Option Pricing Theory

Basics of Financial Engineering | 2025.02.11

Option Pricing Theory is a core area of modern financial engineering that deals with mathematical models and methodologies for determining the fair value of derivatives such as options.

The publication of the Black-Scholes-Merton model in 1973 marked a revolutionary turning point in the development of the modern derivatives market and the field of financial engineering. Since then, various methodologies for option pricing have been developed.

This page will explore the basic concepts of options, key pricing models, risk measurement through Greek letters, volatility estimation, and practical applications of option pricing theory.

1️⃣ Basic Concept of Options

Options are derivative instruments that provide the right to buy or sell a specific asset at a predetermined price at a future specified time. Options are distinguished from futures or forward contracts in that they grant rights to the investor without imposing obligations.

Types and Characteristics of Options

Basic Types of Options

Call Option

• Definition of Call Option
  It is the right to purchase the underlying asset at a predetermined price (strike price) at a future specified time. The buyer of a call option can profit when they expect the price of the underlying asset to rise.

• Profit Structure of Call Options at Expiration:

  Max(S_T - K, 0)

  Here, S_T refers to the underlying asset price at expiration, and K refers to the strike price.

Put Option

• Definition of a Put Option
  It is the right to sell the underlying asset at a predetermined price (strike price) at a specified point in the future. The buyer of a put option can make a profit when they expect the price of the underlying asset to decline.

• Profit structure at expiration for a put option:

  Max(K - S_T, 0)

  Here, S_T is the price of the underlying asset at maturity, and K is the strike price.

Option Exercise Methods

Option Exercise Methods

• European Option
  An option that can be exercised only on the expiration date. European options have relatively simple mathematical modeling, allowing for the existence of analytical solutions such as the Black-Scholes model.

• American Option
  An option that can be exercised at any point before the expiration date. It involves a more complex mathematical problem regarding optimal decision-making at the exercise time.

• Bermudan Option
  An option that can only be exercised at several predetermined points, serving as an intermediate form between European and American options.

• Exotic Options
  A variety of options with more complex structures than standard call/put options, including barrier options, Asian options, lookback options, and digital options.

Factors Affecting Option Prices

Factors Determining Option Prices

• Underlying Asset Price (S)
  The value of a call option increases when the underlying asset price rises, while the value of a put option increases when the underlying asset price falls.

• Strike Price (K)
  A call option is more valuable when the strike price is lower, while a put option is more valuable when the strike price is higher.

• Time to Maturity (T)
  Generally, the longer the time until expiration, the greater the time value of the option, leading to an increase in the option price.

• Volatility (σ)
  The greater the volatility of the underlying asset price, the higher the option price. Higher volatility increases the likelihood that the option will become valuable.

• Risk-free Interest Rate (r)
  Typically, the price of a call option increases when interest rates rise, while the price of a put option decreases when interest rates rise.

• Dividend (q)
  When the underlying asset pays dividends, the price of a call option tends to decrease, and the price of a put option tends to increase.

Components of Option Value

Composition of Option Value

• Intrinsic Value
  The value that can be obtained by exercising the option immediately, calculated as follows:

  Intrinsic Value of Call Option = Max(S - K, 0)  
  Intrinsic Value of Put Option  = Max(K - S, 0)

• Time Value
  The portion of an option's total value that remains after subtracting its intrinsic value, reflecting the possibility that the option may become more valuable during the time remaining until expiration.

  Option's Time Value = Option Price - Intrinsic Value

• In-the-money, At-the-money, Out-of-the-money Options
  In-the-money (ITM): An option that has intrinsic value (+) if exercised immediately (Call option: S > K, Put option: K > S)
  At-the-money (ATM): An option where the underlying asset price is approximately equal to the strike price (S ≈ K)
  Out-of-the-money (OTM): An option that has no intrinsic value if exercised immediately (Call option: S < K, Put option: K < S)

The main goal of option pricing theory is to develop a mathematical model that determines the fair value of an option by considering these various factors. In the next section, we will examine the most widely known option pricing model, the Black-Scholes model.

2️⃣ Black-Scholes Option Pricing Model

The Black-Scholes-Merton model, commonly referred to as the Black-Scholes model, is an option pricing model developed in 1973 by Fischer Black, Myron Scholes, and Robert Merton, serving as the foundation of modern financial engineering.

Key Assumptions of the Black-Scholes Model

Key Assumptions of the Black-Scholes Model

• The price of the underlying asset follows a Geometric Brownian Motion
  The changes in the price of the underlying asset are modeled by the following stochastic differential equation:

  dS = μSdt + σSdW

  Here, S is the price of the underlying asset, μ is the expected return of the underlying asset, σ is the volatility, and W is the Wiener process.

• The risk-free interest rate (r) is constant and known.
  The same risk-free interest rate applies for all maturities and does not change until maturity.

• The volatility (σ) of the underlying asset is constant and known.
  The volatility of the underlying asset price remains constant until the expiration of the option.

• The market is efficient and there are no transaction costs.
  There are no arbitrage opportunities, and there are no transaction costs such as trading commissions or taxes.

• The underlying asset does not pay dividends.
  If there are dividends, the model can be extended and adjusted.

• Continuous trading is possible.
  Portfolios can be adjusted at any time, and short selling of the underlying asset is allowed.

Black-Scholes Partial Differential Equation and Solution

The Black-Scholes model is developed by constructing a risk-free hedging portfolio, which leads to the derivation of the partial differential equation (PDE) that option prices must satisfy.

Black-Scholes Partial Differential Equation

∂V/∂t + (1/2)σ²S²(∂²V/∂S²) + rS(∂V/∂S) - rV = 0

Here:

• V is the option price
• t is time
• S is the underlying asset price
• σ is the volatility of the underlying asset
• r is the risk-free interest rate

By solving this partial differential equation with appropriate boundary conditions, you can determine the prices of European call and put options:

Black-Scholes Formula

European call option price:

C = S₀N(d₁) - Ke^(-rT)N(d₂)

European put option price:

P = Ke^(-rT)N(-d₂) - S₀N(-d₁)

Here:

• S₀ is the current underlying asset price
• K is the strike price
• r is the risk-free interest rate
• T is the time until expiration (in years)
• N(x) is the cumulative distribution function of the standard normal distribution
• d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)
• d₂ = d₁ - σ√T

Extensions of the Black-Scholes Model

The basic Black-Scholes model has been extended in various ways:

Major Extensions of the Black-Scholes Model

• Inclusion of Dividend Payments
  When the underlying asset pays dividends at a continuous dividend rate q, the Black-Scholes formula is modified as follows:

  C = S₀e^(-qT)N(d₁) - Ke^(-rT)N(d₂)  
  P = Ke^(-rT)N(-d₂) - S₀e^(-qT)N(-d₁)

  Here, d₁ = [ln(S₀/K) + (r - q + σ²/2)T] / (σ√T)

• Garman-Kohlhagen Model for Forex Options
  In the case of forex options, both domestic and foreign interest rates must be taken into account. The foreign interest rate acts similarly to a dividend yield.

• Stochastic Volatility Model
  In actual markets, volatility is not constant and changes over time. Models such as Hull-White and Heston consider the probabilistic nature of volatility.

• Jump-Diffusion Model
  This models situations where the price of the underlying asset includes jumps, such as Merton's jump-diffusion model and Bates model.

Limitations and Criticism of the Black-Scholes Model

Limitations of the Black-Scholes Model

• Unrealistic assumption of constant volatility
  The phenomenon of 'volatility smile/skew' observed in actual markets contradicts the assumption of constant volatility in the Black-Scholes model.

• Thick tail distribution
  The actual return distribution exhibits fatter tails than a normal distribution, but the Black-Scholes model does not take this into account.

• Extreme market events
  The Black-Scholes model may underestimate the likelihood of extreme events such as market crashes.

• Liquidity risk
  Risks arising from a lack of market liquidity are not reflected in the model.

• Ignoring transaction costs
  Actual markets incur transaction costs such as trading fees, taxes, and bid-ask spreads.

• Impossibility of continuous hedging
  In reality, continuously rebalancing a portfolio is not feasible.

Despite these limitations, the Black-Scholes model remains a fundamental model widely used for option pricing and risk management, serving as the starting point for many extension models.

3️⃣ Binomial Option Pricing Model

The Binomial Model is one of the most widely used methods for option pricing, alongside the Black-Scholes Model. This model was developed in 1979 by John Cox, Stephen Ross, and Mark Rubinstein.

Basic Concepts of the Binomial Model

Key Ideas of the Binomial Model

• Discrete Time Framework
  Continuous time is divided into several discrete time intervals.

• Assumption of Binomial Distribution
  At each time interval, the price of the underlying asset can only change to one of two values (up or down).

• Risk-Neutral Pricing
  Option prices are calculated as the present value of expected returns under the risk-neutral probability.

• No-Arbitrage Principle
  The option pricing is determined under the assumption that there are no arbitrage opportunities.

Single Period Binomial Model

The simplest binomial model is the single period model, which considers only one period from the present moment (t=0) to maturity (t=T).

Single Period Binomial Model

• Underlying Asset Price Fluctuation
  The current underlying asset price S₀ will either rise to S₀u or fall to S₀d at maturity. Here, u > 1 is the upward multiplier and 0 < d < 1 is the downward multiplier.

• Risk-Neutral Probability
  According to the no-arbitrage condition, the risk-neutral probability p is calculated as follows:

  p = (e^(rT) - d) / (u - d)

  Here, r is the risk-free interest rate, and T is the time period.

• Option price
  The call option price C₀ and the put option price P₀ are calculated as follows:

  C₀ = e^(-rT) [p × max(S₀u - K, 0) + (1-p) × max(S₀d - K, 0)]  
  P₀ = e^(-rT) [p × max(K - S₀u, 0) + (1-p) × max(K - S₀d, 0)]

  Here, k is the Strike Price.

Multi-Period Binomial Model

Multi-Period Binomial Model

• Construction of the Binomial Tree
  The binomial tree is constructed by extending the single-period model over multiple periods. At each node, the underlying asset price either increases or decreases.

• Selection of Model Parameters
  In the Cox-Ross-Rubinstein (CRR) method, the following parameters are used:

  u = e^(σ√Δt)  
  d = 1/u = e^(-σ√Δt)  
  p = (e^(rΔt) - d) / (u - d)

  Here, σ is volatility and Δt is the time interval for each step.

• Backward Induction
  Option prices are calculated starting from the final nodes (maturity) of the tree and moving backward:

  1. Calculate the option value at each node at maturity (intrinsic value).
  2. Move back to previous time points, calculating the option value at each node as the weighted average of the values of its subsequent nodes.
  3. Repeat this process until reaching the initial node (current).

• European Options and American Options
  For European options, only the present value of expected returns needs to be calculated, but for American options, the possibility of early exercise at each node must be considered.

  Option Value of Each Node = max(Intrinsic value, Holding value)

Advantages and Disadvantages of the Binomial Model

Advantages

1. Easy Intuitive Understanding
   The process of option pricing can be easily understood visually through the tree structure.
2. Flexibility
   Various situations, such as American options, dividend payments, and changes in volatility, can be easily modeled.
3. Convergence with the Black-Scholes Model
   By increasing the number of steps, the results of the binomial model converge to those of the Black-Scholes model.

Disadvantages

1. Computational Complexity
   As the number of steps increases, the amount of calculations grows exponentially.
2. Choice of Model Parameters
   Results can vary depending on how the up/down factors and probabilities are chosen.
3. Limitations in Reflecting Real-World Constraints
   Since continuous market conditions are approximated with a discrete model, certain errors may occur.

The binomial model is particularly useful for pricing American options or complex options with early exercise features, and it is also widely used for educational purposes.

4️⃣ Greek Letters and Risk Management

To measure and manage the risk of an options portfolio, sensitivity indicators known as Greek letters (Greeks) are used. The Greeks measure how the price of an option responds to changes in various factors.

Delta (Δ)

Delta (Δ)

• Definition
  The sensitivity of the option price to changes in the underlying asset price:

  Δ = ∂V/∂S

  Here, V is the option price, and S is the underlying asset price.

• Delta in the Black-Scholes model

  Call Option: Δᶜ = e^(-qT)N(d₁)  
  Put Option: Δᵖ = e^(-qT)(N(d₁) - 1) = -e^(-qT)N(-d₁)

  Here, q represents the dividend rate.

• Characteristics and Applications

  • The delta of a call option ranges between 0 and 1, while the delta of a put option ranges between -1 and 0.
  • Delta hedging: Holding the underlying asset in the opposite position by the amount of delta to hedge against the risk of price fluctuations in the underlying asset.
  • The closer the option is to being in-the-money, the absolute value of delta is closer to 1, and the closer the option is to being out-of-the-money, the absolute value of delta is closer to 0.

Gamma (Γ)

Gamma (Γ)

• Definition
  The rate of change of delta with respect to changes in the price of the underlying asset:

  Γ = ∂²V/∂S² = ∂Δ/∂S

• Gamma in the Black-Scholes Model

  Call/Put Option: Γ = e^(-qT)N'(d₁)/(S₀σ√T)

  Here, N'(x) is the probability density function of the standard normal distribution.

• Characteristics and Applications

  • The gamma of all options is positive.
  • The larger the gamma, the faster the delta changes, which requires frequent rebalancing.
  • Gamma hedging: It manages the risk of delta changes by keeping the total gamma of the portfolio close to zero.
  • At-the-money options have larger gamma, while deep in-the-money or out-of-the-money options have smaller gamma.

Vega (ν)

Vega (ν)

• Definition
  Sensitivity of option prices to changes in volatility:

  ν = ∂V/∂σ

• Vega in the Black-Scholes Model

  Call/Put Option: ν = S₀e^(-qT)N'(d₁)√T

• Characteristics and Utilization

  • The vega of all options is positive (option prices increase as volatility rises).
  • At-the-money options have a larger vega, and the longer the time to expiration, the larger the vega.
  • Vega hedging: Managing the risk of changes in volatility by keeping the total vega of the portfolio close to zero.
  • Vega is not an official Greek letter but is commonly treated as such, being an abbreviation for Vega (V).

Theta (Theta, Θ)

Theta (Theta, Θ)

• Definition
  The rate of change of an option's price over time:

  Θ = ∂V/∂t

• Theta in the Black-Scholes Model

  Call Option: Θᶜ = -S₀e^(-qT)N'(d₁)σ/(2√T) 
                    - rKe^(-rT)N(d₂) 
                    + qS₀e^(-qT)N(d₁)  
  Put Option : Θᵖ = -S₀e^(-qT)N'(d₁)σ/(2√T) 
                    + rKe^(-rT)N(-d₂) 
                    - qS₀e^(-qT)N(-d₁)

• Characteristics and Utilization

  • Generally, the theta of an option is negative (the value of the option decreases as time passes).
  • This is referred to as time value decay or time decay.
  • For at-the-money options, the absolute value of theta is larger, and the impact of theta increases as expiration approaches.
  • Time Management: An options selling position generates positive theta income, while a buying position incurs negative theta costs.

Rho (ρ)

Rho (ρ)

• Definition
  Sensitivity of the option price to changes in the risk-free interest rate:

  ρ = ∂V/∂r

• Rho in the Black-Scholes Model

  Call Option: ρᶜ = KTe^(-rT)N(d₂)  
  Put Option : ρᵖ = -KTe^(-rT)N(-d₂)

• Characteristics and Utilization

  • The gamma of call options is generally positive, while the gamma of put options is negative.
  • The longer the maturity, the greater the absolute value of gamma, and the more in-the-money the option, the greater the impact of gamma.
  • Interest Rate Risk Management: Managing risk to interest rate changes by adjusting the total gamma of the portfolio.

Risk Management Using Greek Letters

Greek letters are used as essential tools in managing an options portfolio.

Delta-Neutral Portfolio

Delta-Neutral Portfolio

• Concept
  A strategy that hedges the risk from fluctuations in the underlying asset's price by setting the total delta of the portfolio to 0.

• Implementation Method
  Hold a positive amount of the underlying asset that is opposite to the delta of the options position.

  Example: To hedge the delta of 100 contracts of a call option (delta 0.6), sell 60 units of the underlying asset.

• Limitations
  Delta changes as the price of the underlying asset changes (gamma effect), so continuous rebalancing is required.

Gamma Hedging

Gamma Hedging

• Concept
  A strategy to keep the total gamma of a portfolio close to zero to mitigate changes in delta.

• Implementation Method
  Adjust the portfolio's gamma by adding options with different expirations or strike prices.

  Example: Selling options with different gamma to hedge a long position in options with high gamma.

• Advantages
  Enhances the stability of a delta-neutral portfolio and reduces the frequency of rebalancing.

Vega Hedging

Vega Hedging

• Concept
  A strategy to keep the total vega of a portfolio close to zero to manage risks associated with changes in volatility.

• Implementation Method
  Adjust the portfolio's vega by adding options with different expirations or strike prices.

  Example: Selling short-term options to hedge a long position in options with high vega.

• Importance
  Managing vega risk is crucial as volatility can change rapidly under market stress conditions.

Practical Considerations for Greek Letter Management

Practical Considerations for Greek Letter Management

• The Need for Dynamic Hedging
  All Greek letters change over time and depending on market conditions, necessitating regular portfolio rebalancing.

• Balancing with Transaction Costs
  Excessive rebalancing can increase transaction costs, making it important to find an optimal balance between risk exposure and transaction costs.

• Cross-Greek Letter Impact
  Adjusting one Greek letter can affect others, requiring a comprehensive approach.

• Stress Testing
  Scenario analysis to evaluate the effectiveness of Greek letter-based hedging strategies under extreme market conditions is crucial.

Risk management through Greek letters is essential for the management of financial institutions' options portfolios, allowing systematic control over risks arising from changes in various market factors.

5️⃣ Implied Volatility and Volatility Estimation

The most important yet challenging parameter to estimate in option pricing is volatility. There are two main approaches to estimating volatility: Implied Volatility and Historical Volatility.

Implied Volatility

Concept and Calculation of Implied Volatility

• Definition
  It is the volatility value obtained by back-calculating when the market price of an option is substituted into an option pricing model (primarily the Black-Scholes model).

• Calculation Method
  Find the implied volatility σᵢₘₚ that makes the observed market option price Vᵐᵏᵗ equal to the theoretical price V(σ) calculated using the Black-Scholes model:

  Vᵐᵏᵗ = V(σᵢₘₚ)

  In general, solutions are obtained through numerical methods (such as the Newton-Raphson method).

• Characteristics

  • Implied volatility reflects market participants' expectations of future volatility.
  • Even for options with the same underlying asset and expiration, implied volatility can vary depending on the strike price.

Volatility Smile and Skew

In actual markets, contrary to the assumptions of the Black-Scholes model, the implied volatility of options with the same expiration shows different patterns depending on the strike price. This phenomenon is referred to as Volatility Smile or Volatility Skew.

Volatility Smile and Skew

• Volatility Smile
  Primarily observed in the foreign exchange options market, it shows a U-shaped pattern where the implied volatility of in-the-money (ITM) and out-of-the-money (OTM) options is higher than that of at-the-money (ATM) options.

• Volatility Skew (or Smirk)
  Mainly observed in the stock options market, it exhibits an asymmetric pattern where implied volatility increases as the strike price decreases (for OTM puts/ITM calls).

• Causes

  1. Market Crash Risk
     Investors pay a premium for OTM put options in anticipation of a market downturn.
  2. Leverage Effect
     As stock prices fall, the debt ratio of companies increases, leading to higher volatility.
  3. Jump Risk
     Stock prices can experience sudden jumps rather than changing continuously.
  4. Investor Preferences
     Risk-averse investors prefer hedging against downside risks.

• Modeling Approaches

  • Stochastic Volatility Models (such as the Heston model)
  • Jump-Diffusion Models (such as Merton's Jump-Diffusion model)
  • Local Volatility Models (such as the Dupire model)

Volatility Surface

Volatility Surface

• Definition
  It is a three-dimensional surface that represents the implied volatility for all options with different strike prices and maturities.

• Characteristics

  • The volatility surface changes over time and reflects market conditions and investor sentiment.
  • Generally, the volatility smile/skew is more pronounced in short-term options and becomes flatter as the maturity extends.

• Utilization

  • Pricing exotic options
  • Developing risk management and hedging strategies
  • Identifying market imbalances and trading opportunities

Historical Volatility

Concept and Calculation of Historical Volatility

• Definition
  Volatility estimated from past underlying asset price data.

• Standard Calculation Method

  1. Calculation of daily returns of the underlying asset (r_t):

  r_t = ln(S_t / S_{t-1})

  2. Calculation of the standard deviation of daily returns (σ_d)
  3. Annualization:

  σ_a = σ_d × √252

  (252 is the typical number of trading days in a year)

• Alternative methods

  • Exponentially Weighted Moving Average (EWMA): Assigning higher weights to recent data
  • GARCH model: Modeling volatility clustering and mean reversion characteristics
  • Powered Volatility: Using indices other than absolute returns or squares
  • Realized Volatility: Utilizing ultra-high-frequency data

Volatility Prediction Models

Major Volatility Prediction Models

• ARCH/GARCH family of models

  • ARCH (AutoRegressive Conditional Heteroskedasticity)
  • GARCH (Generalized ARCH)
  • EGARCH (Exponential GARCH): Modeling asymmetric reactions of volatility
  • GJR-GARCH: Reflecting the leverage effect

  These models effectively capture the phenomenon of volatility clustering.

• Stochastic Volatility Models

  • Heston model
  • SABR (Stochastic Alpha, Beta, Rho) model

  Assumes that volatility itself follows a stochastic process.

• Implied Volatility-Based Prediction

  • Utilizing volatility indices such as the VIX index
  • Utilizing the prices of volatility derivatives (volatility swaps, variance swaps)

  Leverages market participants' expectations of future volatility.

• Machine Learning-Based Models

  • Neural Networks
  • Support Vector Machines
  • Random Forests

  Can capture nonlinear patterns to enhance prediction accuracy.

Volatility estimation and prediction play a crucial role in various areas of financial engineering, including option pricing, risk management, derivative structuring, and trading strategy development.

6️⃣ Monte Carlo Simulation and Numerical Methods

In situations where options with complex structures or analytical solutions do not exist, numerical methods are used to calculate option prices. Notable methods include Monte Carlo Simulation, Finite Difference Methods, and Binomial/Trinomial Trees.

Monte Carlo Simulation

Basic Principles of Monte Carlo Simulation

Concept of Monte Carlo Simulation

It is a method that probabilistically generates multiple future paths of the underlying asset price to calculate the expected returns of the option.

Advantages of Monte Carlo Simulation

1. Effective for high-dimensional problems (multiple underlying assets, path-dependent options, etc.)
2. Relatively simple to implement
3. Flexible modeling of various stochastic processes
4. Easy to estimate errors

Disadvantages of Monte Carlo Simulation

1. Can have high computational costs
2. Slow convergence speed (error ∝ 1/√N, where N is the number of simulations)
3. Difficult to handle early exercise features like American options

Steps to Implement Monte Carlo Simulation

Steps to Implement Monte Carlo Simulation

1. Setting Up the Stochastic Process
   Establish the stochastic process that the underlying asset price follows. Under the Black-Scholes model assumptions, this is geometric Brownian motion:

   dS = rSdt + σSdW

   Here, r is the risk-free interest rate, σ is the volatility, and W is the Wiener process.

2. Path Generation
   Generate the underlying asset price path at discrete time intervals:

   S_{t+Δt} = S_t × exp((r - σ²/2)Δt + σ√Δt × Z)

   Here, Z is a random number drawn from the standard normal distribution.

3. Option value calculation
   Calculate the option's payoff at maturity for each path:

   European Call Option: CT = max(ST - K, 0)
   European Put Option:  PT = max(K - ST, 0)

4. Present Value Calculation
   Calculate the average of the maturity yields across all paths and discount it to present value:

   C₀ = e^(-rT) × (1/N) × Σ CT^i

   Here, N is the number of simulations, and CT^i is the terminal yield of the i-th path.

Monte Carlo Simulation Improvement Techniques

Monte Carlo Simulation Improvement Techniques

• Variance Reduction Techniques
  Methods to enhance simulation efficiency:

  1. Control Variates
     Utilizes the correlation with similar problems that have known solutions to reduce error.
  2. Antithetic Variates
     Uses -Z for each random variable Z to increase the sample size and reduce variance.
  3. Importance Sampling
     Allocates more samples to the area of interest to improve efficiency.
  4. Quasi-Random Sequences
     Implements low-discrepancy sequences such as Sobol and Halton to improve convergence speed.

• American Option Valuation
  Methods to address the early exercise possibility of American options:

  1. Least-Squares Monte Carlo (LSM)
     Longstaff-Schwartz Algorithm: Models optimal exercise decisions by estimating the expected value of holding through regression analysis at each point in time.
  2. Stochastic Mesh Method
     Considers the impact of decisions at each point on all subsequent paths.

Finite Difference Method

Basic Principle of Finite Difference Method

Concept and Main Methods of Finite Difference Method

• Concept
  It is a method to numerically solve the Black-Scholes Partial Differential Equation (PDE) by discretizing space and time to approximate using difference equations.

• Main Methods

  1. Explicit Method
     It directly calculates the value at the next time point from the values at the current time point. It is simple to implement but has stability constraints.
  2. Implicit Method
     It solves a system of equations to determine the value at the next time point. It is unconditionally stable but more complex to compute.
  3. Crank-Nicolson Method
     This is a weighted average of the explicit and implicit methods, providing a good balance between accuracy and stability.

Advantages and Disadvantages of Finite Difference Method

Advantages

1. Can compute Greek letters simultaneously
2. Can naturally handle early exercise features, such as American options
3. Boundary conditions can be explicitly set

Disadvantages

1. Curse of dimensionality (computational effort increases exponentially in high-dimensional problems)
2. Difficult to apply to complex stochastic processes or path-dependent options

In financial engineering practice, appropriate numerical methods are selected based on the characteristics of the problem, or multiple methods are combined for use. These numerical methodologies are essential tools for pricing complex structured derivatives and managing risk.

7️⃣ Practical Application of Option Pricing Theory

Beyond the theoretical understanding of option pricing theory, there are additional considerations and challenges when applying it to real financial markets.

Pricing Options in the Real Market

Differences Between Theoretical Models and Actual Market Prices

Actual market prices may differ somewhat from the predictions of theoretical models due to factors such as:

1. Transaction Costs
   There are transaction costs such as trading fees, taxes, and bid-ask spreads.

2. Liquidity Premium
   Illiquid options reflect a premium in their prices due to a lack of liquidity.

3. Market Frictions
   There are market friction factors such as borrowing constraints and short-selling restrictions.

4. Irrational Behavior
   Behavioral biases of investors can influence prices.

Reflection of Volatility Smile/Sku

When determining option prices in the market, it's essential to consider volatility smile/skew instead of constant volatility:

1. Local Volatility Model
   Uses different volatility values for each strike price and expiration.

2. Volatility Surface Calibration
   Calibrates the model to fit the observed volatility surface in the market.

Options Investment and Trading Strategies

Options can be used for various investment purposes, and there are multiple strategies that combine options.

Basic Strategies

Directional Strategies: Strategies based on views about market direction

1. Long Call
   Seeks profits through leverage in rising markets (unlimited upside potential, limited loss)

2. Long Put
   Seeks profits or protects portfolios in falling markets (significant downside potential, limited loss)

3. Short Call
   Seeks premium income in declining or sideways markets (limited profit, unlimited loss potential)

4. Short Put
   Seeks premium income in rising or sideways markets (limited profit, significant loss potential)

Volatility Strategy: A strategy based on views of market volatility

1. Straddle
   Buying/Selling both a call and a put option with the same strike price (betting on significant price movements)

2. Strangle
   Buying/Selling both a call and a put option with different strike prices (lower cost than straddle with a wider breakeven point)

3. Butterfly
   A strategy that bets on low volatility and narrow price movements

4. Condor
   A strategy similar to the butterfly but with a wider profit range

Compound Option Strategies

Vertical Spread: A combination of options with the same expiration but different strike prices

1. Bull Call Spread
   Buying a lower strike call + Selling a higher strike call (betting on limited upward movement)

2. Bear Put Spread
   Buying a higher strike put + Selling a lower strike put (betting on limited downward movement)

3. Bear Call Spread
   Selling a lower strike call + Buying a higher strike call (betting on limited downward movement)

4. Bull Put Spread
   Selling a lower strike put + Buying a higher strike put (betting on limited upward movement)

Synthetic Positions: Combining options with underlying assets to replicate the payoff structure of other financial products

1. Synthetic Long Stock
   Call Buy + Put Sell

2. Synthetic Short Stock
   Put Buy + Call Sell

Other Complex Options Strategies

• Calendar Spread
  A combination of options with the same strike price but different expirations, utilizing time decay and changes in volatility.
• Diagonal Spread
  A combination of options with different strike prices and different expirations (a combination of vertical spread and calendar spread).
• Ratio Spread
  A strategy that involves unequal quantities of bought and sold options.

Applications of Option Pricing Theory by Industry

Applications in the Financial Industry

Investment Banks and Securities Firms

• Designing and pricing structured products
• Developing trading desk strategies
• Risk management and hedging
• Market making activities

Asset Management Firms

• Portfolio protection strategies
• Yield enhancement strategies
• Alpha-seeking strategies
• Risk parity portfolio construction

Hedge Funds

• Quantitative trading strategies
• Relative value trading
• Volatility trading
• Event-driven strategies

Insurance Companies

• Valuation of minimum guaranteed values for variable annuities/variable insurance
• Asset-liability management (ALM)
• Risk transfer solutions

Applications in Corporate Finance

Risk Management

• Hedging against exchange rate, interest rate, and commodity price risks
• Cost reduction and establishment of stable financial planning

Capital Raising

• Convertible Bonds
• Bonds with Warrants
• Structured Debt Products

Mergers and Acquisitions (M&A)

• Valuation of Corporate Value
• Valuation of Contract Terms (so-called 'Real Options')

Incentive Design

• Executive compensation through stock options
• Performance-linked compensation systems

Challenges and Directions for Development in Options Pricing Theory

Challenges of Modern Options Pricing Theory

Challenges of Modern Options Pricing Theory

• Model Risk
  A risk that arises when the assumptions of an options pricing model do not align with reality, particularly evident in extreme market scenarios.

• Parameter Uncertainty
  There exists uncertainty due to estimation errors in key parameters such as volatility.

• Liquidity Risk
  In stressful situations, it may become difficult to adjust hedge positions.

• Market Microstructure
  Micro-level factors such as execution costs, bid-ask spreads, and market impact can influence strategy performance.

• Behavioral Bias
  Irrational behaviors of investors can affect market prices.

Direction of Development in Option Pricing Theory

Integration of Machine Learning and Big Data

• Improvement in volatility prediction accuracy
• Discovery of trading signals through nonlinear pattern recognition
• Optimization of transaction costs and enhancement of execution algorithms

High-Performance Computing

• Strengthening real-time risk management
• Efficient implementation of complex models
• Large-scale portfolio optimization

Development of Stochastic Models

• Development of more realistic models such as multifactor models and jump processes
• Improvement in model risk assessment and management methodologies
• Strengthened linkages with macroeconomic variables

Response to Regulatory Changes

• Management of counterparty risk
• Pricing suitable for central clearing counterparty (CCP) environments
• Optimization of regulatory capital

Option pricing theory has been a core driving force of financial innovation over the past 50 years, and it will continue to evolve through new challenges and technological advancements in the future.

8️⃣ Conclusion: The Importance of Options Pricing Theory

Options pricing theory is a fundamental theory that underlies modern financial markets, bringing revolutionary changes to the financial industry alongside the development of derivatives markets. Since the emergence of the Black-Scholes model, methodologies for options pricing have continuously evolved, enabling investors to manage risks more effectively and capitalize on various market opportunities.

The importance of options pricing theory can be seen in the following aspects:

Importance of Options Pricing Theory

• Foundation of Risk Management
  Options pricing theory provides a systematic methodology for quantifying and managing financial risk. Sensitivity analysis using Greek letters and hedging strategies have become the cornerstone of financial institutions' risk management systems.

• Driving Force of Financial Innovation
  Options pricing theory has provided the theoretical foundation for the development of new financial products and the expansion of markets. The design of various structured products and customized derivatives has become possible, contributing to increased market completeness.

• Information Efficiency of Asset Prices
  The information embedded in options prices (such as implied volatility) reflects the future expectations of market participants, which enhances the information efficiency of asset prices.

• Academic Influence
  Options pricing theory has facilitated interaction between various academic fields beyond finance, including economics, mathematics, physics, and computer science, becoming a prime example of interdisciplinary research.

• Expansion of Practical Applications
  The concepts of options pricing theory are expanding beyond traditional finance into various areas such as corporate decision-making (real options), energy markets, and insurance product design.

In the upcoming foundational series on financial engineering, we will delve deeper into topics such as portfolio theory, risk management, time series analysis, and financial engineering programming based on options pricing theory. A solid understanding of options pricing theory will be an essential foundation for learning more complex financial engineering concepts.

9️⃣ References and Recommended Materials

For a deeper understanding of option pricing theory, the following materials may be helpful:

Recommended Books

• "Options, Futures, and Other Derivatives" (Author: John C. Hull, Publisher: Pearson)
• "Option Volatility & Pricing" (Author: Sheldon Natenberg, Publisher: McGraw-Hill)
• "Dynamic Hedging" (Author: Nassim Nicholas Taleb, Publisher: Wiley)
• "The Volatility Surface" (Author: Jim Gatheral, Publisher: Wiley)
• "Option Pricing Models and Volatility Using Excel-VBA" (Authors: Fabrice D. Rouah, Gregory Vainberg, Publisher: Wiley)
• "Derivatives Markets" (Author: Robert L. McDonald, Publisher: Pearson)
• "Stochastic Calculus for Finance I & II" (Author: Steven E. Shreve, Publisher: Springer)

Online Resources

• CBOE Options Education Center: https://www.cboe.com/education/
• Korea Exchange Derivatives Education: https://main.krxverse.co.kr/krx-academy/derivative/det-guide
• Korea Financial Investment Association Education Center: https://www.kifin.or.kr/
• Coursera: "Financial Engineering and Risk Management" (Columbia University)
• edX: "MITx: Mathematical Methods for Quantitative Finance"
• Khan Academy - Options, Swaps, Futures: https://www.khanacademy.org/economics-finance-domain/core-finance/derivative-securities

Major Journal

• Journal of Derivatives
• Quantitative Finance
• Journal of Computational Finance
• Mathematical Finance
• Review of Derivatives Research
• Journal of Financial Economics
• Journal of Finance
• Review of Financial Studies

Free Software and Library

• Python: QuantLib-Python, PyQL, py_vollib
• R: RQuantLib, fOptions
• Excel: DerivaGem (Hull)
• MATLAB: Financial Toolbox
• Interactive Option Calculators: https://www.optionseducation.org/toolsoptionquotes/optionscalculator

Disclaimer

• The content of this blog is written for educational and informational purposes and should not be considered an investment recommendation or a substitute for financial advice. For actual financial decisions, please seek the advice of a professional.