Black-Scholes options pricing

Options are the world's most widely used derivative to help manage asset price risk. In this exercise you'll price a European call option on IBM's stock using the Black-Scholes option pricing formula. IBM_returns data has been loaded in your workspace.

First you'll compute the volatility sigma of IBM_returns, as the annualized standard deviation.

Next you'll use the function black_scholes(), created for this and the following exercises, to price options for two different volatility levels: sigma and two times sigma.

The strike price K, i.e. the price an investor has the right (but not the obligation) to buy IBM, is 80. The risk-free interest rate r is 2% and the market spot price S is 90.

You can find the source code of the black_scholes() function here.

This exercise is part of the course

Quantitative Risk Management in Python

Exercise instructions

Compute the volatility of IBM_returns as the annualized standard deviation sigma (you annualized volatility in Chapter 1).
Calculate the Black-Scholes European call option price value_s using the black_scholes() function provided, when volatility is sigma.
Next find the Black-Scholes option price value_2s when volatility is instead 2 * sigma.
Display value_s and value_2s to examine how the option price changes with an increase in volatility.

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Compute the volatility as the annualized standard deviation of IBM returns
sigma = np.sqrt(____) * IBM_returns.____

# Compute the Black-Scholes option price for this volatility
value_s = black_scholes(S = 90, X = 80, T = 0.5, r = 0.02, 
                        sigma = ____, option_type = "call")

# Compute the Black-Scholes option price for twice the volatility
value_2s = ____(S = 90, X = 80, T = 0.5, r = 0.02, 
                sigma = ____, option_type = "call")

# Display and compare both values
print("Option value for sigma: ", ____, "\n",
      "Option value for 2 * sigma: ", ____)

Edit and Run Code

This exercise is part of the course

Quantitative Risk Management in Python

AdvancedSkill Level

4.8+

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Risk management begins with an understanding of risk and return. We’ll recap how risk and return are related to each other, identify risk factors, and use them to re-acquaint ourselves with Modern Portfolio Theory applied to the global financial crisis of 2007-2008.

Exercise 1: Welcome!Exercise 2: Portfolio returns during the crisis Exercise 3: Asset covariance and portfolio volatility Exercise 4: Risk factors and the financial crisis Exercise 5: Frequency resampling primer Exercise 6: Visualizing risk factor correlation Exercise 7: Least-squares factor model Exercise 8: Modern portfolio theory Exercise 9: Practice with PyPortfolioOpt: returns Exercise 10: Practice with PyPortfolioOpt: covariance Exercise 11: Breaking down the financial crisis Exercise 12: The efficient frontier and the financial crisis

Now it’s time to expand your portfolio optimization toolkit with risk measures such as Value at Risk (VaR) and Conditional Value at Risk (CVaR). To do this you will use specialized Python libraries including pandas, scipy, and pypfopt. You’ll also learn how to mitigate risk exposure using the Black-Scholes model to hedge an options portfolio.

Exercise 1: Measuring Risk Exercise 2: VaR for the Normal distribution Exercise 3: Comparing CVaR and VaR Exercise 4: Which risk measure is "better"?Exercise 5: Risk exposure and loss Exercise 6: What's your risk appetite?Exercise 7: VaR and risk exposure Exercise 8: CVaR and risk exposure Exercise 9: Risk management using VaR & CVaR Exercise 10: VaR from a fitted distribution Exercise 11: Minimizing CVaR Exercise 12: CVaR risk management and the crisis Exercise 13: Portfolio hedging: offsetting risk Exercise 14: Black-Scholes options pricing

Current Exercise

Exercise 15: Options pricing and the underlying asset Exercise 16: Using options for hedging

In this chapter, you’ll estimate risk measures using parametric estimation and historical real-world data. You'll then discover how Monte Carlo simulation can help you predict uncertainty. Lastly, you’ll learn how the global financial crisis signaled that randomness itself was changing, by understanding structural breaks and how to identify them.

Exercise 1: Parametric Estimation Exercise 2: Parameter estimation: Normal Exercise 3: Parameter estimation: Skewed Normal Exercise 4: Historical and Monte Carlo Simulation Exercise 5: Historical Simulation Exercise 6: Monte Carlo Simulation Exercise 7: Structural breaks Exercise 8: Crisis structural break: I Exercise 9: Crisis structural break: II Exercise 10: Crisis structural break: III Exercise 11: Volatility and extreme values Exercise 12: Volatility and structural breaks Exercise 13: Extreme values and backtesting

It's time to explore more general risk management tools. These advanced techniques are pivotal when attempting to understand extreme events, such as losses incurred during the financial crisis, and complicated loss distributions which may defy traditional estimation techniques. You’ll also discover how neural networks can be implemented to approximate loss distributions and conduct real-time portfolio optimization.

Exercise 1: Extreme value theory Exercise 2: Block maxima Exercise 3: Extreme events during the crisis Exercise 4: GEV risk estimation Exercise 5: Kernel density estimation Exercise 6: KDE of a loss distribution Exercise 7: Which distribution?Exercise 8: CVaR and loss cover selection Exercise 9: Neural network risk management Exercise 10: Single layer neural networks Exercise 11: Asset price prediction Exercise 12: Real-time risk management Exercise 13: Wrap-up and Future Steps