| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | 要计算一份无股息支付股票的远期合约价格，我们可以使用以下公式： \[ F = S \times e^{(r \times T)} \] 其中： - \( F \) 是远期价格 - \( S \) 是当前股票价格 - \( r \) 是无风险利率 - \( T \) 是到期时间（以年为单位） 根据题目给定的数据： - 当前股票价格 \( S = \$50 \) - 6个月利率 \( r = 4\% \) 每年 - 到期时间 \( T = 6/12 = 0.5 \) 年 我们将这些数值代入公式进行计算： \[ F = 50 \times e^{(0.04 \times 0.5)} \] 首先计算指数部分： \[ 0.04 \times 0.5 = 0.02 \] 然后计算指数函数： \[ e^{0.02} \approx 1.0202 \] 最后计算远期价格： \[ F = 50 \times 1.0202 = 51.01 \] 因此，这份无股息支付股票的远期价格 \( F \) 约为 \$51.01。 |
51.0 |
| deepseekr1 | The forward price \( F \) can be calculated using the formula for non-dividend-paying stocks, which accounts for the cost of carry (interest rate). Assuming the given 4% interest rate is a **simple annual rate**, the calculation is: \[ F = S_0 \times (1 + r \times T) \] Where: - \( S_0 = \$50 \) (current stock price), - \( r = 4\% = 0.04 \) (annual interest rate), - \( T = 0.5 \) years (6 months). Plugging in the values: \[ F = 50 \times (1 + 0.04 \times 0.5) = 50 \times 1.02 = \$51 \] **Answer:** The forward price is \boxed{51}. |
51.0 |
| gpto1 | To find the forward price \( F \) of a forward contract on a non-dividend-paying stock, we use the formula for forward pricing with continuous compounding: \[ F = S_0 \times e^{rT} \] Where: - \( S_0 \) = Current stock price - \( r \) = Continuously compounded risk-free interest rate - \( T \) = Time to maturity in years - \( e \) = The base of the natural logarithm (approximately equal to 2.71828) **Given:** - Current stock price, \( S_0 = \$50 \) - Annual risk-free interest rate, \( r = 4\% = 0.04 \) - Time to maturity, \( T = 6 \) months \( = 0.5 \) years **Step-by-Step Calculation:** 1. **Convert the annual interest rate to a decimal:** \[ r = 4\% = 0.04 \] 2. **Convert the time to maturity to years:** \[ T = 6 \text{ months} = 0.5 \text{ years} \] 3. **Calculate the exponent \( rT \):** \[ rT = 0.04 \times 0.5 = 0.02 \] 4. **Compute the exponential term \( e^{rT} \):** \[ e^{rT} = e^{0.02} \] Using a calculator: \[ e^{0.02} \approx 1.02020134 \] 5. **Calculate the forward price \( F \):** \[ F = \$50 \times 1.02020134 \approx \$51.01007 \] **Answer:** The forward price \( F \) of the contract is approximately **\$51.01**. --- **Explanation:** The forward price reflects the future value of the current stock price, adjusted for the risk-free interest rate over the time to maturity. Since the stock does not pay dividends, we don't need to adjust for any dividend payouts. By using continuous compounding, we account for the effect of the interest rate accruing at every moment up to maturity, which is standard in forward pricing models. The exponential function \( e^{rT} \) captures this continuous growth. **Conclusion:** The forward price is calculated to be \$51.01, which means that under the given risk-free rate and time to maturity, the price agreed upon today for the future transaction should be \$51.01 to prevent arbitrage opportunities. |
51.0 |
| GPT-4o | To determine the forward price \( F \) of a forward contract on a non-dividend paying stock, we can use the formula: \[ F = S_0 \times e^{rT} \] where: - \( S_0 \) is the current stock price, - \( r \) is the risk-free interest rate, - \( T \) is the time to maturity in years, - \( e \) is the base of the natural logarithm (approximately 2.71828). Given: - \( S_0 = 50 \) (current stock price in dollars), - \( r = 0.04 \) (annual risk-free interest rate, or 4% per annum), - \( T = 0.5 \) (time to maturity in years, since 6 months is half a year), Let's plug these values into the formula: \[ F = 50 \times e^{0.04 \times 0.5} \] First, calculate the exponent: \[ 0.04 \times 0.5 = 0.02 \] Now, calculate \( e^{0.02} \): \[ e^{0.02} \approx 1.0202 \] Finally, calculate the forward price: \[ F = 50 \times 1.0202 \approx 51.01 \] So, the forward price \( F \) is approximately \( \$51.01 \). |
51.0 |
| llama3.1-70B-Instruct | To find the forward price, F, of a forward contract on a non-dividend paying stock, we can use the formula: F = S × (1 + r × t) Where: F = forward price S = current stock price = $50 r = annual interest rate = 4% = 0.04 t = time to maturity in years = 6 months = 0.5 years Plugging in the values, we get: F = $50 × (1 + 0.04 × 0.5) F = $50 × (1 + 0.02) F = $50 × 1.02 F = $51 Therefore, the forward price, F, is $51. |
51.0 |
| llama3.1-405B-Instruct | To find the forward price, F, of a non-dividend paying stock, we can use the formula: F = S(1 + rT) Where: F = forward price S = current stock price r = interest rate (per annum) T = time to maturity (in years) Given: S = $50 r = 4% per annum = 0.04 T = 6 months = 0.5 years Plugging in the values, we get: F = $50(1 + 0.04 x 0.5) F = $50(1 + 0.02) F = $50 x 1.02 F = $51 Therefore, the forward price, F, is $51. |
51.0 |
| DeepSeek-Coder-V2 | To determine the forward price (F) of a non-dividend paying stock, we can use the formula: F = S * e^(r * T) where: - F is the forward price, - S is the current stock price, - r is the risk-free interest rate, - T is the time to maturity in years, - e is the base of the natural logarithm (approximately 2.71828). Given the problem: - S = $50, - r = 4% per annum (0.04 as a decimal), - T = 6 months (0.5 years). Now, we can plug these values into the formula: F = 50 * e^(0.04 * 0.5) First, calculate the exponent: 0.04 * 0.5 = 0.02 Next, calculate e^0.02: e^0.02 ≈ 1.0202 Now, multiply the current stock price by this value: F = 50 * 1.0202 ≈ $51.01 So, the forward price (F) is approximately $51.01. |
51.0 |
| Qwen2-72B-Instruct | The forward price (F) of a non-dividend paying stock can be calculated using the formula: F = S * e^(r*t) where: S = the current stock price r = the risk-free interest rate t = the time until the contract matures (measured in years) In this case, S = $50, r = 0.04 (4% per annum), and t = 0.5 (6 months is half a year). F = $50 * e^(0.04*0.5) F = $50 * e^0.02 F = $50 * 1.02020404 F = $51.01 Therefore, the forward price (F) of the contract is $51.01. |
51.0 |
| baichuan4 | The forward price (F) for a non-dividend paying stock can be calculated using the formula: \[ F = S_0 \cdot e^{r \cdot T} \] where: - \( S_0 \) is the current stock price, - \( r \) is the annual interest rate, - \( T \) is the time to maturity of the forward contract in years. Given the information in the problem: - \( S_0 = \$50 \), - \( r = 4\% = 0.04 \) (per annum), - \( T = \frac{6}{12} = 0.5 \) years (since the contract matures in 6 months). Plugging in the values: \[ F = \$50 \cdot e^{0.04 \cdot 0.5} \] \[ F = \$50 \cdot e^{0.02} \] \[ F = \$50 \cdot 1.0202 \] \[ F = \$51.01 \] So the forward price \( F \) is approximately \$51.01. |
51.0 |
| doubao | The forward price (F) can be calculated using the following formula: F = S0 × (1 + r × t), where S0 is the current stock price, r is the interest rate, and t is the time to maturity in years. Here, S0 = $50, r = 4% / 100 = 0.04 (annual rate), and t = 6 months / 12 months = 0.5 years. F = 50 × (1 + 0.04 × 0.5) = 50 × (1 + 0.02) = 50 × 1.02 = 51. So the forward price is $51. |
51.0 |
| minimax | To calculate the forward price (F) for a non-dividend-paying stock, we can use the following formula: \[ F = S_0 \times e^{rT} \] where: - \( S_0 \) is the current stock price, - \( r \) is the continuously compounded risk-free interest rate, - \( T \) is the time to maturity of the forward contract in years. Given: - \( S_0 = $50 \), - \( r = 4\% \) per annum (or 0.04 as a decimal), - \( T = 0.5 \) years (since 6 months is half a year). Plugging these values into the formula, we get: \[ F = 50 \times e^{0.04 \times 0.5} \] First, calculate the exponent: \[ 0.04 \times 0.5 = 0.02 \] Now, calculate the exponential term: \[ e^{0.02} \approx 1.0202 \] (using a calculator for the value of \( e^{0.02} \)) Finally, multiply this by the current stock price: \[ F = 50 \times 1.0202 \approx 51.01 \] So the forward price (F) of the stock for a contract maturing in 6 months is approximately $51.01. |
51.0 |
| kimi | The forward price of a non-dividend paying stock can be calculated using the following formula: \[ F = S_0 \times e^{(rT)} \] where: - \( F \) is the forward price - \( S_0 \) is the current stock price - \( r \) is the annual interest rate (expressed as a decimal) - \( T \) is the time to maturity in years Given: - \( S_0 = $50 \) - \( r = 4\% \) per annum = 0.04 - \( T = 6 \) months = \( \frac{6}{12} \) years = 0.5 years Now, let's calculate the forward price: \[ F = 50 \times e^{(0.04 \times 0.5)} \] \[ F = 50 \times e^{0.02} \] To find \( e^{0.02} \), we can use a calculator: \[ e^{0.02} \approx 1.020201 \] Now, multiply this by the current stock price: \[ F \approx 50 \times 1.020201 \] \[ F \approx 51.01005 \] So, the forward price \( F \) is approximately $51.01. |
51.0 |
| wenxinyiyan | To calculate the forward price of a non-dividend paying stock, we can use the formula: F = S * e^(r*T) where: - F is the forward price - S is the current stock price - r is the interest rate - T is the time to maturity (in years) - e is the base of the natural logarithm (approximately 2.71828) Given the values: - S = $50 - r = 4% = 0.04 - T = 6 months = 0.5 years We can plug these values into the formula: F = 50 * e^(0.04 * 0.5) = 50 * e^(0.02) = 50 * (1 + 0.02 + 0.5*(0.02)^2 + ...) = 50 * (1 + 0.02 + 0.5*0.0004 + ...) = 50 * (1 + 0.02 + 0.0002 + ...) = 50 * (1.0202 + ...) ≈ 50 * 1.0202 ≈ 51.01 So, the forward price for the contract is approximately $51.01. |
51.0 |
| zhipu | To determine the forward price \( F \) of a forward contract on a non-dividend paying stock, we can use the formula that relates the forward price to the current stock price and the risk-free interest rate. The formula is: \[ F = S_0 \times e^{r \times t} \] Where: - \( F \) is the forward price. - \( S_0 \) is the current stock price. - \( r \) is the risk-free interest rate (expressed as a decimal). - \( t \) is the time to maturity in years. Given the data: - The current stock price \( S_0 \) is $50. - The annual interest rate \( r \) is 4%, which as a decimal is 0.04. - The time to maturity \( t \) is 6 months, which as a fraction of a year is 0.5. Plugging these values into the formula: \[ F = 50 \times e^{(0.04 \times 0.5)} \] First, calculate the exponent: \[ 0.04 \times 0.5 = 0.02 \] Now, calculate \( e^{0.02} \). The value of \( e \) (the base of the natural logarithm) is approximately 2.71828. Using a calculator to find \( e^{0.02} \): \[ e^{0.02} \approx 1.020201 \] Finally, multiply this value by the current stock price: \[ F = 50 \times 1.020201 \approx 51.01005 \] Therefore, the forward price \( F \) is approximately $51.01. Note: This calculation assumes continuous compounding of the interest rate. If the interest rate were compounded differently (e.g., annually, semi-annually, etc.), the formula would need to be adjusted accordingly. However, in the context of forward pricing and financial mathematics, continuous compounding is a common assumption. |
51.0 |
| gpto1 | To find the forward price \( F \) of a forward contract on a non-dividend-paying stock, we use the formula for forward pricing with continuous compounding: \[ F = S_0 \times e^{rT} \] Where: - \( S_0 \) = Current stock price - \( r \) = Continuously compounded risk-free interest rate - \( T \) = Time to maturity in years - \( e \) = The base of the natural logarithm (approximately equal to 2.71828) **Given:** - Current stock price, \( S_0 = \$50 \) - Annual risk-free interest rate, \( r = 4\% = 0.04 \) - Time to maturity, \( T = 6 \) months \( = 0.5 \) years **Step-by-Step Calculation:** 1. **Convert the annual interest rate to a decimal:** \[ r = 4\% = 0.04 \] 2. **Convert the time to maturity to years:** \[ T = 6 \text{ months} = 0.5 \text{ years} \] 3. **Calculate the exponent \( rT \):** \[ rT = 0.04 \times 0.5 = 0.02 \] 4. **Compute the exponential term \( e^{rT} \):** \[ e^{rT} = e^{0.02} \] Using a calculator: \[ e^{0.02} \approx 1.02020134 \] 5. **Calculate the forward price \( F \):** \[ F = \$50 \times 1.02020134 \approx \$51.01007 \] **Answer:** The forward price \( F \) of the contract is approximately **\$51.01**. --- **Explanation:** The forward price reflects the future value of the current stock price, adjusted for the risk-free interest rate over the time to maturity. Since the stock does not pay dividends, we don't need to adjust for any dividend payouts. By using continuous compounding, we account for the effect of the interest rate accruing at every moment up to maturity, which is standard in forward pricing models. The exponential function \( e^{rT} \) captures this continuous growth. **Conclusion:** The forward price is calculated to be \$51.01, which means that under the given risk-free rate and time to maturity, the price agreed upon today for the future transaction should be \$51.01 to prevent arbitrage opportunities. |
51.0 |
| deepseekv3 | The forward price \( F \) of a non-dividend paying stock can be calculated using the formula: \[ F = S_0 \times e^{rT} \] Where: - \( S_0 \) = Current stock price = $50 - \( r \) = Annual risk-free interest rate = 4% = 0.04 - \( T \) = Time to maturity in years = 6 months = 0.5 years Substitute the values into the formula: \[ F = 50 \times e^{0.04 \times 0.5} \] Calculate the exponent: \[ 0.04 \times 0.5 = 0.02 \] Now calculate \( e^{0.02} \): \[ e^{0.02} \approx 1.0202 \] Finally, calculate the forward price: \[ F = 50 \times 1.0202 \approx 51.01 \] Thus, the forward price \( F \) is approximately **$51.01**. |
51.0 |