a robust market development Additionally, the proposed pricing strategies are applicable not only in Vietnam but also in numerous global markets, irrespective of their liquidity and developmental status.
Thesis structure
The upcoming chapters of this thesis are structured as follows: Chapter 2 presents the necessary theoretical background and literature review, Chapter 3 outlines the methodology and data utilized, Chapter 4 showcases the results corresponding to each research objective, and Chapter 5 concludes the thesis.
THEORETICAL FRAMEWORK AND LITERATURE REVIEW
Core concepts
Before delving into the main focus of this thesis, it is essential to clarify several key definitions, beginning with the fundamental concept of an option and the methods used to determine its price.
Derivatives are unique financial instruments that do not possess intrinsic value; instead, their worth is derived from underlying assets such as stocks or bonds These contracts can be effectively priced, managed for risk, and traded in various forms, including forwards, futures, swaps, and options.
An option is a derivative instrument defined as a contract between a buyer and a seller, granting the buyer the right to trade a specific underlying asset at a predetermined price on a specified date Notably, the buyer is not obligated to execute the trade, allowing the contract to expire without transaction There are two main types of options: a "call option," which allows the buyer the right to purchase an asset, and a "put option," which provides the buyer the right to sell an asset.
To purchase an option, the buyer pays a fee known as the option price or premium The option holder has the right to buy or sell the specified underlying asset at a predetermined price, known as the exercise or strike price, upon contract expiration This process is referred to as exercising an option If the option is exercised, the seller, or writer, must fulfill their obligations by selling (for a call option) or buying (for a put option) the underlying asset at the agreed price Additionally, options are categorized as either European or American, each with distinct characteristics.
2.1.1.1 European call and put options
A European call option is a financial derivative that grants the holder the right, without any obligation, to purchase an underlying asset at a predetermined strike price (K) before a specified expiration time (T) The value of the option at expiration is determined by the price of the underlying asset (XT) at that time, defining its payoff.
0, 𝑖𝑓 𝑋 𝑇 ≤ 𝐾 where (ã) + denotes the maximum function and is less verbose to write (𝑋 𝑇 − 𝐾) + rather than 𝑚𝑎𝑥(𝑋 𝑇 − 𝐾, 0) In other words, in the first case, where XT is higher than
K, the option holder can buy the underlying at price K and sell it right after and earn
XT, making a profit of XT - K This is the case when the European call option is exercised Otherwise, the option is not exercised, hence the profit is zero
A European put option is a derivative contract that grants the holder the right, but not the obligation, to sell a unit of the underlying asset at a predetermined strike price K, with an expiration time T The payoff structure of the European put option can be expressed mathematically.
A European put option is exercised when the underlying asset price (XT) falls below the strike price (K), allowing the option holder to sell at K and repurchase at XT, resulting in a profit of K - XT If XT is greater than or equal to K, the option remains unexercised, resulting in no payoff.
More generally, we can consider European options as derivatives with expiration time
The function h(x) represents the non-negative payoff of a derivative, which is determined by the stock price XT at expiration time T European options are classified as path-independent since their payoff h(XT) relies solely on the stock price at the time of expiration.
Before expiration (t < T), the value of the derivative, or option price, varies with t and the stock price at that time, Xt Therefore, option price at time t and for a stock price
Xt = x can be written as P(t, x):
The option price, represented by the formula P(t, x) = E[e^(-r_c(T-t)) h(X_τ)], reflects the expected value of the option’s discounted payoff, where r_c denotes the continuously compounded risk-free rate used for discounting future cash flows Understanding this pricing function is essential for accurately pricing the option.
An American option is a type of derivative contract that allows the holder to exercise the option at any time before its expiration time, denoted as T The exercise time, represented by τ, is crucial as market movements are unpredictable, requiring the holder to make a decision on whether to exercise the option at any time t, where t is less than or equal to T The payoff function of the option, denoted as h, determines the value at the exercise time τ, calculated as h(Xτ), with Xτ representing the stock price at that moment.
The payoff for an American call option is calculated as h(Xτ) = (Xτ – K) +, where K represents the strike price and τ is the exercise time determined by the option holder, with τ ≤ T In contrast, the payoff for an American put option is given by h(Xτ) = (K – Xτ) +, and exercising this option occurs only when K > Xτ The notation ( ) + indicates the maximum function To price an American option at time t = 0, one approach is to determine the maximum expected value of the discounted payoff across all potential exercise times τ ≤ T, similar to the method used for European options.
Option premiums are influenced by several key factors, including the underlying asset's price (St), the exercise price (K), the time remaining until contract expiration (T), the interest rate (rC), and the asset's volatility (σ), as illustrated by the Black-Scholes formula for European call options.
𝑑 2 = 𝑑 1 − 𝜎√𝑇 − 𝑡 with N(d1), N(d2) as cumulative normal probabilities and rc as continuously compounded risk-free rate
Similarly, the Black-Scholes formula for the European put option is
The classic Black-Scholes option pricing model
This section introduces a model proposed by Samuelson and utilized by Black and Scholes, which consists of two types of assets: a riskless asset, represented by a bond, and a risky asset The price of the bond, denoted as βt, at time t is characterized by an ordinary differential equation.
𝑑𝛽 𝑡 = 𝑟𝛽 𝑡 𝑑𝑡, (1.1) where r, is the instantaneous interest rate for lending or borrowing money Setting β0
= 1, we have βt = e rt for t ≥ 0 The price of the risky asset, denoted by Xt, evolves according to the following stochastic differential equation
𝑑𝑋 𝑡 = 𝜇𝑋 𝑡 𝑑𝑡 + 𝜎𝑋 𝑡 𝑑𝑊 𝑡 (1.2) where μ is a constant mean return rate, σ > 0 is a constant volatility, and (Wt)t ≥ 0 is a standard Brownian motion The following sections will expand upon this model
Brownian motion is a stochastic process, i.e., a collection of random variables, whose definition, existence, properties and applications have been subject of numerous studies since the nineteenth century
Brownian motion is a real-valued stochastic process characterized by continuous trajectories and independent, stationary increments Standard Brownian motion is defined by specific properties that highlight its unique behavior in the realm of probability and statistics.
● for any 0 < t1 < ã ã ã < tn, the random variables (𝑊 𝑡 1 ,𝑊 𝑡 2 − 𝑊 𝑡 1 , ,𝑊 𝑡 𝑛 −
● for any 0 ≤ s < t, the increment W t −Ws is a normal random variable with mean 0 and variance 𝐸{(Wt - Ws) 2 }= t −s In particular, Wt follows the normal distribution with mean 0 and variance t
In the context of probability theory, a Brownian motion is defined on the probability space (Ω, ℱt, ℙ), where the increasing family of σ-algebras ℱt represents all available information about the Brownian motion up to time t, including sets of probability zero This collection of σ-algebras is known as the natural filtration of the Brownian motion A stochastic process (Xt)t≥0 is considered adapted to this filtration if each random variable Xt is ℱt-measurable for every time t, indicating that it cannot access information beyond the current time.
In order to summarize the independence of the Brownian increments and their normal distribution, one can use the conditional characteristic function For 0 ≤ s < t and u ∈
If Wt represents a Brownian motion, the independence of Wt – Ws from the past σ-algebra ℱs indicates that the left side of equation (1.3) simplifies to 𝐸{𝑒 𝑖𝑢(𝑊 𝑡 −𝑊 𝑠 )}, which is the characteristic function of a Normal random variable with a mean of 0 and a variance of t – s This expression is equivalent to the right side of the equation Conversely, if equation (1.3) is satisfied, it confirms that the continuous process Wt is indeed a standard Brownian motion.
The independence of increments in Brownian motion makes it an ideal candidate for defining independent increments dWt, which are Gaussian random variables with a mean of 0 and variance dt However, the trajectories of Wt exhibit unbounded variation, as illustrated by a simple computation involving a subdivision of the interval [0, t] Assuming an evenly spaced subdivision, where each sub-interval is defined by ti – ti-1 = t/n, we can analyze the characteristics of these increments further.
As n approaches infinity, the expected value of |W1| tends toward positive infinity, suggesting that the integral with respect to dWt cannot be defined using conventional calculus methods In the following section, we will outline a framework for defining such integrals.
Let a fixed time period represented by T We can have (Xt)0≤ t ≤T as a continuous stochastic process adapted to (ℱt) 0≤ t ≤T This leads to
By the rule of iterated conditional expectations and the independent increments property of the Brownian motion, we can see that with t0 < t1 < ã ã ã < tn = t
The Brownian increments in the equation are forward in time, and the sum on the right converges to a specific value.
The stochastic integral of (Xt) with respect to the Brownian motion (Wt) is obtained as the limit in the mean-square sense (L 2 (Ω))
Considering stochastic integral as a function of time, it defines a continuous square integrable stochastic process such that
0 } (1.6) and enjoys the martingale property, i.e., for the stochastic integral Y t =∫ 𝑋 0 𝑡 𝑢 𝑑𝑊 𝑢 , the conditional expectation of Y t given 𝐹 𝑠 is 𝑌 𝑠
0 𝑃 − 𝑎 𝑠., for s ≤ t, (1.7) which follows from the definition (1.5) The quadratic variation ⟨Y⟩t of the stochastic integral Y t =∫ 𝑋 0 𝑡 𝑢 𝑑𝑊 𝑢 is
〈𝑌〉 𝑡 = ∑ 𝑛 𝑖=1 (𝑌 𝑡 𝑖 − 𝑌 𝑡 𝑖−1 ) 2 = ∫ 𝑋 0 𝑡 𝑠 2 𝑑𝑠 (1.8) in the mean-square sense
Stochastic integrals represent continuous martingales that are zero-mean and square integrable Conversely, any continuous martingale that maintains these properties can be classified as a Brownian stochastic integral.
In the Black–Scholes model, the price of a risky asset (Xt) evolves continuously, where the return dXt/Xt has a mean of μdt, indicating a constant rate of return μ This return is accompanied by independent random fluctuations represented by σdWt, with σ reflecting the asset's volatility and dWt denoting increments of Brownian motion.
The stochastic differential equation (1.2) features a right-hand side that represents both a riskless return component and a risky return component, offering a natural financial interpretation This equation can also be expressed in integral form.
𝑋 𝑡 = 𝑋 0 + 𝜇 ∫ 𝑋 0 𝑡 𝑠 𝑑𝑠+ 𝜎 ∫ 𝑋 0 𝑡 𝑠 𝑑𝑊 𝑠 (1.10) where the last integral is a stochastic integral that has been described in the previous section which is square integrable and X0 is the initial value
Equation (1.10), is an example of stochastic differential equation based on Brownian motion:
In the Black–Scholes model, μ (t, x) = μx and σ (t, x) = σx; these are independent of t, differentiable in x, and linearly growing at infinity (since they are linear)
While one may try to express Xt/X0 as the exponential of (μt + σWt) based on (1.9), it is not correct as stochastic differentials cannot use the chain rules For example,
The equation \( W_t^2 \) does not equal \( 2 \int_0^t W_s \, dW_s \) as one might anticipate, since the expectation of the latter is zero, while \( E\{W_t^2\} = t \) as stated in (1.7) This inconsistency is addressed by Itō's lemma, also known as Itō's formula, which will be discussed in the following section.
We define a new stochastic process, g(Wt), based on a function of Brownian motion, Wt, where the function g is twice differentiable and bounded, along with its derivatives To compute the differential dg(Wt), we apply the chain rule and consider a subdivision of the interval [0, t] as t0 = 0 < t1 < < tn = t.
We can then apply Taylor’s formula to each term in order to obtain
+ 𝑅 where R consists of all the higher-order terms
In the case of a differentiable process (Wt), the limit as the mesh size of the subdivision approaches zero would only involve the first sum, resulting in the classical chain rule dg(Wt) = g′(Wt)Wt′dt However, since (Wt) is not differentiable in the context of Brownian motion, the first sum converges to a stochastic integral as described by equation (1.5).
𝑡 0 whereas the second sum converges to the following integral
𝑡 0 as can be seen by comparing it in L 2 with 1
2∑ 𝑛 𝑖=1 𝑔 ′′ (𝑊 𝑡 𝑖−1 )(𝑡 𝑖 − 𝑊 𝑡 𝑖−1 ) We note that the higher-order terms in R converge to zero and as such do not contribute to the limit, which is
2∫ 𝑔′′(𝑊 0 𝑡 𝑠 )𝑑𝑠 (1.13) The above equation is known as the Itō’s lemma It is very often written in a differential form:
We derive a formula for dg(Xt), where Xt represents the solution to a stochastic differential equation Additionally, we present a general formula for a time-dependent function g.
𝜕𝑥 2 (𝑡, 𝑋 𝑡 )𝑑〈𝑋〉 𝑡 (1.15) where dXt is given by the stochastic differential equation (1.11) and
0 is the quadratic variation of Xt In terms of dt and dWt the above formula can be expressed as
𝜕𝑥𝑑𝑊 𝑡 (1.16) with all the partial derivatives of g evaluated at (t, Xt)
As an application, let us compute the differential of the discounted price That is, we let g (t, Xt) = e −rt Xt :
We note that the second derivative of g(t, x) with respect to x is zero Assuming the price Xt id given by (1.2) with μ (t, x) = μx and σ (t, x) = σx, we obtain
For a discount rate of μ = r, the discounted price 𝑋̃ 𝑡 = 𝑒 −𝑟𝑡 𝑋 𝑡 qualifies as a martingale, as the drift term in the stochastic differential equation vanishes Consequently, any stochastic process that adheres to a driftless stochastic differential equation is classified as a martingale.
Stochastic volatility and stochastic option pricing models
Volatility is often viewed as a random variable, particularly highlighted by events like the Black Monday crash in October 1987, which exemplifies its unpredictable nature in equity markets Despite its significance, the advantages of this perspective may not be immediately apparent, especially given the widespread reliance on the Black-Scholes model for options pricing in financial markets.
Stochastic volatility models effectively account for the varying Black-Scholes implied volatilities observed in options with different strikes and expirations, commonly known as the "volatility smile." Unlike local volatility models that merely fit the smile, stochastic volatility models incorporate realistic dynamics of the underlying asset, providing a more coherent and self-consistent explanation for these variations.
Traders utilizing the Black-Scholes model for hedging often need to frequently adjust their volatility assumptions to align with market prices, leading to significant fluctuations in their hedge ratios This challenge can be effectively addressed by employing stochastic volatility models.
Mixture distributions with varying variances exhibit fat tails and a pronounced central peak, indicating that variance functions as a random variable The phenomenon of volatility clustering suggests that volatility is auto-correlated, which, in a stochastic volatility model, stems from the mean reversion of volatility.
Mean reversion in volatility and interest rates is supported by Gatheral (2006), who argues that analyzing a century's worth of volatility data for a company suggests that without mean reversion, the likelihood of extreme volatility values—ranging from one to one hundred percent—would significantly increase, which is unlikely Therefore, it is reasonable to conclude that mean reversion exists, indicating that variance must also be mean-reverting, and this principle can be utilized in valuation equations.
In this section, I refer to Gatheral (2006) Let us have stock price S and its variance v which meet the conditions stated by the following set of stochastic differential equations:
The stochastic process governing stock price returns is represented by the equation dv = αSvt dt + ηβSvt dZ, where μt denotes the instantaneous drift, η signifies the volatility of volatility, and ρ indicates the correlation between stock price returns and variance changes, with dZ1 and dZ2 as Wiener processes This model aligns with the framework established by Black and Scholes (1973), ensuring that it converges to the Black-Scholes model when η approaches zero, which is crucial for practitioners who interpret option prices through the lens of Black-Scholes implied volatility Additionally, the variance dynamics described in the model follow a broad stochastic process without specific assumptions about the functional forms of α(ã) and β(ã).
In the Black-Scholes model, the stock price is the only variable that is random, allowing hedging through stock alone However, in a stochastic volatility model, it is essential to hedge volatility movements alongside the stock to create a risk-free replicating portfolio This involves establishing a portfolio Π that consists of an option valued at V(S, v, t), a quantity Δ of the stock, and a quantity Δ1 of an additional asset whose value V1 is influenced by volatility.
The change in this portfolio over a time interval dt is given by
+ − − + − where, we have eliminated the dependence on t of the state variables S t and v t and the dependence of α and β on the state variables
To make the replicating portfolio risk-free, we have to eliminate both the dS and dv terms Hence we set
The return on a risk-free portfolio is equal to the risk-free rate, denoted as r By rearranging the equation, we can group all V terms on the left side and all V1 terms on the right side.
The left-hand side depends solely on V, while the right-hand side relies exclusively on V1 For both sides to be equal, they must correspond to a common function f of the independent variables S, V, and t This leads us to the conclusion that both expressions must equal the same function.
(2.3) where, we have assumed that an arbitrary function f of S, v and t is ( − v )where α and β are the drift and volatility terms from the stochastic differential equation in
(2.2) φ (S, v, t) represents the market price of volatility risk To see why, I refer to Gatheral
(2006) again Consider portfolio Π1 which consist of a delta-hedged option V We have
Since the option is delta-hedged, the coefficient of dS is zero and we are left with
V V V V V V vS v S v rS rV dt dv t S v S v S v v V S v t dt dZ v
In our analysis, we utilized both the partial differential equation and the stochastic differential equation to determine that the additional return per unit of volatility risk, represented as dZ², is expressed by φ(S, v, t) dt This concept parallels the established Capital Asset Pricing Model, leading us to define φ as the market price of volatility risk.
Now, we can define risk-neutral drift as follows:
The risk-neutral stochastic differential equation for v, represented as dv = α' dt + β v dZ2, allows us to derive the same results without including a term for the market price of risk, highlighting the concept of risk neutrality in financial modeling.
The Heston (1993) model corresponds to choosing α(S, v t , t) = − λ (v t − v) and β(S, v, t) = 1 in equations (2.1) and (2.2) These stochastic differential equations then become:
( ) 2 t t t t t t t t dS S dt v S dZ dv v v dt v dZ
The equation dZ/dt = -λ(v_t - v) describes the speed of reversion of the instantaneous variance process v_t to its long-term mean v This process follows the square root dynamics outlined by Cox, Ingersoll, and Ross (1985).
In order to obtain the partial differential equation for pricing European options in the Heston model, one can substitute α(S, v t , t) = − λ (v t − v) and β(S, v, t) = 1 in the partial differential equation (2.3):
The solution to the above equation will be presented in the Methodology and Data specialized topic.
The implied volatility surface
2.4.1 Implied volatility and volatility smile
Volatility, in this context, refers to the standard deviation of the underlying asset's rate of return, indicating the strength of price fluctuations over a specific period Unlike other pricing factors, volatility is unobservable and cannot be directly measured, making it challenging to accurately estimate its value compared to observable stock price movements.
Volatility can be estimated in two ways The first method is the historical volatility
This article discusses two methods for estimating volatility in financial markets The first method involves calculating the standard deviation of historical returns over a specific time period The second method, which is the main focus of this thesis, is implied volatility (IV) This approach relies on the assumption that the market price of an option aligns with its theoretical fair value derived from an option pricing model, such as Black-Scholes By using the market option premium in the model, we can reverse-engineer the calculations to determine the volatility that corresponds to that specific premium, resulting in the implied volatility value.
The Black-Scholes model assumes that volatility remains constant throughout the life of an option, irrespective of the exercise price If this assumption were accurate, plotting implied volatility (IV) against exercise price would yield a straight horizontal line However, actual IV values do not conform to this pattern; instead, they create a curved plot This indicates that implied volatility does indeed vary with exercise price, leading to the conclusion that the relationship is not linear.
“volatility smile” (or smirk) In addition, a similar phenomenon happens when one plots the IV of an option across maturities, where the IV values are shown to vary
The term structure of implied volatility (IV) highlights the variations in volatility across different exercise prices and option lifetimes To tackle the challenge of a non-flat volatility surface, stochastic option pricing models are employed.
2.4.2 Deriving an expression of implied volatility
Based on Dupire's work from 1998, we can derive a general path-integral representation of the Black-Scholes implied variance This derivation begins with the assumption that the stock price, denoted as St, follows a specific stochastic differential equation represented as dSt = μ(t)dt + σ(t)dZt.
S = + where the volatility σt can be stochastic
For any fixed K and T, we define the gamma in the Black-Scholes model as:
and define the “Black-Scholes forward implied variance” as follows:
In the context of pathwise analysis, for a sufficiently smooth function f(St, t) representing the stock price St and any realization {σt} of the volatility process, the change from the initial to the final value of the function f(St, t) can be determined through antidifferentiation By utilizing Itō’s lemma, we derive the necessary results for this calculation.
The non-discounted value of a European call option, represented as C(S0, K, T), is determined by the expected final payoff calculated under the risk-neutral measure, as per standard assumptions.
Now, C BS (S t , K, σ(t), T − t) must satisfy the Black-Scholes equation and from the definition of σ(t), we obtain:
Using the above equation to substitute for the time derivative 𝜕 𝐶 𝐵𝑆
(3.4) where the second equality uses the fact that S t enjoys the martingale property
The final term in equation (3.4) offers a valuable economic insight, representing the anticipated profit from selling a European call option with an implied volatility of σ This profit is achieved through delta-hedging using the deterministic forward variance v K,T, while the actual realized volatility is σ t.
From the definition (3.2) of v K,T (t), we have:
The equation \( E[S_t^2 \Gamma_{BS}(S_t, \bar{\sigma}(t)) | \mathcal{F}_0] v_{K,T}(t) = E[\sigma_t^2 S_t^2 \Gamma_{BS}(S_t, \bar{\sigma}(t)) | \mathcal{F}_0] \) allows for the elimination of the second term in equation (3.4) Here, \( \bar{\sigma}(0) \) represents the Black-Scholes implied volatility at time 0 for a European option with a strike price K and a time to expiration T.
Thus, the above equation (3.5) expresses implied variance as the time-integral of the expected instantaneous variance 𝜎 𝑡 2
To calculate the Black-Scholes implied volatility of an option, one must average all possible realized volatilities across various stock price paths, with each scenario weighted by the option's gamma The delta hedger's profitability over a time interval T is directly related to the option gamma and the difference between expected instantaneous variance (local variance) and realized instantaneous variance At the initiation of delta hedging, only the stock price at that moment is relevant, and only paths that conclude at the strike price contribute to the average, as the gamma value for other paths is zero.
Basic facts about machine learning
Machine learning, as defined by Burkov (2019), is a specialized area of computer science that enables computers to learn from examples to accomplish various tasks This learning process involves the development of algorithms that utilize diverse sources of data, including real-life and artificial events Essentially, machine learning involves constructing models based on datasets to effectively solve problems or achieve specific objectives.
There are many different types of machine learning Some of those types are referred to including supervised, semi-supervised, unsupervised and reinforcement
Supervised learning utilizes a dataset composed of labeled examples, represented as {(xi, yi)} for i = 1 to N, where each xi is a feature vector comprising D dimensions that capture specific characteristics of the example, such as height, age, weight, and gender Each feature, denoted as x(j), maintains consistent values across the dataset, ensuring that every instance of x(i)(2) shares the same information, like height The labels, yi, can take various forms, including finite classes {1, 2, , C}, real numbers, or vectors, categorizing the examples accordingly In essence, supervised learning employs an initial dataset alongside algorithms to predict the labels of feature vectors, making it applicable for both classification and regression tasks.
Unsupervised learning involves analyzing a dataset composed of unlabeled examples, represented as feature vectors The primary goal of unsupervised learning algorithms is to transform these vectors into new representations or values to perform specific tasks, such as clustering, dimensionality reduction, or outlier detection Clustering aims to identify the cluster to which each feature vector belongs, while dimensionality reduction seeks to create a simplified feature vector with fewer dimensions than the original In outlier detection, the algorithm generates a numerical value that indicates how distinct a feature vector is compared to others in the dataset.
Semi-supervised learning utilizes both labeled and unlabeled examples, with the latter typically outnumbering the former This approach aims to enhance algorithm performance by leveraging the additional information provided by unlabeled data While it may seem that increasing the number of unlabeled examples complicates the learning process, it actually helps create a more accurate representation of the underlying probability distribution of the labeled data Ultimately, semi-supervised learning offers greater flexibility than traditional supervised learning, enabling the processing of large datasets without the necessity for extensive prior labeling.
Reinforcement learning is a method that enables machines to interact with a simulated environment by perceiving its state as a vector of features In each state, the machine can perform various actions, leading to different rewards and potentially transitioning to new states The goal of reinforcement learning algorithms is to help machines develop a policy, which is a function that determines the optimal action based on the current state’s feature vector An action is deemed optimal if it maximizes the expected average reward This approach is particularly effective for problems involving sequential decision-making and long-term objectives, such as in gaming, robotics, resource management, and logistics.
A machine learning model can effectively label new data by leveraging decision boundaries that separate distinct classes When training examples are randomly selected, new negative examples are likely to be situated near existing negative ones, and the same applies to positive examples This proximity enables the decision boundary to accurately differentiate between new positive and negative instances While less common scenarios may lead to errors, they are expected to be fewer than correct predictions due to their lower likelihood Ultimately, increasing the diversity of training examples reduces the chances of significant divergence in new data, enhancing the model's predictive accuracy.
Machine learning encompasses a variety of algorithms with diverse applications, including significant research in option pricing, as explored in Section 2.6.4 This article will also provide an in-depth analysis of the K-nearest neighbor (KNN) algorithm and its weighted variant (WKNN) in Chapter 3, specifically in Section 3.2.2.2.
Literature review and research contributions
The Black-Scholes model is a widely used method for determining the theoretical fair price of stock options, but it has limitations due to its reliance on certain assumptions Notably, it assumes that the logarithm of stock returns follows a normal distribution and that volatility remains constant throughout the option's lifespan Over the years, various models have been developed to address these limitations by incorporating stochastic volatility, such as those proposed by Scott (1987), Hull and White (1987), and Wiggins (1987) However, these advanced models still exhibit their own shortcomings.
The Heston model (1993) offers a distinct alternative to the Black-Scholes model by incorporating the correlation between the underlying asset's spot price and its volatility, along with stochastic interest rates, which Black-Scholes assumes to be constant This model is particularly advantageous for pricing currency and bond options, although it remains applicable solely to European options.
Building on the research of Stein and Stein (1991) and Heston (1993), Bates (1996) developed a method for pricing Deutsche Mark American options that integrates stochastic volatility with a jump-diffusion process This approach considers both systematic volatility and jump risk, while also analyzing skewness and kurtosis in option price distributions to test their consistency.
Brigo and Mercurio (2007) introduced the Black-Scholes-Hull-White model, which combines the traditional Black-Scholes framework with Hull and White’s stochastic interest rate process to enhance the pricing accuracy of European options with longer maturities In contrast, Grzelak and Oosterlee (2011) developed the Heston-Hull-White model by integrating Hull and White’s stochastic interest rate process into Heston’s model, offering an alternative approach to option pricing.
In 2014, Pacati et al proposed the Heston ++ model, which enhances the fitting of volatility surfaces while maintaining the affine structure of the original Heston model This improvement is achieved by incorporating a single additional time-dependent parameter, allowing for a minor modification of the Heston model and ensuring that the calculation process remains efficient and straightforward.
A comparison of existing stochastic models is essential to understand their effectiveness in option pricing Previous studies by Bakshi et al (1997), Kim and Kim (2004), and Moyaert and Petitjean (2011) have evaluated the empirical performance of stochastic volatility models using index options data Additionally, Lassance and Vrins (2018) focused on the pricing and hedging performance of these models with individual stock options data This article aims to enhance the existing literature by providing a more detailed comparison using a cross-section of individual stock options.
This section of the thesis enhances the current literature by connecting model performance to the features of options market data, which are encapsulated by the shape of the implied volatility surface and the moments of the risk-neutral return distribution.
2.6.2 Methods of determining implied volatility surface
Regarding IV, a variety of frameworks has been proposed to model it They can be split into four classes: general equilibrium, principal component, parametric modeling and machine learning
Firstly, there are plenty of literature that deals with general equilibrium models In particular, the works of David and Veronesi (2000), Guidolin and Timmermann
In their studies, Garcia et al (2003), Hibbert et al (2008), and Bernales and Guidolin (2015) introduced rational asset pricing models that theoretically calibrate to the observed asymmetric implied volatility (IV) shape and its temporal evolution, highlighting the influence of stochastic volatility and investor uncertainty regarding economic fundamentals However, Bedendo and Hodges (2009) noted that these general equilibrium models pose challenges in practical application, particularly for forecasting and risk management purposes.
Another approach is determining the IV surface using a purely data-driven method, through the use of unobservable latent statistical factors, as in Skiadopoulos et al
Chalamandaris and Tsekrekos (2010) utilized principal component factors to generate out-of-sample forecasts of the implied volatility (IV) shape However, a limitation of their approach is that these forecasts rely on a consistently sized daily IV shape dataset, which ensures that each observation date includes identical moneyness and maturity data.
The third methodology employs deterministic parametric specifications that analyze the cross-section of available options at a given time These specifications link the maturity and moneyness of option contracts to the implied volatility (IV) shape, with notable contributions from Dumas et al (1998) and Peùa et al (1999) Additionally, Goncalves and Guidolin (2006) developed a two-stage framework to enhance the forecasting of Dumas et al.'s findings.
(1998) cross-sectional maturity and moneyness coefficients Furthermore, Chalamandaris and Tsekrekos (2011) extended their work by explicitly modeling the
IV term structure by employing Nelson and Siegel (1987) factors
The fourth methodology involves employing machine learning and non-parametric techniques to model implied volatility (IV) shapes, as demonstrated in studies by Cont and Da Fonseca (2002), Fengler et al (2003, 2007), Audrino and Colangelo (2010), and Fengler and Hin (2015) This approach favors non- or semi-parametric techniques over parameterized models to mitigate the risk of overfitting while maintaining complexity Notably, only Audrino and Colangelo (2010) successfully generated out-of-sample forecasts of the IV shape by utilizing regression trees and a cross-validation strategy for surface predictions.
This thesis proposes various parametric and non-parametric approaches to simplify the methods used for pricing options in emerging and illiquid markets.
2.6.3 Option pricing in illiquid markets
Options remain a relatively new product in various markets, often first explored through futures Pricing options poses challenges for emerging markets, particularly those with limited liquidity compared to larger markets This illiquidity not only complicates transactions but also makes market prices more susceptible to fluctuations from large trades, resulting in decreased price stability Research by Çetin et al (2006) highlights the critical role of liquidity in option pricing, noting that liquidity's impact increases quadratically as more options are hedged, with its effects varying based on the option's moneyness.
Research has shown that pricing options in low liquidity conditions can significantly impact their replication Liu and Yong (2005) examined how suboptimal liquidity in the underlying market affects the replication of European options, revealing that the "excess replicating cost" due to illiquidity increases quadratically with the number of options to be replicated This means that participants must purchase more stock and borrow additional funds to replicate call options, or short sell more stock and lend more money for put options Notably, the study found that even minor price effects can lead to substantial excess costs A more recent study by Cruz and Ševčovič further explores these dynamics.
In 2019, researchers explored a concept where underlying assets adhere to a Lévy stochastic process with jumps, utilizing a generalized version of the Frey-Stremme option pricing model They developed a fully nonlinear partial integro-differential equation to effectively price options, taking into account the impact of large-volume market participants.