The Journal of finance purchasing a units of the option at a price of Pr-1. The value of a is chosen so that the portfolio payoffs are the same in both states, or Vi=Hi +av lving for a we obtain the number of units of the option to be held in the rtfolio per $1 invested in the stock. A negative value of a implies that the option is sold short (written)with the proceeds being used to partially fund the purchase of the stock The time t- 1 value of portfolio is 1 +aP-I. The end-of-period value is given by either side of (1). Discounting the left-hand side by the riskless interest rate, R, and setting the discounted value equal to the present value of the portfolio, a pricing equation for the option is obtained Hi +av 1 +aP- Substituting the value of a from(2)into(3), the price of the option can be solved in terms of its end-of-period values. +R-Hd)+v( P (H-H;)(1+R) Equation 4 is a recursive relationship that can be applied at any time t-l to determine the price of the option as a function of its value at time t Note that in equation (4)we make a notational distinction between an option,s value (V) and its price(P). Assuming that an investor will exercise an option when it is in his best interest to do so V,= MAXIPI, VEXER where VEXER, is the value of exercising the option at time t The distinguishing feature among American and European puts and calls is in he definition of their exercisable values. American options can be exercised at any time whereas European options can only be exercised at maturity. Calls are options to buy stock at a set price whereas puts are options to sell. Letting S, represent the time t price of the stock, X the option,s exercise price, and t the maturity date of the option, we obtain American Call VEXEl t, Put VEXER,=X-S, for all t Call VEXER, =S,-X for t=T vEⅹER,=0 for t< T Put veXer=X-s, for t VEXER,=0 Recognizing that for both American and Eupopean puts and calls Pr=0