Summary
This chapter goes off a delta neutral strategy. With the use of a theoretical model to price an option (weather it be right or wrong) the idea is to remain delta neutral for the open position. We'd make a play on weather the current option is above or below our calculated theoretical value(TV). If it were above our TV we would consider selling the contract in a way, if the it were below our TV we'd consider a play relating to buying a contract. Once whichever position is entered we have to remain delta neutral by making adjustments to the position so that our contracts approach expiration. As it approaches expiration the buying and selling of the underlying (our adjustments throughout the contract) will equate to our profit, or difference in pricing due to volatility. This chapter goes through hedging for a delta neutral strategy. The big assumption through this is that our theoretical value helps us identify a contract volatility deviation and that our model is correct.

1. The process of delta hedging an option position to remain delta neutral is an important part using a theoretical pricing model. In a delta neutral position all the deltas add up to approximately zero, or at least as close to zero as is practically possible.
\[ \begin{array}{|l|l||l|l|} \hline \text{} & \text{July 70} & \text{July 80}& \text{July 90}\\ \hline \text{call delta} & \text{70} & \text{41} & \text{18} \\ \hline \hline \text{put delta} & \text{-30} & \text{-59} & \text{-82} \\ \hline \end{array} \] a. You buy 25 July 80 calls. From the table of delta values above, what do you need to do (buy? sell? how many?) to hedge your position as close to delta neutral as possible using each of the following contracts? \[ \begin{array}{|l|l|l|l|} \hline \textbf{i. } & \text{underlying contract} & \text{ 10.25} & \text{-10}\\ \hline \hline \textbf{ii.} & \text{July 90 call} & \text{56.94} & \text{-57}\\ \hline \hline \textbf{iii. } & \text{July 80 put} & \text{17.37} & \text{+17}\\ \hline \end{array} \] Answer:
$$ \text{Delta position on the initial +25 contracts of the July 80 Calls is +1025} $$ $$ \textbf{i. } \frac{1025}{100} = 10.25 \text{ ,1 underlying contract represents 100 delta} $$ $$ \text{Since our original position is +1025 delta we need to sell roughly 10 contracts to remain Delta neutral} $$ $$ \textbf{ii. } \frac{1025}{18} = 56.94, \text{Sell roughly 57 July 90 Call contracts to remain delta neutral of original position} $$ $$ \textbf{iii. } \frac{1025}{59} = 17.37, \text{Buy roughly 17 July 80 Puts to remain delta neutral} $$ b. You sell 50 July 70 puts. What do you need to do (buy? sell? how many?) to hedge your position a close to delta neutral as possible using each of the following contracts. \[ \begin{array}{|l|l|l|l|} \hline \textbf{i. } & \text{July 70 call} & \text{ 21.42} & \text{-21}\\ \hline \hline \textbf{ii.} & \text{July 80 put} & \text{25.42} & \text{+25}\\ \hline \hline \textbf{iii. } & \text{July 90 put} & \text{18.30} & \text{+18}\\ \hline \end{array} \] Answer:
$$ \text{Same as previous part as we are trying to calculate an opposing delta to our original} $$ $$ \text{delta position of +1500 from selling 50 July 70 puts} $$ c. You sell 15 underlying contracts. You would like to hedge half your delta position using July 80 puts and the other half using July 90 calls. As close as possible, how many of each contract do you need to buy or to sell?
Answer:
$$ \text{We have an initial delta of -1500 from selling 15 underlying contracts} $$ $$ \text{Objective is to hedge our -1500 delta with half the deltas coming from the July 80 puts and half using July 90 calls} $$ $$ \text{July 80 puts, } \frac{750}{59} = 12.7 \text{ meaning selling roughly 13 giving us delta of +767} $$ $$ \text{July 90 calls, } \frac{750}{18} = 41.6 \text{ meaning buying roughly 41 giving us delta of +738} $$ $$ \text{Summing these two positions we get a net delta of +5 which is near delta neutral} $$

2. In each of the option positions below, how many underlying contracts do you need to buy or sell in order to create an initial delta neutral position? Then, as the delta changes, how many underlying contracts do you need to buy or sell in order to maintain a delta neutral position, or a position that is as close to delta neutral as possible? Delta values are given in the whole number format.
a. long 10 calls
\[ \begin{array}{|l|l|l|} \hline \text{Current Option Delta} & \text{Current Total Delta Position} & \text{Underlying Contracts to Buy or Sell}\\ \hline \hline \text{70} & \text{+700} & \text{-7}\\ \hline \hline \text{50} & \text{-200} & \text{+2}\\ \hline \hline \text{40} & \text{-100} & \text{+1}\\ \hline \hline \text{80} & \text{+400} & \text{-4}\\ \hline \end{array} \] Explanation:
Initially we calculate the first row. We are long 10 70 calls giving us a delta of +700. We need to sell 7 underlying contracts (-700 delta) to remain delta neutral. Then in the second row we see our option delta decreases to 50. Since we are still short 7 underlying contracts we have a current delta position of -200. To remain delta neutral we need to buy 2 underlying contracts (+200). The next row we see that the delta decreases once again to 40. We are short 5 underlying contracts giving us a total delta of -100. To get back to delta neutral we need to buy 1 underlying contract. On the final row we see that the option delta goes to 80. We are now +400 delta meaning we have to sell 4 underlying contracts to get back to delta neutral. Overall idea is to go from row to row keeping track of the total position delta. When the total delta is unbalances adjustments in terms of buying or selling underlying contracts are done to get back to being delta neutral.

b. short 30 calls \[ \begin{array}{|l|l|l|} \hline \text{Current Option Delta} & \text{Current Total Delta Position} & \text{Underlying Contracts to Buy or Sell}\\ \hline \hline \text{28} & \text{-840} & \text{+8}\\ \hline \hline \text{40} & \text{-400} & \text{+4}\\ \hline \hline \text{63} & \text{-690} & \text{+7}\\ \hline \hline \text{52} & \text{+340} & \text{-3}\\ \hline \end{array} \] c. long 62 puts \[ \begin{array}{|l|l|l|} \hline \text{Current Option Delta} & \text{Current Total Delta Position} & \text{Underlying Contracts to Buy or Sell}\\ \hline \hline \text{-65} & \text{-4030} & \text{+40}\\ \hline \hline \text{-71} & \text{-402} & \text{+4}\\ \hline \hline \text{-33} & \text{+2354} & \text{-24}\\ \hline \hline \text{-49} & \text{-1038} & \text{+10}\\ \hline \end{array} \] d. short 44 puts \[ \begin{array}{|l|l|l|} \hline \text{Current Option Delta} & \text{Current Total Delta Position} & \text{Underlying Contracts to Buy or Sell}\\ \hline \hline \text{-95} & \text{+4180} & \text{-42}\\ \hline \hline \text{-77} & \text{-812} & \text{+8}\\ \hline \hline \text{-40} & \text{-1640} & \text{+16}\\ \hline \hline \text{0} & \text{-1800} & \text{+18}\\ \hline \end{array} \]

3. In theory an option's value should be equal to the present value of its intrinsic value at expiration, plus the present value of all the cash flows resulting from the delta neutral dynamic hedging process.
In the two scenarios below, assume that the option is hedged at each given underlying price and delta. If interest rates are zero (no present valuing is necessary), what are the values of each option?

a. 55 call \[ \begin{array}{|l|l|l|l|l|} \hline \text{Underlying Price} & \text{Delta} & \text{Hedge} & \text{Cash Flow} & \text{Total Hedge}\\ \hline \hline \text{53.70 (initial)} & \text{45 (initial)} & \text{Sell 0.45} & \text{+24.17} & \text{-0.45}\\ \hline \hline \text{55.70} & \text{57} & \text{Sell 0.12} & \text{+6.68} & \text{-0.57}\\ \hline \hline \text{53.40} & \text{40} & \text{Buy 0.17} & \text{-9.08} & \text{-0.40}\\ \hline \hline \text{57.10} & \text{69} & \text{Sell 0.29} & \text{+16.56} & \text{-0.69}\\ \hline \hline \text{55.00} & \text{51} & \text{Buy 0.18} & \text{-9.90} & \text{-0.51}\\ \hline \hline \text{51.50 (underlying price at expiration)} & \text{} & \text{Buy 0.51} & \text{-26.27} & \text{0}\\ \hline \end{array} \] Explanation:
An initial 55 call is bought giving us a positive delta. The rows are how the underlying behaves as time passes all the way up to expiration. Initially the underlying was trading at 53.70 with a positive delta of 45. To be delta neutral we need hedge in such a way that we have a net delta of 0. To do this we need to sell x amout of underlying contract at the price of 53.70. If we sell teh 0.45 underlying contracts we get a cash flow of +24.17. The total hedge column is the delta achieved by our hedge which in this case is -0.45. We then move onto the next row where the underyling is now trading at 55.70 with a delta of 57. The price of the underyling has increased along with our delta meaning our positions are no longer delta neutral, we currently have a +0.12 delta. To become delta neutral we sell 0.12 underlying contracts giving us more cash flow of +6.68. Our total hedge delta is now -0.57. The price moves down now to 53.40 with a delta of 40 on the third row. We are now -0.17 delta meaning we have to buy underlying contracts of 0.17 to get back to delta neutral. This requires us to spend meaning we get a negative cash flow of -9.08 (-0.17*53.40). On the fourth row our underling is trading at 57.10 with a delta of 69. Giving us a delta of +0.29. To get back to delta neutral we have to sell 0.29 underlying at 57.10. This gives us a cash flow of +16.56 and a new total hedge delta of -0.69. Our fifth row shows the underlying price at 55.00 with a delta of 51. We are now net -0.18 delta meaning we have to buy 0.18 underlying contracts at 55.00 to remain delta neutral. This costs us money giving us a cash flow of -9.90. We then arrive on the final row which is the underlying price at expiration where it is trading at 51.50 and no delta (or 0) as our 55 call is out of the money. Our total delta is still -0.51 meaning we have to buy 0.51 underling contracts to close out this trade at the price of 51.50. This costs us -26.27.
Now having going through this process we have to find out what we made. To do this we sum up the value of our 55 call (intrinsic value) and add it to our total cash flows. $$ \text{Total Value} = \text{intrinsic value of our 55 Call} + \Sigma \text{Cash Flow} $$ $$ \text{Total Value} = $0 + (24.17 +6.68 - 9.08 +16.56 -9.90 -26.27) $$ $$ \text{Total Value} = $2.16 $$ As the price changes we hedge accordingly. Our initial 55 call ends up expiring worthless but since we hedged we made $2.16. This is under the assumption that we found a theoretical edge and made moves to take advantage of the arbitrage which is fully discovered whent the option expires.

b. 70 put \[ \begin{array}{|l|l|l|l|l|} \hline \text{Underlying Price} & \text{Delta} & \text{Hedge} & \text{Cash Flow} & \text{Total Hedge}\\ \hline \hline \text{68.42 (initial)} & \text{-65 (initial)} & \text{Buy 0.65} & \text{-44.47} & \text{+0.65}\\ \hline \hline \text{69.71} & \text{52} & \text{Sell 0.13} & \text{+9.06} & \text{+0.52}\\ \hline \hline \text{71.29} & \text{-33} & \text{Sell 0.19} & \text{+13.55} & \text{+0.33}\\ \hline \hline \text{70.19} & \text{-46} & \text{Buy 0.13} & \text{-9.12} & \text{+0.46}\\ \hline \hline \text{71.02} & \text{-28} & \text{Sell 0.18} & \text{+12.78} & \text{+0.28}\\ \hline \hline \text{68.51 (underlying price at expiration)} & \text{} & \text{Sell 0.28} & \text{+19.18} & \text{0}\\ \hline \end{array} \] $$ \text{Total Value} = \text{intrinsic value of our 70 Put} + \Sigma \text{Cash Flow} $$ $$ \text{Total Value} = $1.49 + (-44.47+9.06+13.55-9.12+12.78+19.18) $$ $$ \text{Total Value} = $2.47 $$