Code: https://github.com/felipeOL10/Option_data_Russell2000/blob/main/Option_data_analysis.ipynb

# The Black-Scholes model

The Black-Scholes model is a funtion of time, stock price, implied volatility and the interest rate.

The sensitivity of an option to the variables above is called the "Greeks".

## 1st order greeks:

Delta: The delta of a call option indicates how much its price changes in response to changes in the underlying asset price. It can range from 0 to 1 for a call option, and from -1 to 0 for a put option. The delta tells us how closely the option price tracks the movements of the underlying asset.

Theta: represents the time decay of an option's value. It tell us how much the option price decreases over time due to the passage of time. In theory limits for theta are not explicitly defined because they depend on multiple factors like time to expiration, volatility, and interest rates.

Rho: It indicates how much the option price changes when interest rates fluctuate. Values for rho generally range from 0 to positive infinity, and for put options, it ranges from negative infinity to 0.

Vega: Shows how much the option price changes in response to changes in the volatility of the underlying asset. The limits for vega are not explicitly defined as they depend on the pricing model and other factors. Vega is typically positive for both call and put options.

## 2nd order greeks

Gamma: Indicates how quickly the delta of an option adjusts to movements in the underlying asset. Gamma limits are not specifically stated, but it tends to be higher for options that are close to the current market price and decreases as options move further away from that price.

Volga: Volga, also known as vega convexity, captures the sensitivity of vega to changes in volatility. It shows how vega itself changes as volatility fluctuates. Limits for volga are not explicitly defined and depend on the pricing model and other factors.

Vanna: shows how the delta of the option changes in response to shifts in volatility. The values vanna can take are not explicitly stated and depend on the pricing model and other factors.

## Analysing volatility

In the Black-Scholes model, the volatility is assumed to be constant. Therefore, by ploting the volatility of the option we should get a flat line, as well as a flat volatility surface. However, as we can see on the plots below this is not in accordance with reality.

*   The smile: It describes the situation where implied volatility tends to be higher for both out-of-the-money (OTM) and in-the-money (ITM) options compared to at-the-money (ATM) options with the same expiration date. This results in a U-shaped curve when plotting implied volatility against strike price. The smile indicates that the market assigns higher probabilities to extreme price movements or "fat tails" in the underlying asset.

![](/images/newplot-cd3429bc.png)The Skew: In a volatility skew, the implied volatility tends to be higher for OTM options compared to ATM options. This means that the market expects larger price movements in one direction (e.g., downside) compared to the other direction. The skew is often observed in equity markets where there may be a higher demand for protective put options, leading to increased implied volatility for OTM puts.
![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%2004eedde.png)

## Analysing the greeks

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20042634.png)

*   The delta shows that ITM options have higher deltas then OTM options, in fact, this is expected because the delta can also be represented as the probability of a options expire ITM;

*   The theta is negative for both ITM and OTM options, especilly for OTM options with more negative values. This is because ITM options already have intrinsic value, reducing the impact of time decay on their overall value. OTM options, on the other hand, have higher Theta values and are more affected by time decay due to their higher likelihood of expiring worthless if they remain out of the money.

*   For the vega, ITM option have higher value than OTM, meaning that changes in implied volatility have a more significant impact on the value of ITM options. This is because ITM options have a higher likelihood of being exercised and, therefore, their value is more influenced by changes in volatility.

*   In term of the rho, both ITM and OTM options benefit with rising interest rates.

*   ITM options exhibit higher Gamma values, meaning their Delta and value change more rapidly in response to changes in the underlying asset's price. OTM options, on the other hand, have lower Gamma values and are less affected by small changes in the underlying price.

*   OTM options show higher volgas than ITM, meaning their Vega and value change more rapidly in response to changes in implied volatility.

*   ITM options have higher Vanna values compared to OTM options. This means that as implied volatility changes, ITM options experience larger shifts in their Delta, resulting in a more significant impact on their value.

![](/images/newplot-5b0a66a7.png)

*   Delta is higher for lower strike prices and lower for higher strike prices. This means that in-the-money call options have a higher sensitivity to changes in the underlying asset's price compared to out-of-the-money call options.

*   Theta is higher for options with shorter time to expiration and decreases as the time to expiration increases. Therefore, options with higher strike prices (OTM) tend to have higher Theta values compared to options with lower strike prices (ITM). OTM call options are more susceptible to time decay.

*   Options with lower strike prices tend to have higher Rho values compared to options with higher strike prices.

*   OTM options are more sensitive to changes in implied volatility given by the evga. This could be due to higher implied volatility for OTM options, smaller intrinsic value in OTM options, higher reliance on implied volatility for OTM options' value, or market expectations of future volatility.

*   Strike prices closer to the current price have higher Gamma values compared to options with strike prices further away.

*   The closer to the market price the lower the sensitivity to changes in the vega. Also, OTM options show the highest values, meaning that are the most sensitive to changes in the implied volatility.

*   Options with higher Vanna values near at-the-money will have larger changes in their Delta for small price movements in the underlying asset and changes in implied volatility.



Analysis of Russell 2000 options data using web scrapping and the Black and Scholes model



Code: https://github.com/felipeOL10/Historical_VaR_cpp

The goal of the project was to further develop my C++ skills and to create a program that allows to calculate the historicall VaR for a given csv file.


Historical VaR using C++

## Context

##### This project is a part of my thesis, "Bitcoin and Ethereum capabilities against investment philosophis".

## Tools

This part of the thesis was developed using Python.

## Goal

The goal of this part of my thesis was to better understand the implication of adding cryptocurrencies to personalized investment strategies, namely, value investing, growth investing, real estate, bonds and ESG.

## Methodology

The portfolios optimizations as well as the performance ratios were obtained using the pyfolioopt (https://pyportfolioopt.readthedocs.io/en/latest/) model and by using the code in my github repository (https://github.com/felipeOL10/Portfolio_opt_thesis/blob/main/Portfolio_optimization.ipynb)

Note: the following pictures were taken to the teses itself.

Portfolio optimizations

*   Mean variance portfolio optimization

*   Quadratic utility

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20222650.png)

*   Efficient CVaR

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20222744.png)

*   Efficient CDaR

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20222810.png)

Performance ratios:

*   Sharpe ratio
*   Sortino ratio

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20223002.png)

## Data used

*   Bitcoin and Ethereum as them were, at the time of the study, controlling close to 60% of the market share.

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20223245.png)

*   BlackRock etf as representative of the investment philosophies

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20223313.png)

The data used was from Yahoo Finance between [10/11/2017 and 19/07/2022]().

## Data analysis

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20223547.png)

The summary statistics reveal some differences between assets. The annual return of cryptocurrencies is substantially bigger than other assets, where Bitcoin has a return of 25.4% and Ethereum of 33.6%, thus reflected in a much higher annual volatility than other assets.

Furthermore, growth investing has an average of 47.9% annual expected returns compared to value investing with an average of 31.6%. Moreover, large-cap growth was the dominating type of investment during this period with 11.8%, on average. Conversely, the small-cap value was the lowest annual return with 4.2%. In addition, mid-capitalization performed better than small-cap. In terms of annual volatility, small-cap has the highest value with 27.2% for both value and growth and large-cap was the least risky investment in terms of growth and value with 21% and 23.9% for large-cap value and growth, respectively.

In terms of the other investment types, ESG was the highest performer with 9.5%, outperforming every investment philosophy except for large-cap growth firms. However, it achieved such performance with lower annual volatility than most investment philosophies. Furthermore, dividend investing had an annual return of 4.1%, outpacing real estate and bond investing. Bond investing was the only asset class with negative returns, however, it had the lowest recorded volatility with 7.5% on average.

In general, every risk measurement shows similar results, where Bitcoin and Ethereum present higher values and bond investing the lower values, suggesting similar results as with the descriptive analysis of returns. The mean absolute deviation shows that bond investing was 0.002 and 0.003 from the mean returns (AGG and LQD, respectively), and Bitcoin and Ethereum were 0.033 and 0.043 from the mean. Looking at the downside risk of an investment, measurements such as semi-deviation, Value at Risk (VaR), Conditional Value at Risk (CVaR) or expected shortfall (ES), Drawdown at Risk (DaR), and Conditional Drawdown at Risk (CDaR) can be very informative. In terms of the semi-deviation, it looks at the dispersion of an asset from an observed mean, which shows that for most values the semi-deviation was less than the standard deviation, meaning that negative returns have less extreme values. The historical VaR, where in the case of Ethereum and Bitcoin 9.6% and 7.5%, suggests that at a 95% confidence level the cryptocurrencies will not have larger losses than 9.6% and 7.5% in a day contrasting to bond investing where the losses do not pass the 0.4% in a day. The CVaR, represents the expected loss if the worst case happens, reaching 15.6% and 12.1% in the case of Bitcoin and Ethereum. The maximum drawdown shows that Ethereum and Bitcoin dropped 273.5% and 177.4% from the highest point in the sample and the average drawdown was 115% and 69.2% respectively. Furthermore, the DaR indicates that Bitcoin and Ethereum take the longest to recoup their losses and the CDaR shows that both assets also have a bigger average of drawdowns.

## Dynamic beta

![](/images/Picture2.png)

As expected, the results show similar behavior of the risk measurements. Both Bitcoin and Ethereum show much larger values than one, indicating much higher riskiness than the overall market. Also, most investment strategies are relatively riskier than the overall market, yet to a lower extent than cryptocurrencies.

## Portfolio analysis

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20225248.png)

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20225432.png)
![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20225512.png)

From the tables above we can conclude

1.  Firstly,annualized returns are better with only one or both cryptocurrencies, except for the minimum variance portfolios and very risk averse investors relative to the benchmark portfolios (only ETFs). However, in most cases, this was at the expense of an increase in annualized volatility (Baur et al., 2022).

2.  Secondly, when comparing portfolios with Bitcoin and with Ethereum, when investors seek performance, Ethereum-only portfolios are better, although, with higher risk aversions, investors should focus more on Bitcoin.

3.  Thirdly, the investors that benefit the most from the adoption of cryptocurrency are bond investors with the highest improvement in the Sharpe ratio, followed by real estate, dividend investors, value investors, growth investors, and ESG investors. However, for bond and real estate to get such improvement, investors should only invest in cryptocurrency.

4.  Fourthly, bond and real estate should, in most cases, allocate close to 50% in each cryptocurrency, dividend investors with Wbtc=0.389 and Weth=0.418, value investors should allocate Wbtc=0.348 and Weth=0.383 and ESG and growth investors are the ones that allocate the lowest percentage in cryptocurrencies with Wcryptocurrencies=0.091 for ESG and Wbtc=0.220 and Weth=0.246 for growth investing. Therefore, the allocations found for growth investing are relatively close to what Conlon et al. (2020) found for CSI 300 of 16% and 14% for Bitcoin and Ethereum, respectively.

Overall, the portfolio results are in line with those presented by Baur et al. (2022), Borri (2019), Demiralay and Bayracı (2021), and Kajtazi and Moro (2019).


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Analysis of Russell 2000 options data using web scrapping and the Black and Scholes model

Historical VaR using C++

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