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BFC3340 Derivatives
创建日期：2024-04-16 15:52:05
标签： BFC3340
Derivatives

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BFC3340 - Individual Assignment BFC3340 Derivatives 2 – Individual Assignment Due date: 6:00PM, 24th of September 2021 Instructions: ➔ This is an individual assessment, no part is to be completed as a group. ➔ Attempt all three questions outlined in this document. ➔ Marked out of a Total of 20 , which accounts for 20% of Overall Assessment for the unit. Individual question marks are outlined in this document. ➔ To submit your assignment; ◆ Go to “Submit Individual Assignment Here” under “Assessments” on Moodle, ◆ Multiple files can be submitted; ● Document (pdf or word) outlining your solution to Questions 1 ● Output Files relating to your solution for Questions 2 and 3 ○ Details regarding acceptable output formats are outlined in the questions. ➔ No cover sheet is required. Submission file names should begin with your student ID. ➔ 2 Marks out of 20 deducted per day for late submission without prior approval. Question 1 - Black-Scholes-Merton PDE Proof (5 Marks) An investment firm is developing a new Exotic Derivative Contract. This contract will pay off stock price at expiry squared, S T 2 , given stock price is less than the strike price, K . That is defined by the following function: Given that the underlying stock is assumed to follow Geometric Brownian Motion; (A) Use Risk-Neutral Valuation to derive the fair price of the security at time t in terms of the stock price, S , at time t . This will be referred to as G. (2 Marks) ( HINT: You will first need to derive the stochastic process that is followed by Y t = S t 2 , Your derivation in (a) should show that t also follows a GBM) (B) Determine whether or not this value satisfies the Black-Scholes-Merton Partial Differential Equation; (2 Marks) (C) Explain, in words, what the result of part (B) means for this contract's tradeability . (1 Mark) SUBMISSION: Please type your response for this question and submit it as a PDF. It is strongly encouraged that you use some kind of Equation Editor or LaTeX to generate the response to this question, in order to ensure it is legible and clear. Handwritten submissions will have a 1 mark deduction , and may not be accepted at all if they are illegible. If you would like guidance on how to utilise such an editor/language, feel free to attend consultation. BFC3340 - Individual Assignment Question 2 - Monte Carlo Simulation to Value Exotic Contracts (10 Marks) Considering an underlying stock that follows Geometric Brownian Motion; This underlying stock has a spot price of $27, and is known to have an Expected Return of 15% per annum and a Volatility of 31% per annum. The risk-free rate in this market is 1.5%. There exists a set of Exotic Options contracts on this underlying, including; ● Asian Call - Pays off the difference between the Strike Price and the Arithmetic (Simple) Average Stock Price for the life of the option, as long as it is above 0 ; Payoff AC = Max(0, S Avg - K), ● Floating Lookback Option - At expiry, looks back at the history/pathway of Stock Price, and sets the payoff of the option equal to the Highest Stock Price that occurred during the life of said option less the Terminal Stock Price ; S Mx = Max(S i ), i =0, 1, 2, … , T Payoff FLB = Max(S Mx - S T , 0) ● Rebate Option - A fixed cash payment if the asset price reaches the barrier; Down - Threshold, H, is below S 0 ; payment is triggered when S t moves through Up - Threshold, H, is above S 0 ; payment is triggered when S t moves through H This contract has a maturity of 15 Months and the Strike price, where relevant, is $27. Using a Monte Carlo Simulation and the Antithetic Variable Technique , where we have daily steps per path and 100 paths; (a) Price an Asian Call , as described above. (2 Marks) (b) Price a Floating Lookback Option , as described above. (2 Marks) (c) Price an Up Rebate Option, with a payoff of $120 at the Threshold of H = $50 (2 Marks) (d) Price a European Call Option with the same strike price, underlying, and maturity. (1 Mark) (e) Price an American Call Option with the same strike price, underlying, and maturity. (1 Mark) (f) Compare the options and their prices in (a) through (e) and explain their differences. Discuss what they might be used to hedge. (2 Marks) SUBMISSION: Your final product can be submitted as any of the following: ● Excel File with VBA (.xlsm) - Make sure to comment your code so that it is transparent as to how it functions. Output should be into clearly labelled cells and Visuals/Graphs are heavily encouraged. ● Matlab Script (.m) - Make sure to comment your code so that it is transparent as to how it functions. Outputs and final answers should be printed clearly, Visuals/Graphs are heavily encouraged. If you have any questions regarding submission format, feel free to ask staff for guidance. BFC3340 - Individual Assignment Question 3 - VaR Simulation (5 Marks) A Financial Institution has written 20 European Call Options on one stock and 45 European Put Options on another stock in order to produce income. For the Call Option ; the current underlying stock price is $33.50, the strike price is $37, the volatility is 28% per annum, and the time to maturity is four months. For the Put Option ; the current underlying stock price is $18.25, the strike price is $15, the volatility is 19% per annum, and the time to maturity is ten months. Neither stock is expected to pay a dividend, and the risk‐free rate is 2.5% per annum. The correlation between stock price returns is 53%. Calculate a 3‐day 99% VAR; (a) Using the Partial simulation approach with a Quadratic Model (2 Marks) (b) Using the Full Simulation approach (2 Marks) Both methods should be conducted with 500 random samples. (c) For the simulation conducted in part (b), what is the Expected Shortfall? (1 Mark) HINT: You will need to simulate 2 sets of random daily returns, one for each stock. For the second stock, you will need to factor in the correlation of 0.53. To simulate correlated normal random variables; First simulate 2 sets of independent standard normal variables, with a mean 0 and standard deviation of 1. The daily return of the first stock can be found via; The daily return of the second stock can be found via; As per normal practice, use zero for the mean daily returns . SUBMISSION: Your final product can be submitted as any of the following: ● Excel File with VBA (.xlsm) - Make sure to comment your code so that it is transparent as to how it functions. Output should be into clearly labelled cells and Ensure there are variables stored that can be inspected to view calculation steps. ● Matlab Script (.m) - Make sure to comment your code so that it is transparent as to how it functions. Outputs and final answers should be printed clearly, ensure there are variables stored that can be inspected to view calculation steps. If you have any questions regarding the submission format, feel free to ask staff for guidance. 

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