Replicating portfolios for Call and Put options

As we discussed during yesterday’s class meeting, the replicating portfolios for long calls and long puts resemble the replicating portfolios for long forwards and short forwards. Specifically, the replicating portfolio for a long call comprises a margined investment in the underlying, whereas the replicating portfolio for the long put involves shorting the underlying and lending money. The primary difference between the replicating portfolios for forward contracts vis-a-vis replicating portfolios for options is that replicating portfolios for options involve fractional long/short positions in the underlying combined with fractional short/long positions in a riskless bond, whereas replicating portfolios for forward contracts require unitary positions (i.e., 100% long/short position in the underlying combined with 100% short/long position in a riskless bond).

My teaching note entitled “A Simple Model of a Financial Market” lays out the logic behind these claims concerning replicating portfolios for calls and puts. Suppose two states of the economy occur 1 period from now: good (g) and bad (b). In state g, a share of stock is worth $2 whereas in state b, a share of stock is worth $0.50; today’s price for a share of stock is {S_0} = \$1. Also suppose that today’s and next period’s riskless bond price is {B_0} = {B_1} = \$1. Furthermore, a call option with an exercise price of K = $1 is worth {C_{1,g}} = \$1 and {C_{1,b}} = \$0. Let {\Delta} correspond to the number of shares of stock in the replicating portfolio, and {\beta} correspond to the number of bonds. It follows that the value of the replicating portfolio in state g, {V_{1,g}} = \Delta({S_{1,g}}) + {\beta}({B_1}) = \Delta(\$2) + {\beta}(\$1) = \$1, and the value of the replicating portfolio in state b, {V_{1,b}} = \Delta({S_{1,b}}) + {\beta}({B_1}) = \Delta(\$.5) + {\beta}(\$1) = 0. Thus we have two equations in two unknowns:

{V_{1,g}} = \Delta (\$2) + \beta(\$1) = \$1, and {V_{1,b}} = \Delta (\$0.50) + \beta(\$1) = \$0.

By subtracting the equation for {V_{1,b}} from the equation for {V_{1,g}}, we obtain \Delta = 2/3, and {\beta} = -1/3, which implies that {V_0} = \Delta ({S_0}) + \beta ({B_0}) = 2/3(1) - 1/3(1) = \$1/3. Since the payoff on the replicating portfolio is identical to the payoff on the long call position, it follows that {V_0} = {C_0} = \$1/3 represents the arbitrage-free price for the call option.

Next, consider the replicating portfolio for the put option with an exercise price of K = $1. Since {S_{1,g}} = \$2 and {S_{1,b}} = \$0.50, this implies that {P_{1,g}} = \$0 and {P_{1,b}} = \$0.50. In state g, the value of the replicating portfolio {V_{1,g}} = \Delta({S_{1,g}}) + {\beta}({B_1}) = \Delta(\$2) + {\beta}(\$1) = \$0, and the value of the replicating portfolio in state b, {V_{1,b}} = \Delta({S_{1,b}}) + {\beta}({B_1}) = \Delta(\$.5) + {\beta}(\$1) = \$0.50 Thus we have two equations in two unknowns:

{V_{1,g}} = \Delta (\$2) + \beta(\$1) = \$0, and {V_{1,b}} = \Delta (\$0.50) + \beta(\$1) = \$0.50.

By subtracting the equation for {V_{1,b}} from the equation for {V_{1,g}}, we obtain \Delta = -1/3, and {\beta} = 2/3, which implies that {V_0} = \Delta ({S_0}) + \beta ({B_0}) = -1/3(1) + 2/3(1) = \$1/3. Since the payoff on the replicating portfolio is identical to the payoff on the long put position, it follows that {V_0} = {P_0} = \$1/3 represents the arbitrage-free price for the put option.

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