1. Case & Hedging Examples

2. Case: Keller Investments Appropriateness of Mutual Fund (MF) use of Options Appropriateness of Mutual Fund (MF) use of Options Market Pricing and B-S Variants Market Pricing and B-S Variants

3. Delta Adjusting Positions Delta Adjustment Delta Adjustment SpreadsSpreads StraddlesStraddles Consider our strategy of a Call Bullspread: Consider our strategy of a Call Bullspread: A long Call w/ Lo-X and a short Call w/ Hi-X.A long Call w/ Lo-X and a short Call w/ Hi-X. What we are interested in is 1-to-1 movement of Spread Value and underlying asset price movement. What we are interested in is 1-to-1 movement of Spread Value and underlying asset price movement. With Deltas < 1, Spread Delta << 1. With Deltas < 1, Spread Delta << 1.

4. Delta Adjusting Positions Consider our strategy of a long Straddle: Consider our strategy of a long Straddle: A long Put and a long Call, both at the same exercise price.A long Put and a long Call, both at the same exercise price. What we are interested in is the Stock price movement, either way, and with symmetric returns. What we are interested in is the Stock price movement, either way, and with symmetric returns.

5. Straddle Example: Delta Adjusting for Delta Neutrality Intel at $20, with riskless rate at 3% and time to maturity of 3 months. Volatility for Intel is 35%. Intel at $20, with riskless rate at 3% and time to maturity of 3 months. Volatility for Intel is 35%. Calls (w/ X=20) at $1.47 Calls (w/ X=20) at $1.47 Puts (w/ X=20) at $1.32 Puts (w/ X=20) at $1.32

6. Delta - Neutral Straddle Example Buy 10 calls and 10 puts Buy 10 calls and 10 puts Cost = (10 * $1.47 * 100) + (10 * $1.32 * 100)Cost = (10 * $1.47 * 100) + (10 * $1.32 * 100) Cost = 2790Cost = 2790

7. Delta - Neutral Straddle Example Intel  $22, C = $2.78, P = $0.63 Intel  $22, C = $2.78, P = $0.63 Value = (10 * 2.78 * 100) + (10 *.63 * 100)Value = (10 * 2.78 * 100) + (10 *.63 * 100) Value = $3410Value = $3410 Gain = $620Gain = $620 Intel  $18, C = $0.59, P = $2.45 Intel  $18, C = $0.59, P = $2.45 Value = (10 * 0.59 * 100) + (10 * 2.45 * 100)Value = (10 * 0.59 * 100) + (10 * 2.45 * 100) Value = $3040Value = $3040 Gain = $250Gain = $250 More Gain to upside so actually BULLISH! More Gain to upside so actually BULLISH!

8. Delta - Neutral Straddle Example Delta of Call is 0.5519 Delta of Call is 0.5519 Delta of Put is -0.4481 Delta of Put is -0.4481 Note: Position Delta = Note: Position Delta = (10*100*.5519) + (10*100* -0.4481) = +103.72  BULLISH! Delta Ratio is: Delta Ratio is: 0.4481 / 0.5519 = 0.812 0.4481 / 0.5519 = 0.812 which means we will need.812 calls to each put (or 8 calls and 10 puts).

9. Delta - Neutral Straddle Example Buy 8 calls and 10 puts Buy 8 calls and 10 puts Cost = (8 * $1.47 * 100) + (10 * $1.32 * 100)Cost = (8 * $1.47 * 100) + (10 * $1.32 * 100) Cost = 2496Cost = 2496 Note: Position Delta = (8*100*.5519) + (10*100* -0.4481) = -6.65  Roughly Neutral

10. Delta - Neutral Straddle Example Intel  $22, C = $2.78, P = $0.63 Intel  $22, C = $2.78, P = $0.63 Value = (8 * 2.78 * 100) + (10 *.63 * 100)Value = (8 * 2.78 * 100) + (10 *.63 * 100) Value = $2854Value = $2854 Gain = $358Gain = $358 Intel  $18, C = $0.59, P = $2.45 Intel  $18, C = $0.59, P = $2.45 Value = (8 * 0.59 * 100) + (10 * 2.45 * 100)Value = (8 * 0.59 * 100) + (10 * 2.45 * 100) Value = $2922Value = $2922 Gain = $426Gain = $426 Now Gains roughly symmetric; delta- neutral Now Gains roughly symmetric; delta- neutral