Excercises SMRA
An illustrative Exercises
The daily log returns are independent and normally distributed. 𝐸[𝑟𝑡 ] = 0.1, 𝜎[𝑟𝑡 ] = 0.1. Suppose you buy
900$. Compute:
1.
2.
3.
4.
The probability that after one trading day your investment is worth less than 900$
The probability that after one trading day your investment is worth less than 700$
The probability that after one trading day your investment is worth more than 1000$
The probability that after one trading day your investment is worth less than 500$
𝑟𝑡 ∼ 𝑁(0.1, 0.12 )
𝑃0 = 900
𝑟𝑡 = log⁡(1 + 𝑅𝑡 )
𝑒 𝑟𝑡 = 1 + 𝑅𝑡 < − − 𝑤𝑒⁡𝑤𝑖𝑙𝑙⁡𝑢𝑠𝑒⁡𝑡ℎ𝑖𝑠
𝑦 ∼ 𝑁(𝜇, 𝜎 2 ) → 𝑒 𝑦 ∼ 𝑙𝑜𝑔𝑁(𝜇, 𝜎 2 )
𝑥 ∼ log N(𝜇, 𝜎 2 ) → log(𝑋) ∼ 𝑁(𝜇, 𝜎 2 )
𝑛𝑜𝑟𝑚𝑎𝑙: (−∞, +∞)
𝑙𝑜𝑔𝑁 ∼ (0, +∞)
𝑃𝑡 ∼ 𝐹
𝑒 𝑟𝑡 (𝑘) = 1 + 𝑅𝑡 (𝑘) =
𝑃𝑡
𝑃𝑡−𝑘
𝑒 𝑟𝑡 (𝑘) → 𝑟𝑡 (𝑘)𝑖𝑠⁡𝑎⁡𝑛𝑜𝑟𝑚𝑎𝑙, 𝑠𝑜⁡𝑒 𝑟𝑡 (𝑘) ⁡𝑖𝑠⁡𝑎⁡𝑙𝑜𝑔𝑁𝑜𝑟𝑚𝑎𝑙
𝑒 𝑟𝑡 (𝑘) ∙ 𝑃𝑡−𝑘 = 𝑃𝑡 ⁡𝑤ℎ𝑒𝑛⁡𝑡 = 𝑘
𝑃𝑡 ∼ 𝑙𝑜𝑔𝑁(𝑡𝜇 + log(𝑃0 ) , 𝑘𝜎 2 )
Part 1
𝑃(𝑃1 < 900) → 𝑖𝑡⁡𝑖𝑠⁡𝑙𝑖𝑘𝑒⁡𝑐𝑜𝑚𝑝𝑢𝑡𝑖𝑛𝑔⁡𝑡ℎ𝑒⁡𝐶𝐷𝐹⁡𝑜𝑓⁡𝑎⁡𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛.+
𝑃1 ∼ 𝑙𝑜𝑔𝑁(1 ∙ 0.1 + log(900) , 1 ∙ 0.12 )⁡(𝑖𝑛⁡𝑅⁡𝑤𝑒⁡𝑢𝑠𝑒⁡𝑝𝑛𝑜𝑟𝑚(900, 𝜇, 𝜎))
𝑃(𝑃1 < 900) = Φ (
log(900) − 𝜇
)⁡(𝑛𝑜𝑟𝑚𝑎𝑙⁡𝐶𝐷𝐹)
𝜎⁡
𝑃(log(𝑃1 ) < log(900)) → 𝑃(𝑁𝑂𝑅𝑀𝐴𝐿 < log⁡(900))
𝑃(𝑁(𝜇, 𝜎 2 ))
log(900) − (0.1 + log⁡(900))
0.1
) = Φ (− ) = Φ(−1) = 1 − 𝜙(1) = 𝑓𝑟𝑜𝑚⁡𝑡𝑎𝑏𝑙𝑒𝑠
0.1
0.1
= 1 − 0.84 = 0.16
𝑃(𝑃1 < 900) = Φ (
Part 2
900
log (
) − 0.1
log(700) − (0.1 + log⁡(900))
700
𝑃(𝑃1 < 700) = Φ (
) = Φ(−3.5)
) == 1 − Φ (
0.1
0.1
= 1 − Φ(3.5)
Part 3
𝑃(𝑃1 > 1000) = 1 − 𝑃(𝑃1 < 1000) = 1 − Φ (
log(1000) − (0.1 + log(900))
)
0.1
Part 4
A trading week, k=7
𝑃7 ∼ log N(7 ∙ 0.1 + log(900) , 7 ∙ 0.1⁡) → 𝐿𝑜𝑔𝑁(𝜇, 𝜎)
𝑃(𝑃7 < 500) = ⁡Φ (
log(500) − (0.1 + log(900))
)
√7 ∙ 0.1
2
𝑉(𝑃7 ) = (𝑒 𝜎 − 1) ∙ 𝑒^(2𝜇 + 𝜎 2 )⁡
2
𝑉[𝑃𝑡 ] = (𝑒 7∙0.1 − 1) ∙ 𝑒^(2𝜇 + 𝜎 2 )