Option Basics - Part II. Call Values When Future Stock Prices Are Certain to Be Above the Exercise Price Suppose the value of a stock one year from today.

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Option Basics - Part II

Call Values When Future Stock Prices Are Certain to Be Above the Exercise Price Suppose the value of a stock one year from today (S 1 ) will be either 50$ or $70. The price of the stock today equals $45. The one-year Treasury Bill currently yields 6%. What is the value today of a call that is “at the money”, with an exercise price is $45? What is the process? We try to combine an investment in a risk-free asset and a call option to replicate the payoffs of just holding the stock. If we can replicate the payoffs, we know that the combination of the risk-free asset and the call must have the same value today as the stock.

Call Values When Future Stock Prices Are Certain to Be Above the Exercise Price $45 $70 $50 $70 $50 Risk Free Investment plus Call = $45 How much do we invest in the risk-free asset and the call?

Call Values When Future Stock Prices Are Certain to Be Above the Exercise Price The Values One Year From Today for the Stock (S 1 ) and the Call (S 1 -E,0) State Value of Stock(S 1 ) Value of Call (S 1 -E,0) Good $70 $70-$45 Bad $50 $50-$45 We can replicate the payoffs for the stock if we add back the value of the exercise price with certainty one year from today. What amount do we need to invest in the risk-free asset today to be able to have the value of the exercise price one year from now? What is the value of the call?

Call Values When Future Stock Prices Are Certain to Be Above the Exercise Price The amount invested in the risk-free asset today to guarantee that we will have the exercise price a year from today is $45/(1.06) = $42.45 What is the value of the call? It is the value of the stock today, less the amount invested in the risk-free asset. That is $45-$42.45 = $2.55 Are the arbitrage conditions met?

Call Values When Future Stock Prices Are Certain to Be Above the Exercise Price What determines the value of the call? Other things constant, the value of the call increases with the value of the underlying stock (S 0 ), the risk-free rate (R f ), and the time before expiration (t), but it decreases with with the exercise price (E). What about the variance?

Call Values When the Variance of Future Stock Prices Increases and Their Value Can be Below the Exercise Price Suppose the value of a stock one year from today (S 1 ) will be either $40 or $75. The price of the stock today equals $45. The one-year Treasury Bill currently yields 6%. What is the value today of a call that is “at the money”, with an exercise price is $45? State Value of Stock(S 1 ) Value of Call (S 1 -E,0) Good $75 $75-$45$30 Bad $40 $40-$45$ 0 How can investors replicate the payoffs of the stock now? They need to get $40 with certainty and make up the difference between $40 and $75 with the call options. How will they do that?

Call Values When Future Stock Prices Can be Below the Exercise Price Investors invest enough today so they will have $40 with certainty one year from today, I.e., they invest $40/1.06 = $37.74 Then, they invest the rest of the $45 in call options. How many options will they have to buy? –One call option will give them $30 if the option is in the money. –However, investors need $35 to replicate the payoffs of the stock. Therefore, they need to buy 1.16 (=$35/$30) options. As before, the value of the stock today must equal the value of the investment in the risk-free asset and the investment in the options. Thus, –$45 = 1.16*C 0 + $37.74; Therefore, C 0 = $6.23

Call Values When Future Stock Prices Can be Below the Exercise Price and Take On Any Positive Value This more general relationship is given by the Black-Scholes Option Pricing Model. Where N(d 1 ) and N(d 2 ) are probabilities that a standard normally distributed random variable is less than or equal to N(d 1 ) or N(d 2 ), where and

Call Values When Future Stock Prices Can be Below the Exercise Price and Take On Any Positive Value Suppose the current stock price is $45, the risk-free rate is 6% per year, the annual standard deviation of returns on the stock is 75%, and the exercise price of the option is $45. What is the value of the call option today?