JustHedge Corporation has the following portfolio of options, all written on the same stockwhich currently sells for £1,025 per share, has a volatility of 25% p.a. and pays no dividends –recall that 1 contract = 100 shares:Option type Position StrikeTime to Expiration(years)Number ofContractsCall Long 1,000 0.5 50Put Long 1,200 0.25 100BE332-6-AU/6i) Using the table below and the fact that delta is given by∆ = 𝑁𝑁 �𝑙𝑙𝑙𝑙( 𝑆𝑆0/𝐾𝐾) + (𝑟𝑟𝑓𝑓 + 𝜎𝜎 2/2)𝑇𝑇𝜎𝜎√𝑇𝑇 �determine the delta of the entire position assuming a continuously compounded risk-free rate of 10% p.a. (show all the details of your calculations and display the resultswith four decimal places)
Question
JustHedge Corporation has the following portfolio of options, all written on the same stockwhich currently sells for £1,025 per share, has a volatility of 25% p.a. and pays no dividends –recall that 1 contract = 100 shares:Option type Position StrikeTime to Expiration(years)Number ofContractsCall Long 1,000 0.5 50Put Long 1,200 0.25 100BE332-6-AU/6i) Using the table below and the fact that delta is given by∆ = 𝑁𝑁 �𝑙𝑙𝑙𝑙( 𝑆𝑆0/𝐾𝐾) + (𝑟𝑟𝑓𝑓 + 𝜎𝜎 2/2)𝑇𝑇𝜎𝜎√𝑇𝑇 �determine the delta of the entire position assuming a continuously compounded risk-free rate of 10% p.a. (show all the details of your calculations and display the resultswith four decimal places)
Solution
To calculate the delta of the entire position, we need to calculate the delta for each option and then sum them up.
The formula for delta is given by:
∆ = N(d1) for a call option ∆ = -N(-d1) for a put option
where d1 = [ln(S0/K) + (rf + σ^2/2)T] / σ√T
Here, S0 is the current stock price, K is the strike price, rf is the risk-free rate, σ is the volatility, and T is the time to expiration.
Let's calculate the delta for each option:
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Call option: d1 = [ln(1025/1000) + (0.10 + 0.25^2/2)0.5] / (0.25√0.5) ∆ = N(d1) = N(0.139) = 0.5553 (from standard normal distribution table) Total delta for call option = 0.5553 * 50 = 27.765
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Put option: d1 = [ln(1025/1200) + (0.10 + 0.25^2/2)0.25] / (0.25√0.25) ∆ = -N(-d1) = -N(-0.139) = -0.4447 (from standard normal distribution table) Total delta for put option = -0.4447 * 100 = -44.47
The delta of the entire position is the sum of the deltas of the individual options, which is 27.765 - 44.47 = -16.705.
So, the delta of the entire position is -16.705.
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