At time 0 the position of the option writer is Bank X Shares of SPX Y X comes

At time 0 the position of the option writer is bank x

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At time 0, the position of the option writer is: Bank: X 0 Shares of SPX: Y 0 X 0 comes from depositing the option premium being netted with the use of funds to purchase Y 0 shares of SPX, and Y 0 is Δ computed with parameters at time 0. At time dt, the position becomes Bank: X 1 Shares of SPX: Y 1
FIN311 Tutor-Marked Assignment SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS) Page 7 of 8 Where Y 1 is Δ computed with parameters at time dt, and X 1 is the funds in the bank obtained from growing, over the time interval of length dt, the deposit X 0 , which is netted with the use of funds to purchase the requisite shares of SPX to obtain Y 1 shares, i.e. X 1 = X 0 (1 + r dt) – (Y 1 – Y 0 )S dt At time 2dt = T, when the option matures, the position becomes Bank: X 2 Shares of SPX: Y 2 where X 2 and Y 2 are defined as above. At this juncture, the option writer nets his position together with the requirement to settle his short position in the call option, obtaining X 2 + Y 2 – max(0, S T – K) as his P&L. Your Python code should compute and output the positions of the option writer as time evolves from start to maturity as well as the P&L, given the input parameters, for general values of N (not just N=2 as explained here). (10 marks) (d) Assume that SPX evolves over the period [0, T] randomly, so that at each time point idt (i=0,1,…,N), there is a snapshot value S idt This sequence of snapshot values constitute a random trajectory of SPX over the period, in reflection of the random nature of how prices and indices evolve in the financial markets. Particularly it is common to assume that the random evolution is computed by the following formula: S (i+1)dt = S idt (1 + r dt + σ √𝑑𝑡 Z) for i = 0, 1, 2, …, N, and where Z is a random variable that follows the standard normal distribution. Write a Python program to generate such a random sequence given the input parameters S 0 , r, dt and σ , and plot a sample of trajectory that is obtained. (5 marks)
FIN311 Tutor-Marked Assignment SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS) Page 8 of 8 (e) Calculate the P&L of the trading strategy if S 0 = K = 3000, T = 3/12, σ = 20%, r = 2%, N = 100 and assuming that SPX evolves according to the formula given in (d) for 1000 iterations. Plot the obtained P&Ls into a histogram and comment on it. (5 marks) (f) Discuss the significance of the magnitude of the P&Ls obtained in (e) from the simulations in the context of the Black-Scholes Theory. (5 marks) ---- END OF ASSIGNMENT ----

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• Spring '20
• Implied volatility, Strike price, SPX

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