GR5260 Exam 1: Answers and Rubrics Question 1
a) 𝑆(𝑡)=𝑆(0)exp((𝜇 −𝜎12)𝑡+𝜎√𝑡𝑍) 11121
𝑆2(𝑡) = 𝑆2(0)exp((𝜇2 − 𝜎22) 𝑡 + 𝜎2√𝑡𝑊) 2
b) 𝑉 =1∑𝑁 𝑓(𝑧,𝑤)where 𝑁 𝑁 𝑗=1 𝑗 𝑗
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𝑓(𝑧, 𝑤) = 𝑒−𝑟𝑇𝑚𝑎𝑥(𝑆2(0) exp ((𝜇2 − 𝜎22) 𝑇 + 𝜎2√𝑇𝑤) − 𝑆1(0) exp ((𝜇1 − 𝜎12) 𝑇 + 𝜎1√𝑇𝑧) − 𝐾, 0) 22
𝐸[𝑉 ]=1∑𝑁 𝐸[𝑓(𝑧,𝑤)]=1∑𝑁 𝐸[𝑓(𝑍,𝑊)]=𝐸[𝑓(𝑍,𝑊)]=𝐸[𝑉] 𝑁 𝑁 𝑗=1 𝑗 𝑗 𝑁 𝑗=1
Thus, 𝑉 is an unbiased estimator of 𝑉. 𝑁
c) 𝜀𝑁 = 1.96𝑠𝑁 where √𝑁
𝑠 =√ 1 ∑𝑁 [𝑓(𝑧,𝑤)−𝑚 ]2 and𝑚 =1∑𝑁 𝑓(𝑧,𝑤)(seedefinitionof𝑓(𝑧,𝑤)in2.b) 𝑁𝑁−1𝑗=1𝑗𝑗𝑁 𝑁𝑁𝑗=1𝑗𝑗
e) Confidence interval output is optional
2pts: correctly obtaining z,w
1pt : correctly obtaining S1, S2
1pt : correctly obtaining option value 1pt : correctly return value, CI
f) Confidence interval output is optional
1pt : correctly obtaining stock prices
2pts: correctly obtaining antithetical draw payoffs 1pt : correctly obtaining option value
1pt : correctly return value
2pts: correctly calling previous functions to get values
[1pt for each_value in three_values if each_value is correct]
1pt : correctly obtaining stock prices 2pts: correctly obtaining Joe’s payoffs 1pt : correctly obtaining option value 1pt : correctly return value
h) No, Joe’s method would not work unless 𝜌 = 0, because 𝑉 is a biased estimator of 𝑉.
𝑐𝑜𝑟𝑟(−𝑍, 𝑊) = 𝑐𝑜𝑟𝑟(𝑍, −𝑊) = −𝜌 and 𝑐𝑜𝑟𝑟(𝑍, 𝑊) = 𝑐𝑜𝑟𝑟(−𝑍, −𝑊) = 𝜌
Case 1: 𝜌 = 0
𝐸[𝑓(𝑍, 𝑊)] = 𝐸[𝑓(−𝑍, 𝑊)] = 𝐸[𝑓(𝑍, −𝑊)] = 𝐸[𝑓(−𝑍, −𝑊)] and thus, 𝐸[𝑉 ] = 𝐸[𝑉]
Case 2: 𝜌 ≠ 0
𝐸[𝑓(−𝑍, 𝑊)] = 𝐸[𝑓(𝑍, −𝑊)] and 𝐸[𝑓(𝑍, 𝑊)] = 𝐸[𝑓(−𝑍, −𝑊)], but 𝐸[𝑓(𝑍, 𝑊)] ≠ 𝐸[𝑓(−𝑍, 𝑊)] in
𝐸[𝑉 ] ≠ 𝐸[𝑉] in general.
Question 2
2pts: reading the csv inputs
[2pts for each_value in [EOD value, daily pnl, total pnl] if each_value is correct] 2pts: print a result dataframe
1pt : dropping trade price column
3pts: correctly calling groupby method with correct columns 1pt : applying sum method to the groupby object
Question 3
2 points: 1 point : 1 point : 1 point : 1 point : 1 point : 1 point : 1 point : 1 point :
correctly declaring the class method
correctly calling self.period_dates to get coupon dates correctly obtaining yield
correctly obtaining coupon amount
checking value date against maturity
correctly specifying the loop
correctly getting annualized time
correctly implement duration formula
correctly return duration
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