The derivation of the Biack-Scholes equation shows that a derivative security can be duplicated by constructing a portfolio consisting of an appropriate combination of the underlying security and the risk-free asset We say that this portfolio replicates the derivative security, The proportions of stock and the risk-free asset in the portfolio must be adjusted continuously with time, but no additional money need be added or taken away; the portfolio is self-financing. This replication can be carried out in practice in order to construct a synthetic derivative security using the underlying and the risk-free assets Of course, the required construction is dynamic, since the particular combination must change every period (or continuously in the context of the Black-Scholes framework),

The process for a call option is this: At the initial time, calculate the theoretical price C Devote an amount C to the replicating portfolio. This portfolio should have AS invested in the stock and the remainder invested in the risk-free asset (although this will usually require borrowing, not lending) Then both the delta and the value of the portfolio will match those of the option. Indeed, the short-term behavior of the two will match.

A short time later, delta will be different, and the portfolio must be rebalanced. However, the value of the portfolio will be approximately equal to the corresponding new value of the option, so it will be possible to continue to hold the equivalent of one option. This rebalancing is repeated frequently. As the expiration date of the (synthetic) option approaches, the portfolio will consist mainly of stock if the price of the stock is above K\ otherwise the portfolio's value will tend to zero

Example 13.4 (A replication experiment) Let us construct, experimentally, a synthetic call option on Exxon stock with a strike price of $.35 and a life of 20 weeks We will replicate this option by buying Exxon stock and selling (that is, borrowing) the risk-free asset., In order to use real data in this experiment, we select the 20-week period from May 11 to September 21, 198.3. The actual weekly closing prices of Exxon (with stock symbol XON) are shown in the second column of Table 13,1, The measured sigma corresponding to this period is o ~ 18% on an annual basis, so we shall use that value to calculate the theoretical values of call prices and delta. We assume an interest rate of 10%

Let us walk across the first row of the table., There are 20 weeks remaining in the life of the option. The initial stock price is $35.50 The third column shows that the initial value of the call (as calculated by the Black-Scholes formula) is $2.62. Likewise the initial value of delta is 701, To construct the replicating portfolio we devote a value of $2,62 to it, matching the initial value of the call, This is shown in the column marked "Portfolio value,'1 However, this portfolio consists of two parts, indicated in the next two columns. The amount devoted to Exxon stock is $24.89, which is delta times the current stock value The remainder $2.62 — $24.89 — —$22.27 is devoted to the risk-free asset In other words we borrow $22 27, add $2 62, and use the total of $24 89 to buy Exxon stock.

13.6 REPLICATION, SYNTHETIC OPTIONS, AND PORTFOLIO INSURANCE* 361

TABLE 13 1

An Experiment in Option Replication

TABLE 13 1

An Experiment in Option Replication

Weeks |
XON |
Call |
Portfolio |
Stock |
Bond | |

remaining |
price |
price |
Delta |
value |
portfolio |
portfolio |

20 |
35 50 |
2.62 |
701 |
2 62 |
24 89 |
-22 27 |

19 |
34.63 |
1.96 |
615 |
1.96 |
21 28 |
-19.32 |

18 |
3,3 75 |
I 40 |
515 |
1 39 |
17.37 |
-15.98 |

17 |
34 75 |
1.89 |
618 |
I 87 |
21.47 |
-19.59 |

16 |
33.75 |
1 25 |
498 |
1 22 |
16 79 |
-1558 |

15 |
33.00 |
0.85 |
,397 |
81 |
13 09 |
-1228 |

14 |
33 88 |
I 17 |
.494 |
1.14 |
1674 |
-15 60 |

13 |
34 50 |
1 42 |
565 |
I 41 |
19.48 |
-18.07 |

12 |
33 75 |
0.96 |
456 |
96 |
15 39 |
— 14 43 |

11 |
34.75 |
1 40 |
583 |
I 38 |
20.27 |
— 18.89 |

10 |
34 38 |
i 10 |
522 |
1 13 |
17 94 |
-1681 |

9 |
35.13 |
I 44 |
.624 |
1 49 |
21 92 |
-20.43 |

8 |
36 00 |
1 94 |
743 |
2 00 |
26.74 |
-2475 |

7 |
37 00 |
2 65 |
860 |
2 69 |
31 80 |
-29 11 |

6 |
36.88 |
2 44 |
.858 |
2 53 |
31.65 |
-29.12 |

5 |
38 75 |
4.10 |
979 |
4 08 |
37 92 |
-33 84 |

4 |
37 88 |
3 17 |
.961 |
3 [6 |
36 39 |
-33 23 |

3 |
38 00 |
321 |
980 |
3.22 |
37.25 |
-3403 |

2 |
38 63 |
3.76 |
998 |
3.76 |
38 56 |
-34.79 |

1 |
38.50 |
3 57 |
1.000 |
3 57 |
38 50 |
-34 93 |

0 |
37.50 |
2.50 |
2 50 |

A call on XON with strike price 15 anil 20 weeks to expiration is replicated bv buying XON stock and selling the risk-free asset at ¡0% The portfolio is adjusted each week according to the value of delta at that lime. When the volatility is set at 18% (the actual value during that period), the portfolio value closely matches the Black-Scholes value of the call

A call on XON with strike price 15 anil 20 weeks to expiration is replicated bv buying XON stock and selling the risk-free asset at ¡0% The portfolio is adjusted each week according to the value of delta at that lime. When the volatility is set at 18% (the actual value during that period), the portfolio value closely matches the Black-Scholes value of the call

Now walk across the second row, which is calculated in a slightly different way. The first four entries show that there are 19 weeks remaining, the new stock price is $34 63, the corresponding Black-Scholes option price is $1.96, and delta is now 615. The next entry, "Portfolio value," is obtained by updating from the row above it The earlier stock purchase of $24.89 is now worth (34 63/35.50) x $24.89 ~ $24.28. The debt of $22.27 is now a debt of (1 +0 10/52)$22 27 = $22.31. The new value of the portfolio we constructed last week is therefore now $24.28 — $22.31 = $1.96 (adjusting for the round-off error in the table). This new value does not exactly agree with the current call value (although in this case it happens to agree within the two decimal places shown) We do not add or subtract from the value However, we now rebalance the portfolio by allocating to the stock $21 28 (which is delta times the stock price) and borrowing $19.32 so that the net portfolio value remains at $1.96

Succeeding rows are calculated in the same fashion. At each step, the updated portfolio value may not exactly match the current value of the call, but it tends to be very close, as is seen by scanning down the table and comparing the call and portfolio values The maximum difference is 11 cents At the end of the 20 weeks it happens in this case that the portfolio value is exactly equal (to within a fraction of a cent) to the value of the call.

The results depend on the assumed value of volatility. The choice of a = 18% represents the actual volatility over the .20-week, period, and this choice leads to good results. Study of a longer period of Exxon stock data before the date of this option indicates that volatility is more typically .20% If this value were used to construct Table 13.1, the resulting final portfolio value would be $2.66 rather than $2.50. If a — 15% were used, the final portfolio value would be $2,27.

The degree of match would also be affected by transactions costs The experiment with an Exxon call assumed that transactions costs were zero and that stock could be purchased in any fractional amount In practice these assumptions are not satisfied exactly. But for large volumes, as might be typical of institutional dealings, the departure from these assumptions is small enough so that replication is in fact practical.

Example 13.5 (Portfolio insurance) Many institutions with large portfolios of equities (stocks) are interested in insuring against the risk of a major market downturn, They could protect the value of their portfolio if they could buy a put, giving them the right to sell their portfolio at a specified exercise price K.

Puts are available for the major indices, such as the S&P 500, and hence one way to obtain protection is to buy index puts However, a particular portfolio may not match an index closely, and hence the protection would be imperfect , . Another approach is to construct a synthetic put using the actual stocks in the portfolio and the risk-free asset. Since puts have negative deltas, construction of a put requires a short position in stock and a long position in the risk-free asset. Hence some of the portfolio would be sold and later bought back if the market moves upward This strategy has the disadvantage of disrupting the portfolio and incurring trading costs,

A third approach is to construct a synthetic put using futures on the stocks held in the portfolio instead of using the stocks themselves. To implement this strategy, one would calculate the total value of the puts required and go long delta times this amount of futures. (Since A < 0, we would actually short futures.) The difference i between the value of stock shorted and the value of a put is placed in the risk-free asset. The positions must be adjusted periodically as delta changes, just as in the previous example. This method, termed portfolio insurance, was quite popular with investment institutions (such as pension funds) for a short time until the U S, stock market fell substantially in October 1987, and it was not possible to sell futures in the quantities called for by the hedging rule, resulting in loss of protection and actual losses in portfolio value,

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