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Although this assumption was not considered in the original Black-Scholes paper, I did discuss the incorporation of dividends in the previous section (see Tables 7.7 and 7.8). To summarize, the impact of a continuous dividend rate can be implemented by reducing the interest on the amount borrowed by the dividend amount received (as dividends are paid on the stocks held). As the astute reader will realize, equations (3.7a) and (3.7b) are the consequence of running a delta-hedging program in the presence of continuously compounded dividends and absence of transaction cost.

Arbitrage Opportunities and Constant Risk-Free Rate

The assumption relating to the lack of arbitrage opportunities is a crucial one as only with it were the authors able to remove the notion of risk-preference in the valuation of the options. More precisely, using equation (7.1) the authors assumed that it was possible for the hedger to hold a riskless portfolio comprising the sale of the option and the purchase of the appropriate number of shares which was continuously rebalanced so as not to allow any arbitrage opportunities.

The consequence of applying these assumptions and Ito's lemma to the equation (7.1) is


which shows the absence of μt,T (the hedger's view on the growth rate of the stock price). Solving equation (7.2) using the boundary conditions max(.ST – K, 0) and max(K – ST, 0) resulted in the Black-Scholes formulae given by equations (3.4a) and (3.4b).

The implication of this result is that one can conveniently replace μt,T for rt,T (assuming that rt,T is also constant) to arrive at the same set of equations. Succinctly put, as stated in the previous chapters, to value these options (or even run the delta-hedging program discussed in this chapter) one can assume that for a nondividend-paying stock, lnST is normally distributed with a mean of, and a variance of With this distributional assumption, one can now take the expected value of the payoff associated with the option and discount the result using the continuously compounded risk-free rate so as to obtain equations (3.4a) and (3.4b).

The concept of no-arbitrage and continuous rebalancing of a riskless portfolio that has been discussed in this section only matters if the hedger is interested in dynamically replicating the exposure using delta-hedging or other hedging methodologies. In the event that the hedger wants to take on risks without doing any kind of hedging, the hedger is free to use any value for μt,T (the growth rate of the stock price) or even any price distribution that is reflective of its view on the market.[1]

  • [1] An example of this can be found in the insurance industry. More precisely, it is not uncommon to find an insurance company selling a policy containing an embedded market guarantee which is priced using the insurance company's view on the market growth rate and volatility. In this instance, instead of running a hedging program, the insurance company would set aside the required capital and reserves for the risks it is exposed to.
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