The price calculated from the Black Scholes formula has a theoretical

# The price calculated from the black scholes formula

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The price calculated from the Black-Scholes formula has a theoretical underpinning which is described as follows. Suppose a call option on SPX has the characteristics K (strike price) and T (maturity), the risk- free interest rate is r, the current index level is S 0 , and the volatility of SPX is σ . The option writer writes the option, charges a premium amount V (i.e. the option price), deposits V in a risk-free bank, and carries out the following trading strategy to hedge the risk of being short in the option. The time at which the option is written is time 0, while the option expiry is time T. The period [0, T] is divided equally into N parts, so that each time interval has length dt = T/N.
FIN311 Tutor-Marked Assignment SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS) Page 6 of 8 The strategy involves trading actions at each of the time points 0, dt, 2dt, 3dt, …, N x dt = T, namely, at time i x dt, the option writer: (i) calculates the value of Δ (i.e. Delta) from the Black-Scholes formula (ii) establishes a long position of Δ shares of SPX Finally, at time T, he settles the short option position with the counterparty. The hedging strategy as described above is self-funding in the sense that all funds required to purchase SPX shares are obtained from the savings in the bank from the option premium or are loaned from the bank with full repayment at time T. (Note: As SPX is an index and is not tradeable, we will assume that the trade in shares of SPX are made in terms of an ETF that closely tracks the SPX index.) (a) Find the (mathematical) derivative of the call option price as given by the Black- Scholes formula with respect to S in order to derive the formula for Δ and code it as a Python function. (5 marks) (b) According to the hedging strategy, at time 0, the option writer needs to hold Δ shares of SPX. Assume that S 0 = K = 3000, T = 3/12, σ = 20%, r = 2%. Write down the option writer’s trading position (i.e. amount of cash in bank, number of shares of SPX) at time 0. (5 marks) (c) Use Python to write a script to simulate the hedging strategy given the input that SPX evolves as S 0 , S dt , S 2dt , …, S (N-1)dt , S T which you may assume is given to you as input in the form of a Python list, together with the parameters K, T, r, σ and N. Comment on the number of steps that are required to execute the script. As an illustration, suppose we decide to divide the interval [0, T] into N = 2 equal parts, with time points 0, dt = T/2 and 2dt = T.

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

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