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# (Solved): Option typepositionstrike pricetime to expiration (years) Number of contracts GammaVegacalllong7 ...

 Option type position strike price time to expiration (years) Number of contracts Gamma Vega call long 780 0.5 700 1.35 0.25 put long 750 0.5 500 1.20 0.5 call short 710 0.25 250 0.95 0.9 call long 700 0.25 700 1.20 1.3

There are currently two traded options available in the market:

Option A has a Delta of 0.8, a Gamma of 1.1, and a Vega of 0.45.

Option B has a Delta of 1.15, a Gamma of 0.95, and a Vega of 0.97.

The risk-free rate is 10% per annum and the volatility of the underlying asset is 40%. Based on the information provided, answer the following questions (show all the details of your calculations and present your results with four decimal places):

c) How can the financial institution make this portfolio delta-gamma-vega neutral using traded options A and B?

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Step 1: Calculate the current portfolio's delta, gamma, and vega.

To calculate the current portfolio's delta, gamma, and vega, we sum up the individual values for each option:

Portfolio Delta = (Call Long Delta + Put Long Delta) - Call Short Delta - Call Long Delta = (0.8 * 700 + (-0.8 * 500)) - (-0.95 * 250) - (1.2 * 700) = 560 - (-237.5) - 840 = 637.5

Portfolio Gamma = Call Long Gamma + Put Long Gamma - Call Short Gamma + Call Long Gamma = 1.35 * 700 + 1.2 * 500 - 0.95 * 250 + 1.2 * 700 = 945 + 600 - 237.5 + 840 = 2147.5

Portfolio Vega = Call Long Vega + Put Long Vega + Call Short Vega + Call Long Vega = 0.25 * 700 + 0.5 * 500 + 0.9 * 250 + 1.3 * 700 = 175 + 250 + 225 + 910 = 1560

Delta measures the change in the option price in relation to the change in the underlying asset price. We calculate the delta for each option in the portfolio and sum them up, taking into account the position (long or short) and number of contracts.
Gamma measures the rate of change of an option's delta in response to changes in the underlying asset price. Similar to delta, we calculate the gamma for each option in the portfolio and sum them up.
Vega measures the change in the option price in relation to changes in the underlying asset's volatility. Again, we calculate the vega for each option in the portfolio and sum them up.

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