The current price of a non-dividend paying stock is $50 Use one-step Binomial mode to value a European at-the-money call option on the stock that expires in 12 months with u= 13 and d = 1u The risk-fr

To value the European call option using the one-step Binomial model, we first need to estimate the stock prices at expiration.

Since the stock is non-dividend paying, the two possible stock prices at expiration are:

1. u * $50 = 1.3 * $50 = $65
2. d * $50 = (1/u) * $50 = (1/1.3) * $50 = $38.46 (rounded to two decimal places)

Next, we calculate the option's value at expiration. Since the option is at-the-money, the option's value at expiration is the maximum of 0 or the difference between the stock price and the strike price. However, the strike price is not given in the question, so we cannot calculate the exact option value at expiration.

To calculate the value of the call option, we need to discount the expected option value at expiration back to the present using the risk-free rate. Since the option is European, we only consider the expected option value at expiration, not the early exercise possibility.

The expected option value at expiration is the probability-weighted average of the possible option values at expiration. In this case, since the stock price is at-the-money, the expected option value at expiration is the average of the two possible option values at expiration.

Let p be the probability of an up movement (u) and (1-p) be the probability of a down movement (d). To find the value of p, we use the risk-neutral probability formula:

p = (1 + r - d) / (u - d)

where r is the risk-free rate. Plugging in the values:

p = (1 + 0.07 - 1/1.3) / (1.3 - 1/1.3) = 0.5499 (rounded to four decimal places)

The expected option value at expiration is then:

Expected option value at expiration = p * max(0, $65 - Strike) + (1-p) * max(0, $38.46 - Strike)

Since the strike price is not given, we cannot calculate the exact value of the call option.

Finally, to hedge the trader's position, the number of stocks necessary to buy or sell can be calculated using the delta of the call option. The delta is the sensitivity of the option's value to changes in the stock price. It can be calculated as:

Delta = (Option value at u - Option value at d) / (Stock price at u - Stock price at d)

However, since the exact option value is not given, we cannot calculate the delta or the number of stocks necessary to hedge the trader's position practically