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16. Case Study: Black-Scholes Model with High Volatility

Case Study:
A European call option is priced using the Black-Scholes model. The stock price is ₹200, the strike price is ₹210, the risk-free rate is 7%, the volatility is 50%, and the time to expiration is 1 year.

Question 16:
How does the high volatility of the underlying asset affect the option price?
a) Increases the price of the call option
b) Decreases the price of the call option
c) Has no effect on the price of the call option
d) Causes the price of the call option to become zero
Answer: a) Increases the price of the call option

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17. Case Study: Black-Scholes Model and Interest Rate

Case Study:
You are pricing a European put option with a strike price of ₹120. The current stock price is ₹115, the risk-free rate is 4%, the volatility is 30%, and the time to expiration is 3 months.

Question 17:
How does an increase in the risk-free interest rate affect the price of the put option?
a) The price of the put option increases
b) The price of the put option decreases
c) The price of the put option remains the same
d) The price of the put option becomes negative
Answer: b) The price of the put option decreases

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18. Case Study: Binomial Model with Early Exercise Option

Case Study:
You are valuing an American option using the binomial model. The stock price is ₹80, the strike price is ₹85, and the risk-free rate is 5%. The stock price can increase by 10% or decrease by 8% in each period, and the option expires in 3 periods.

Question 18:
What is the primary reason the American option may be worth more than a European option using the same binomial model?
a) The option can be exercised early
b) The option has a longer time to expiration
c) The volatility is higher
d) The stock price is more volatile
Answer: a) The option can be exercised early

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19. Case Study: Binomial Tree with Two Steps

Case Study:
You are using a two-step binomial tree model to price a European call option. The current stock price is ₹120, the strike price is ₹110, and the risk-free rate is 5%. The stock can move up by 20% or down by 15% in each period, and the option expires in 1 year.

Question 19:
What is the final value of the European call option if the stock price moves up in the first period and down in the second period?
a) ₹15
b) ₹20
c) ₹25
d) ₹30
Answer: b) ₹20

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20. Case Study: Monte Carlo Simulation and Simulated Paths

Case Study:
You are using Monte Carlo simulation to price a complex option. The underlying stock price is ₹200, the strike price is ₹210, the volatility is 40%, the risk-free rate is 5%, and the time to expiration is 1 year. The number of simulated paths is 10,000.

Question 20:
What is the advantage of using Monte Carlo simulation in this case?
a) It is the fastest method for pricing options
b) It allows for modeling path-dependent options
c) It provides an exact result with fewer simulations
d) It does not require assumptions about the underlying asset
Answer: b) It allows for modeling path-dependent options

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21. Case Study: Black-Scholes Model for High Dividend Yield

Case Study:
You are pricing a European call option on a stock with a high dividend yield of 5%. The stock price is ₹100, the strike price is ₹95, the risk-free rate is 4%, the volatility is 20%, and the time to expiration is 1 year.

Question 21:
What impact does the high dividend yield have on the value of the European call option?
a) It decreases the value of the call option
b) It increases the value of the call option
c) It has no impact on the call option value
d) It makes the call option worthless
Answer: a) It decreases the value of the call option

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22. Case Study: Monte Carlo Simulation for Asian Option

Case Study:
You are tasked with pricing an Asian call option using Monte Carlo simulation. The current stock price is ₹120, the strike price is ₹130, the volatility is 25%, the risk-free rate is 6%, and the time to expiration is 6 months.

Question 22:
Why is Monte Carlo simulation particularly useful for pricing an Asian option?
a) Because Asian options have a payoff based on the average price of the underlying asset
b) Because the option has a fixed strike price
c) Because the option has a known expiration date
d) Because Monte Carlo simulations are fast and do not require accurate path data
Answer: a) Because Asian options have a payoff based on the average price of the underlying asset

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23. Case Study: Binomial Model for American Option with Early Exercise

Case Study:
You are using the binomial tree model to value an American call option. The stock price is ₹100, the strike price is ₹90, and the risk-free rate is 3%. The stock price moves up by 15% and down by 10% at each step. The option expires in 2 periods.

Question 23:
What happens to the value of the American call option compared to a European call option in this scenario?
a) The American option value will be the same as the European option value
b) The American option will likely have a higher value due to early exercise possibilities
c) The American option will have a lower value due to higher risk
d) The American option value will be lower due to the exercise feature
Answer: b) The American option will likely have a higher value due to early exercise possibilities

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24. Case Study: Black-Scholes Model with Changing Volatility

Case Study:
You are using the Black-Scholes model to price a European put option. The stock price is ₹150, the strike price is ₹145, the risk-free rate is 5%, and the time to expiration is 6 months. The volatility is expected to increase from 20% to 30% in the next 3 months.

Question 24:
How would the increase in volatility affect the value of the European put option?
a) The value of the put option would decrease
b) The value of the put option would increase
c) The value of the put option would stay the same
d) The value of the put option would become zero
Answer: b) The value of the put option would increase

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25. Case Study: Binomial Tree Model with Option Expiry

Case Study:
You are pricing a European call option with a strike price of ₹105 using the binomial tree model. The stock price is ₹100, the risk-free rate is 5%, and the time to expiration is 1 year. The stock can move up by 10% or down by 5% in each period.

Question 25:
How does the number of time steps affect the accuracy of the option price in the binomial model?
a) The more time steps, the less accurate the price
b) The more time steps, the more accurate the price
c) The number of time steps has no impact on the accuracy
d) The fewer time steps, the more accurate the price
Answer: b) The more time steps, the more accurate the price

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26. Case Study: Black-Scholes Model and Dividend Payments

Case Study:
You are pricing a European call option using the Black-Scholes model. The stock price is ₹250, the strike price is ₹240, the volatility is 20%, the risk-free rate is 4%, and the time to expiration is 6 months. The stock pays quarterly dividends of ₹5.

Question 26:
How should the dividend payments be factored into the Black-Scholes formula?
a) Subtract the dividend payments from the strike price
b) Subtract the present value of dividends from the stock price
c) Add the dividends to the volatility
d) Ignore the dividends in the calculation
Answer: b) Subtract the present value of dividends from the stock price

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27. Case Study: Monte Carlo Simulation and Model Calibration

Case Study:
You are using Monte Carlo simulation to price a European option. The simulation model uses an underlying asset price with a drift of 5%, volatility of 30%, and 1,000 simulations. The simulation is calibrated for 1,000 paths.

Question 27:
What is the purpose of calibrating the Monte Carlo simulation model before running it?
a) To estimate the number of paths needed
b) To ensure the model parameters (such as volatility and drift) match market conditions
c) To reduce the number of simulations required
d) To calculate the option payoff exactly
Answer: b) To ensure the model parameters (such as volatility and drift) match market conditions

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28. Case Study: Binomial Model for Early Exercise of American Option

Case Study:
You are using the binomial tree model to value an American put option with a strike price of ₹80. The stock price is ₹85, and the risk-free rate is 4%. The stock price can move up by 10% and down by 5% at each step. The option expires in 2 periods.

Question 28:
What is the effect of early exercise in the binomial model for American options?
a) The option value may increase if early exercise is beneficial
b) The option value will decrease due to early exercise
c) The option value remains unaffected by early exercise
d) The option value will be the same as a European option
Answer: a) The option value may increase if early exercise is beneficial

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29. Case Study: Black-Scholes Model with Low Volatility

Case Study:
You are valuing a European call option using the Black-Scholes model. The current stock price is ₹200, the strike price is ₹190, the volatility is 5%, and the risk-free rate is 3%. The time to expiration is 1 year.

Question 29:
How would a decrease in volatility affect the price of the call option?
a) The price of the call option would increase
b) The price of the call option would decrease
c) The price of the call option would remain unchanged
d) The price of the call option would become negative
Answer: b) The price of the call option would decrease

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30. Case Study: Monte Carlo Simulation for Path-Dependent Options

Case Study:
You are using Monte Carlo simulation to value a barrier option with a knock-in feature. The underlying asset has a current price of ₹300, a strike price of ₹310, volatility of 25%, and time to expiration of 6 months. The barrier level is ₹280.

Question 30:
What is the primary consideration when simulating paths for a barrier option?
a) The number of paths simulated
b) The underlying asset’s drift
c) The point at which the asset hits the barrier
d) The total duration of the option’s life
Answer: c) The point at which the asset hits the barrier

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31. Case Study: Black-Scholes-Merton Model for Dividends

Case Study:
You are pricing a European put option on a stock with a dividend yield of 3%. The stock price is ₹150, the strike price is ₹160, the risk-free rate is 5%, and the volatility is 25%. The time to expiration is 1 year.

Question 31:
How does a dividend yield affect the value of the European put option using the Black-Scholes-Merton model?
a) It increases the value of the put option
b) It decreases the value of the put option
c) It has no impact on the value of the put option
d) It reduces the volatility of the underlying asset
Answer: b) It decreases the value of the put option

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32. Case Study: Binomial Model with Multiple Steps

Case Study:
You are pricing a European call option using the binomial model with three steps. The current stock price is ₹50, the strike price is ₹55, and the risk-free rate is 6%. The stock can move up by 10% or down by 5% in each step, and the option expires in 1 year.

Question 32:
How does increasing the number of steps in the binomial model improve the accuracy of the option pricing?
a) It leads to a more precise estimate of the option price
b) It increases the complexity of calculations without improving accuracy
c) It decreases the accuracy by introducing more variables
d) It makes the model less flexible in adjusting to market conditions
Answer: a) It leads to a more precise estimate of the option price

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33. Case Study: Monte Carlo Simulation and High Dimensionality

Case Study:
You are pricing a portfolio of options with Monte Carlo simulation. The portfolio includes options on multiple underlying assets with different maturities, volatilities, and strike prices.

Question 33:
What is a potential challenge when using Monte Carlo simulation for high-dimensional options?
a) The simulation becomes less accurate with more variables
b) The computational cost increases exponentially as the number of assets increases
c) The option prices become less sensitive to changes in market conditions
d) The number of paths simulated is irrelevant
Answer: b) The computational cost increases exponentially as the number of assets increases

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34. Case Study: Black-Scholes and Time to Maturity

Case Study:
You are valuing a European call option using the Black-Scholes model. The stock price is ₹100, the strike price is ₹95, the volatility is 30%, the risk-free rate is 5%, and the time to expiration is 1 year.

Question 34:
How does increasing the time to maturity affect the price of the European call option?
a) The price of the call option increases
b) The price of the call option decreases
c) The price of the call option remains the same
d) The price of the call option becomes zero
Answer: a) The price of the call option increases

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35. Case Study: Monte Carlo Simulation for Multi-Asset Options

Case Study:
You are using Monte Carlo simulation to price an option on a portfolio of multiple assets. Each asset has its own volatility and drift, and the assets are correlated.

Question 35:
What is the primary method used to account for correlations between the assets in the Monte Carlo simulation?
a) Use a correlation matrix to adjust the simulated paths
b) Assume the assets are independent
c) Adjust the drift of the assets based on their correlation
d) Simulate the paths without accounting for correlation
Answer: a) Use a correlation matrix to adjust the simulated paths

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36. Case Study: Binomial Tree with Early Exercise Feature

Case Study:
You are using a binomial tree to price an American call option. The strike price is ₹95, the current stock price is ₹90, and the risk-free rate is 5%. The stock price can increase by 10% or decrease by 5% at each step. The option expires in 2 steps.

Question 36:
What is the impact of early exercise on the value of the American call option in this case?
a) The early exercise feature has no impact
b) The option value will increase due to early exercise potential
c) The option value will decrease due to early exercise potential
d) The option will expire worthless
Answer: b) The option value will increase due to early exercise potential

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37. Case Study: Black-Scholes Model with Increasing Stock Price

Case Study:
You are pricing a European call option using the Black-Scholes model. The stock price is ₹130, the strike price is ₹120, the volatility is 25%, the risk-free rate is 5%, and the time to expiration is 6 months.

Question 37:
How does an increase in the stock price affect the value of the European call option?
a) The value of the call option decreases
b) The value of the call option remains unchanged
c) The value of the call option increases
d) The value of the call option becomes zero
Answer: c) The value of the call option increases

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38. Case Study: Monte Carlo Simulation with More Paths

Case Study:
You are using Monte Carlo simulation to value a European option. The strike price is ₹100, the underlying asset is priced at ₹110, and you are using 50,000 paths to simulate.

Question 38:
What happens to the accuracy of the option price if you increase the number of simulation paths?
a) The option price becomes less accurate
b) The option price becomes more accurate
c) The option price remains the same
d) The option price depends only on the volatility
Answer: b) The option price becomes more accurate

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39. Case Study: Black-Scholes Model with Risk-Free Rate Change

Case Study:
You are pricing a European put option using the Black-Scholes model. The stock price is ₹80, the strike price is ₹85, the volatility is 20%, the risk-free rate is 3%, and the time to expiration is 1 year.

Question 39:
What effect does an increase in the risk-free rate have on the price of the European put option?
a) The price of the put option increases
b) The price of the put option decreases
c) The price of the put option stays the same
d) The price of the put option becomes zero
Answer: b) The price of the put option decreases

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40. Case Study: Binomial Model for European Call Option

Case Study:
You are pricing a European call option using a 3-step binomial model. The stock price is ₹150, the strike price is ₹145, and the risk-free rate is 4%. The stock can increase by 10% or decrease by 5% in each period.

Question 40:
How would you calculate the option price using the binomial model?
a) Compute the option price at each final node and work backward
b) Average the strike price and the current stock price
c) Use the current stock price and multiply by the risk-free rate
d) Calculate the intrinsic value at each node and sum them
Answer: a) Compute the option price at each final node and work backward

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41. Case Study: Monte Carlo Simulation for Barrier Options

Case Study:
You are using Monte Carlo simulation to price a knock-out barrier option on a stock with a current price of ₹200, a volatility of 30%, and a risk-free rate of 4%. The barrier level is ₹180.

Question 41:
What is the primary challenge in pricing a knock-out barrier option with Monte Carlo simulation?
a) The option price is highly dependent on the volatility
b) The simulation needs to account for the exact moment the asset reaches the barrier
c) The strike price changes during the simulation
d) The risk-free rate is difficult to model accurately
Answer: b) The simulation needs to account for the exact moment the asset reaches the barrier

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42. Case Study: Black-Scholes Model with Dividends

Case Study:
You are valuing a European call option on a dividend-paying stock. The stock price is ₹120, the strike price is ₹110, the volatility is 25%, and the risk-free rate is 5%. The dividend yield is 2% per year.

Question 42:
What is the effect of the dividend yield on the European call option price?
a) It increases the value of the call option
b) It decreases the value of the call option
c) It has no effect on the call option price
d) It makes the call option more sensitive to stock price changes
Answer: b) It decreases the value of the call option

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43. Case Study: Monte Carlo Simulation with Complex Option Payoff

Case Study:
You are using Monte Carlo simulation to price an exotic option with a complex payoff structure, involving path-dependent features and multiple assets.

Question 43:
What is the primary advantage of using Monte Carlo simulation for complex options?
a) It can handle multiple assets and path-dependent payoffs efficiently
b) It provides an exact solution to the option pricing
c) It is faster than the binomial tree model for complex options
d) It does not require assumptions about volatility
Answer: a) It can handle multiple assets and path-dependent payoffs efficiently

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44. Case Study: Binomial Model with Early Exercise

Case Study:
You are pricing an American option using a binomial model. The option has a strike price of ₹100, and the stock price is ₹95. The risk-free rate is 5%, and the option expires in 2 periods.

Question 44:
What factor makes the American option price higher than that of a European option?
a) The ability to exercise the option early
b) The shorter time to expiration
c) The higher volatility in the stock price
d) The lower strike price
Answer: a) The ability to exercise the option early

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45. Case Study: Black-Scholes Model with Strike Price Adjustments

Case Study:
You are pricing a European put option using the Black-Scholes model. The stock price is ₹200, the strike price is ₹210, the volatility is 25%, the risk-free rate is 3%, and the time to expiration is 6 months.

Question 45:
What would happen to the price of the European put option if the strike price increased?
a) The option price would increase
b) The option price would decrease
c) The option price would stay the same
d) The option price would become zero
Answer: a) The option price would increase

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46. Case Study: Monte Carlo Simulation and Volatility Adjustments

Case Study:
You are using Monte Carlo simulation to price an option. The underlying asset’s volatility changes over time, with a volatility structure that is stochastic.

Question 46:
How does the stochastic volatility impact the Monte Carlo simulation?
a) It simplifies the calculation of option prices
b) It increases the complexity of the simulation, requiring adjustment for volatility changes
c) It reduces the accuracy of the simulation
d) It eliminates the need for a correlation matrix
Answer: b) It increases the complexity of the simulation, requiring adjustment for volatility changes

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47. Case Study: Binomial Tree and Option Valuation

Case Study:
You are valuing a European call option using a binomial tree. The current stock price is ₹100, the strike price is ₹90, the volatility is 20%, and the risk-free rate is 4%. The stock price moves up by 15% and down by 10% each period.

Question 47:
What is the effect of increasing the number of steps in the binomial tree on the option price?
a) It makes the option price less accurate
b) It makes the option price converge to the theoretical price
c) It reduces the computation time
d) It makes the model less accurate
Answer: b) It makes the option price converge to the theoretical price

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48. Case Study: Black-Scholes Model and Time to Expiration

Case Study:
You are pricing a European call option using the Black-Scholes model. The stock price is ₹150, the strike price is ₹145, the volatility is 30%, the risk-free rate is 4%, and the time to expiration is 2 years.

Question 48:
How would increasing the time to expiration affect the value of the European call option?
a) The value of the call option would decrease
b) The value of the call option would increase
c) The value of the call option would stay the same
d) The value of the call option would depend on the volatility
Answer: b) The value of the call option would increase

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49. Case Study: Binomial Model with Multiple Stocks

Case Study:
You are using the binomial model to price a call option on a portfolio of stocks. The portfolio includes stocks with different volatilities and strike prices.

Question 49:
How does the correlation between the underlying stocks affect the option price in the binomial model?
a) The option price is unaffected by the correlation between stocks
b) A negative correlation between stocks will increase the option price
c) A positive correlation between stocks will reduce the option price
d) The correlation affects the option price by altering the payoff structure
Answer: d) The correlation affects the option price by altering the payoff structure

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50. Case Study: Monte Carlo Simulation for Exotic Option

Case Study:
You are pricing an exotic option using Monte Carlo simulation. The option has a complex payoff structure that depends on the average of several underlying assets over time.

Question 50:
What is the primary benefit of using Monte Carlo simulation for this exotic option?
a) It provides an exact solution for complex payoffs
b) It can handle the complex payoff structure and multiple variables
c) It reduces the complexity of the model significantly
d) It eliminates the need to simulate paths for underlying assets
Answer: b) It can handle the complex payoff structure and multiple variables