Hedging the Sale of a Vanilla European-Style Call Option on a Nondividend-Paying Stock

Keeping the stock price, risk-free rate, volatility rate, and expiry the same as in Table 6.1 and setting the dividend rate to 0, one can easily arrive at the results shown in Table 7.2.

As can be seen from Table 7.2, if a unit of the option contract stipulates the delivery of 100 shares at an exercise price of $40/share, then for a unit of call option contract sold, the option seller would receive an up front premium of $687 (=$6.87*100). To be instantaneously riskless at the time of the sale, the option holder has to buy^{[1]} 71.4 (=0.714*100) shares. To acquire these shares, the option seller would need $2856 (=$40*71.4). Since the option writer would receive an amount of $687 for selling this option, only an amount of $2169 needs to be borrowed. Assuming that this can be done at a risk-free rate of 6 percent (which is one of the inputs used to obtain the option premium of $687), one can surmise that the option seller would, in addition to selling a unit of option contract, buy 71.4 shares of stock to have a riskless portfolio.

TABLE 7.3 Premium and Deltas for Vanilla European-Style Call Option on a Nondividend-Paying Stock Six Months into the Contract

Suppose for the moment that the next rebalancing is done 6 months later when the share price is $45 and both the risk-free and the volatility rates remain unchanged. Table 7.3 shows the impact of these changes in the option premium and delta.

As can be seen from Table 7.3, the delta has now changed from 71.4 to 83.4. The consequence of this is that the option writer now has to buy an additional (83.4 – 71.4) = 12 shares. To do this, the option writer has to borrow another 12*$45 = $540 at the continuously compounded risk-free rate of 6 percent.

Table 7.4 captures the impact of rebalancing once every six months (continuing with the scenario in Table 7.3), until the option expires (assuming both the risk-free and volatility rates are kept constant).

The reader should note that to obtain the value of cell E11 in Table 7.4 (i.e., delta value on option maturity), I had to hard-code a delta value of 1 simply because substituting a value of 0 for T -t would yield a #Div/0 comment as an output for cell C11.^{[2]}

TABLE 7.4 Delta Hedging for Vanilla European-Style Call Option on a Nondividend-Paying Stock

Table 7.5 illustrates the settlement value associated with the deltahedging program on option maturity for an option finishing in-the- money.

As can be seen from Table 7.5, the entry in cell J12 refers to the fact that the option has been exercised (as it finished in-the-money). The seller of the option receives a strike price of $40 for each underlying share and would in turn deliver 100 shares to the buyer of the option. The entry in cell J13 refers to the fact that the option seller would have made a profit of about $179 on option maturity should this scenario materialize (funds received from the option exercise less the cumulative amount owing on option maturity just prior to settlement of option).

Table 7.6 shows the illustration associated when the option finishes out- of-the money.

As can be seen in cell J12 of Table 7.6, the option finishing out-of-the- money implies no incoming cash from option exercise. What the hedger is left with is the hedging P&T of $86 (before present valuing).

Since Black-Scholes assumed in their paper that one should arrive at a total hedge profit of 0 whatever the stock-price path realized during the life of the option, it should be rather surprising for the reader to see that the final answers in cell J13 of both Tables 7.5 and 7.6 are not 0. The simple reason for this is that the hedging was not done in continuous time. By doing the hedging over an infinitesimal time interval, one would be able to see a value of 0 in cell J13 – regardless the nature of the path. From a practical implementation standpoint, it is difficult to do this in continuous time. As a consequence, one would need to run a daily simulation 5,000 times to arrive at an average value of 0 – implying that the theoretical option premium charged for the sale of this option is a fair-value premium if interest rates and volatility rates remain constant.

[1] The reason for buying shares as opposed to selling shares comes from the fact that the moment a call is sold, the exposure to the seller of the option is actuallyshares. Since, to neutralize this (i.e., be riskless), the hedger has to acquire shares resulting in a net portfolio delta of 0.

[2] This should not be surprising to the reader, as given the definition for , the denominator would be 0 when. Applying L'Hopital's rule, it can be easily shown that as,. This readily implies that(consistent with what has been hard-coded for delta). Further, should the option finish out-of-the-money, the corresponding delta value would be 0.

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