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DELTA HEDGING

One key assumption used to derive equations (3.4a) and (3.4b) was that the hedger can construct an instantaneously riskless portfolio by buying 1 number of shares for every unit of call option sold at the time of option transaction. By continuously adjusting [1] as the stock price path changes throughout the life of the option so as to maintain the risklessness of the portfolio under a slew of assumptions, Black-Scholes showed that the hedger would end up with a hedge profit and loss (P&L) of zero regardless of the stock price path realized during this rebalancing process. Known as delta hedging, the consequence of this process is the result that theoretically, a fair premium was charged by the hedger for this option.

As in Chapter 5, I will assume that the stock price process follows equation (3.5). In such an instance, as was already discussed, the vanilla European-style options can be priced using equations (3.7a) and (3.7b).

(3.7a)

(3.7b)

where

In the context of my discussion, since the hedger of a call (put) would need to hold() shares for every unit of option sold, and since () represents the sensitivity of the call (put) option price to a change in the underlying stock price, it readily follows from equations (3.7a) and (3.7b) that

(7.1)

(7.2)

Table 7.1 shows the implementation of Equations (7.1) and (7.2).

From Table 7.1, it can be seen from cells B9 (BIO) that if the underlying stock price increases by a dollar, the value of the call (put) option increases (decreases) by $0.63 ($0.32).[2] Furthermore, as was seen in Table 3.4, for selling this call (put) option, the seller of the option would have received an approximate premium of $5.80 ($2.84).

To understand how delta hedging works, I will first start my discussion on the hedging of the sale of a European-style call option on a nondividendpaying stock. Following this, I will revisit the example in the presence of dividends and then discuss the hedging of the sale of European-style vanilla put options.

TABLE 7.1 Deltas for Vanilla European-Style Options

TABLE 7.2 Premium and Deltas for Vanilla European-Style Call Option on a Nondividend Paying Stock at Contract Inception

  • [1] Known as the call option delta, this can be obtained by calculating the sensitivity of the call option premium to a change in the underlying stock price (i.e.,
  • [2] The reader should observe that by the virtue of equations (7.1) and (7.2), the delta of a vanilla European style call (put) option is bounded in the interval [0,1] ([ – 1,0]) – where 1 ( – 1) indicates that the call (put) option is finishing in-the-money if the option is expiring or highly likely to finish in-the-money if the option has not already expired. Additionally, just becauseand , not all option deltas have this property. To see this, consider the binary call option premium in equation (5.1a) where I showed that. The delta for a binary call,, can be shown to be . As can be seen, this is clearly not restricted to the [0,1] interval.
 
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