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Financial Engineering

This lecture focuses on 'The Greeks,' essential tools for measuring the various risks associated with options. With a deep dive into Delta, Gamma, Theta, Vega, and Rho, you'll learn how these concepts relate to the price change of options and strategies for risk management. Delta reflects the option’s sensitivity to the underlying asset's price, while Gamma examines changes in Delta. Theta addresses time decay, and Vega and Rho assess volatility and interest rate effects, respectively. Real-world examples illustrate their application in financial strategies.

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Financial Engineering

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  1. Financial Engineering Lecture 5

  2. Greeks • The Greeks measure the various risks of an option. • Every option has risk related to the variables contained in the price of the option. • Knowing these risks allows us to create strategies and select the best option to include in the strategy. • Delta • Gamma • Theta • Vega • Rho

  3. Greeks Delta - D the rate of price change for an option, relative to the price change in the underlying asset Also called the Hedge Ratio D = N(d1) = DC / DP

  4. Greeks Gamma -G The rate of change in delta, relative to the price change of the asset G = N(d1)p - N(d1)p+1

  5. Greeks Theta - Q The rate of change in the option price relative to a one day change in expiration Also called Time Decay Q = Ct - Ct-1

  6. Greeks Vega - L The rate of change in the option price relative to a 1% change in the volatility L = Cv - Cv+.01

  7. Greeks Rho The rate of change in the option price relative to a 1% increase in the discount rate Rho = Cr - Cr+.01

  8. Greeks Example - original data Call = 1.70 r = 10% Stock = 36 time = 90/365 days Strike = 40 volatility = .40 Delta = N(d1) = .3794 Gamma = N(d1)p - N(d1)p+1 = .3794 - .4329 = -.0535

  9. Greeks example - continued Theta = Ct - Ct-1 = = 1.700 - 1.675 = .0248 (daily) = .0248 x 260 = 6.448 (annual) Vega = Cv - Cv+.01 = 1.70 - 1.7683 = -.0683 Rho = = Cr - Cr+.01 (using NUMA Web Option Calculator) = 1.695-1.725 = .0300

  10. Delta Spread “volatility spread” or “playing the deltas” The “most neutral” position you can create Goal - To capture the time premium, with minimum market risk Neutral Ratio = Delta Long Position Delta Short Position

  11. Delta Spread example price = 44 delta Long Apr 40C = 5 .40 Short Apr 45C = 3 .25 Assume 2:1 ratio Max Profit = Net Credit + (# long x (S2-S1) = 1 + 1 (5) = 6 Break Even = (MP / # naked ) + high strike = (6 / 1) + 45 = 51

  12. Delta Spread example - continued Neutral Ratio = .40 /.25 = 1.60 or 8 to 5 Long 5 = 5x5 = - 25 Short 8= 3x8 = + 24 Net Debit = -1 MP = -1 + 5x5 = 24 BE = 24/3 + 45 = 53

  13. Delta Spread 24 2:1 6 1 -1 40 45 51

  14. Delta Spread 24 8:5 6 1 -1 40 45 51 53

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