Question

Compute with option prices in finance. In this exercise we are going to consider the pricing of so-called Asian options. An Asian option is a financial contract where the owner earns money when certain market conditions are satisfied. The contract is specified by a strike price \(K\) and a maturity time \(T .\) It is written on the average price of the underlying stock, and if this average is bigger than the strike \(K\), the owner of the option will earn the difference. If, on the other hand, the average becomes less, the owner recieves nothing, and the option matures in the value zero. The average is calculated from the last trading price of the stock for each day. From the theory of options in finance, the price of the Asian option will be the expected present value of the payoff. We assume the stock price dynamics given as, $$ S(t+1)=(1+r) S(t)+\sigma S(t) \epsilon(t) $$ where \(r\) is the interest-rate, and \(\sigma\) is the volatility of the stock price. The time \(t\) is supposed to be measured in days, \(t=0,1,2, \ldots\), while \(\epsilon(t)\) are independent identically distributed normal random variables with mean zero and unit standard deviation. To find the option price, we must calculate the expectation $$ p=(1+r)^{-T} \mathrm{E}\left[\max \left(\frac{1}{T} \sum_{t=1}^{T} S(t)-K, 0\right)\right] $$ The price is thus given as the expected discounted payoff. We will use Monte Carlo simulations to estimate the expectation. Typically, \(r\) and \(\sigma\) can be set to \(r=0.0002\) and \(\sigma=0.015 .\) Assume further \(S(0)=100\) a) Make a function that simulates a path of \(S(t)\), that is, the function computes \(S(t)\) for \(t=1, \ldots, T\) for a given \(T\) based on the recursive definition in (8.17). The function should return the path as an array. b) Create a function that finds the average of \(S(t)\) from \(t=1\) to \(t=T\). Make another function that calculates the price of the Asian option based on \(N\) simulated averages. You may choose \(T=100\) days and \(K=102\) c) Plot the price \(p\) as a function of \(N\). You may start with \(N=1000\). d) Plot the error in the price estimation as a function \(N\) (assume that the \(p\) value corresponding to the largest \(N\) value is the "right"price). Try to fit a curve of the form \(c / \sqrt{N}\) for some \(c\) to this error plot. The purpose is to show that the error is reduced as \(1 / \sqrt{N}\). Name of program file: option_price.py. If you wonder where the values for \(r\) and \(\sigma\) come from, you will find the explanation in the following. A reasonable level for the yearly interest- rate is around \(5 \%\), which corresponds to a daily rate \(0.05 / 250=0.0002 .\) The number 250 is chosen because a stock exchange is on average open this amount of days for trading. The value for \(\sigma\) is calculated as the volatility of the stock price, corresponding to the standard deviation of the daily returns of the stock defined as \((S(t+1)-S(t)) / S(t)\). "Normally", the volatility is around \(1.5 \%\) a day. Finally, there are theoretical reasons why we assume that the stock price dynamics is driven by \(r\), meaning that we consider the riskneutral dynamics of the stock price when pricing options. There is an exciting theory explaining the appearance of \(r\) in the dynamics of the stock price. If we want to simulate a stock price dynamics mimicing what we see in the market, \(r\) in Equation \((8.17)\) must be substituted with \(\mu\), the expected return of the stock. Usually, \(\mu\) is higher than \(r \cdot \diamond\)

Short answer

Estimate the Asian option price using Monte Carlo simulations and plot how price and error change with the number of simulations.

Step-by-step solution

Simulate Stock Price Path

To simulate the stock price path, we will create a function that takes parameters for the number of days \(T\), initial stock price \(S(0)\), interest rate \(r\), and volatility \(\sigma\). The function will iterate for each day \(t\) from 1 to \(T\) and calculate \(S(t)\) using the recursive formula: \[ S(t+1) = (1+r) S(t) + \sigma S(t) \epsilon(t) \] where \(\epsilon(t)\) is a randomly generated normal variable with mean 0 and standard deviation 1. The function returns an array of stock prices.

Calculate Average Stock Price

Create a function to compute the average stock price over the period from \(t=1\) to \(t=T\). Given an array of stock prices, this function calculates the average: \[ \text{Average} = \frac{1}{T} \sum_{t=1}^{T} S(t) \] The function returns this average value.

Estimate Asian Option Price

Develop another function to calculate the option price using Monte Carlo simulations. It should simulate \(N\) paths of the stock price, calculate the average price for each path, and then compute the payoff for the option \[ \max \left(\frac{1}{T} \sum_{t=1}^{T} S(t) - K, 0\right) \] The function computes the expected payoff and applies discounting using \[ p = (1+r)^{-T} \times \text{average payoff} \] Return the estimated option price.

Price as a Function of Simulations

Use the function from Step 3 to estimate and plot the option price \(p\) against various values of \(N\) (e.g., \(N = 1000, 2000, \ldots\)). This reveals how the price estimate stabilizes as more simulations are conducted.

Error Analysis and Curve Fitting

Calculate the error in price estimation for increasing values of \(N\). Assume that the price computed with the largest \(N\) is the true value. Plot these errors and fit the curve \(c / \sqrt{N}\) to the error plot. This analysis validates how the error decreases approximately as \(1 / \sqrt{N}\), ensuring reliability as simulations increase.

Key concepts

Monte Carlo simulations

Monte Carlo simulations are a forecasting technique used in various fields, including finance, to approximate the probability of different outcomes. This method involves simulating random variables to compute the possible outcomes of a model or a system. In the context of Asian options, Monte Carlo simulations allow us to estimate the expected present value by generating numerous potential outcomes of stock price paths.

- **Random Sampling**: The core of a Monte Carlo simulation is its reliance on random sampling of input variables. For Asian options, this involves simulating paths of stock prices using random values for the variable \( \epsilon(t) \), which follow a normal distribution.
- **Repetitive Process**: By repeating the process thousands of times, we can see a distribution of potential payoffs. Each repetition calculates a possible path for the stock price over the option's life, allowing us to assess various scenarios of price movements.
- **Accuracy**: As we increase the number of simulations \( N \), the accuracy of our estimation improves. The larger the sample size, the more reliable the simulation results become, which is due to the law of large numbers.

Stock price dynamics

Stock price dynamics describe the movement and fluctuation of stock prices over time. In financial mathematics, this dynamic is often modeled using stochastic processes. For Asian options, the price dynamics are given by a specific equation: \[ S(t+1) = (1 + r) S(t) + \sigma S(t) \epsilon(t) \]This formula considers:
- **Interest rate \( r \)**: Represents the daily risk-free rate, affecting how prices evolve over time by growing at this expected rate.
- **Volatility \( \sigma \)**: Measures the rate at which stock prices fluctuate, adding randomness to the stock price changes.
- **Random shocks \( \epsilon(t) \)**: Modeled as standard normal random variables, introduce unpredictability into stock movements.

These dynamics help us replicate real-world price movements in a simulated environment, essential for evaluating the expected payoff of Asian options.

Expected present value

The expected present value (EPV) is a fundamental concept in finance used to determine the value of an uncertain future cash flow in today's terms. For Asian options, the EPV is calculated to assess the expected payoff of the option, discounted back to the present. \[ p = (1 + r)^{-T} \times \text{average payoff} \]This formula includes:
- **Discounting factor \( (1 + r)^{-T} \)**: Reflects the time value of money, implying that a dollar today is worth more than a dollar tomorrow. In the context of Asian options, we discount the average payoff back to its present value.
- **Average Payoff**: Represents the calculated average outcome of the option's price over numerous simulated paths. This average helps us gauge what the payoff would be under various potential market conditions.

Understanding EPV helps in making decisions regarding the option's worth and its viability as an investment.

Financial mathematics

Financial mathematics involves the mathematical tools and models used to solve problems in finance, like valuing securities, analyzing risk, and optimizing investment strategies. It is a blend of mathematical theory, statistical methods, and economic principles. In the case of Asian options, financial mathematics helps in understanding and implementing the following concepts:
- **Stochastic Processes**: These are used to model the random changes in stock prices. For example, Asian options consider daily price variations based on a stochastic differential equation.
- **Risk-Neutral Valuation**: A concept where securities are evaluated assuming a risk-free rate of return. This simplifies the pricing of options by neutralizing risk preferences.
- **Volatility Modeling**: Using historical data to estimate future price fluctuations ensures accurate pricing estimates for options.

Employing these mathematical foundations allows traders and financial analysts to effectively evaluate financial derivatives and make informed investment decisions.