Derivatives Derivagem Software PDF

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Derivatives Math using derivagem software for Jhon Hull Book. ...

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Part-A 1. Pick call option and put options of 2 companies from yahoo.finance.com. Use derivagem software to calculate the value of a 10-step binomial European options. Also calculate the value of similar American options. Ans: To find out the call options and put options intrinsic value, I choose Apple and Microsoft. The data are collected from Yahoo Finance. The Libor rate which is 0.36436% found from world rates website. Here are the calculations: Apple: The share price on 26th June, 2020 was approximately 364.84. The volatility from historical data of 1 year is 34.95%. With a strike price of $360- and 6-months’ expiration on 18th December, 2020, the call option market price is $35.36 and the put option market price is $34.22. Let’s calculate the intrinsic value:

The intrinsic price of Apple’s European call option is $38.08 and put option is $32.58. Let’s see the American options.

The intrinsic value of American call options is $38.08 and put options are $32.62

Microsoft: The share price on 26th June, 2020 was approximately 200.34. The volatility from historical data of 1 year is 40.27%. With a strike price of $200- and 6-months’ expiration on 18th December, 2020, the call option market price is $18.62 and the put option market price is $20.10. Let’s calculate the intrinsic value of the options:

The intrinsic value of European call options is $22.48 and put options is $21.77. Let’s see the American options.

The intrinsic value of American call options is $22.48 and put options is $21.81

2. Use Black-Scholes Merton model to value of the European call and put option of the 2 companies stock mentioned in Question 1. Apple: So = $364.84, K = $360.00, r = 0.36463%, σ = 34.95%, T = 6 months = 0.5 years d1 = (ln (So / K) + (r + σ2 / 2) T) / σ√T = (ln (364.84/ 360) + (0.0036 + 0.34952 / 2) 0.5) / (0.3495√0.5) = 0.184889431 d2 = (ln (So / K) + (r - σ2 / 2) T) / σ√T = (ln (364.84 /360) + (0.0036 - 0.34952 / 2) 0.5) / (0.3495√0.5) = -0.062244388 N(d1) = 0.57334, N(d2) = 0.47518, N(-d1) = 0.42666, N(-d2) = 0.52482 C = SoN(d1) – Ke-rTN(d2) = 364.84 x 0.57334 – 360e-0.0036 x 0.5 x 0.47518 = 38.36 P = Ke-rTN(-d2) – SoN(-d1) = 360e-0.0036 x 0.5 x 0.52482 – 364.84 x 0.42666 = 32.93

Microsoft: So = $200.34 K = $200.00, r = 0.36463%, σ = 40.27%, T = 6 months = 0.5 years d1 = (ln (So / K) + (r + σ2 / 2) T) / σ√T = (ln (200.34/200) + (0.0036 + 0.40272 / 2) 0.5) / (0.4027√0.5) = 0.154662284 d2 = (ln (So / K) + (r - σ2 / 2) T) / σ√T = (ln (200.34/200) + (0.0036 - 0.40272 / 2) 0.5) / (0.4027√0.5) = -0.130089616 N(d1) = 0.561456, N(d2) = 0.448248, N(-d1) = 0.438544, N(-d2) = 0.551752 C = SoN(d1) – Ke-rTN(d2) = 200.34 x 0.561456– 200e-0.0036 x 0.5 x 0.448248 = 22.99 P = Ke-rTN(-d2) – SoN(-d1) = 200e-0.0036 x 0.5 x 0.551752 – 200.34 x 0.438544= 22.33

3. Based your calculation in 1 & 2, which option is undervalued or overvalued? Ans: Apple: Based on the calculation in question 1, Apple’s call options are undervalued. In question 2, Apple’s call options are slightly over-valued and Put options are at par, rightly valued. Microsoft: Based on the calculation in question 1, Microsoft’s call options are undervalued and put options are slightly overvalued. In question 2, both of Microsoft’s call and put options are rightly valued at par. 4. Pick any European call and put option with same maturity and same strike price from yahoo finance. Use put call parity model to evaluate the parity. If the parity does not hold then show how an arbitrager can make profit. Show the arbitrage strategy and calculate the amount of the profit. Ans: To test whether put call parity holds, I have picked Apple with share price of $364.84, strike price $360.00 and maturity of 6 months expiring on 18th December, 2020 whose call option market price is $35.36 and the put option market price is $34.33. The US Libor – 6 months on 26th June, 2020 is 0.36463%. Apple pays dividends every quarter. As per historical trend, Apple should pay dividends on August 8, 2020 and November 7, 2020 within these 6 months’ period. Historically, each dividend is worth $0.77. For put call parity, LHS = So + P = 364.84 + 34.33= 399.17 RHS = C + Ke-rT + D = 35.36 + 360e-0.0036 x 0.5 + 0.77(e-0.0036 x 0.167 + e-0.0036 x 0.5) = 396.9 So, the put call parity does not hold and there is arbitrage opportunity. The arbitrage can make profit as the RHS is cheaper and the LHS is more expensive.

Arbitrage Strategy: 1. Buy: Call Option 2. Sell put option 3. Short sell the stock Profit: 1. Short Sell the stock + Selling put options – Call Option price (34.33 + 364.84 – 35.36) = 363.81 2. Invest it at a risk-free interest rate = 363.81e0.0036 x 0.5 = 364.46 3. Exercise the call option and dividend payment = 360e-0.0036 x 0.5 + 0.77(e-0.0036 x 0.167 + e-0.0036 x 0.5) = 360.64 Net Profit = 364.46 – 360.64 = $3.82....