PROCESS A: "Driftless" geometric Brownian motion (GBM). familiar process: dS = o S dW with S(0) = 1. PROCESS B: dS = x S² dw for some constant x, with S(0) = 1 the instantaneous return over [t, t+dt] is the random variable: ds/S= (S(t + dt) -S(t))/s(t) [1] Explain why, for PROCESS A, the variance of the instantaneous return is constant (per unit time). Hint: What's the variance of dW? The rest of this problem involves PROCESS B. [2] For PROCESS B, the statement in [1] is not true. Explain why, in PROCESS B, the variance of the instantaneous return (per unit time) depends on the value S(t). "Driftless" means no "dt" term. So it's our is the volatility. Let's manipulate PROCESS B using a change of variable (and Ito's Formula) to see what we come up with. Worth a try. Let Y(t) = 1/Š(t). [3] Apply Ito directly and show that we obtain: dY = (-1/S²)dS + (1/2)(2/S³) (ds)²
PROCESS A: "Driftless" geometric Brownian motion (GBM). familiar process: dS = o S dW with S(0) = 1. PROCESS B: dS = x S² dw for some constant x, with S(0) = 1 the instantaneous return over [t, t+dt] is the random variable: ds/S= (S(t + dt) -S(t))/s(t) [1] Explain why, for PROCESS A, the variance of the instantaneous return is constant (per unit time). Hint: What's the variance of dW? The rest of this problem involves PROCESS B. [2] For PROCESS B, the statement in [1] is not true. Explain why, in PROCESS B, the variance of the instantaneous return (per unit time) depends on the value S(t). "Driftless" means no "dt" term. So it's our is the volatility. Let's manipulate PROCESS B using a change of variable (and Ito's Formula) to see what we come up with. Worth a try. Let Y(t) = 1/Š(t). [3] Apply Ito directly and show that we obtain: dY = (-1/S²)dS + (1/2)(2/S³) (ds)²
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![PROCESS A:
"Driftless" geometric Brownian motion (GBM). "Driftless" means no "dt" term. So it's our
familiar process: dS = o S dW with S(0) = 1. o is the volatility.
PROCESS B: dS = ∞ S² dW_ for some constant x, with S(0) = 1
the instantaneous return over [t, t+dt] is the random variable:
dS/S = (S(t + dt) - S(t))/S(t)
[1] Explain why, for PROCESS A, the variance of the instantaneous return is constant (per
unit time). Hint: What's the variance of dW?
The rest of this problem involves PROCESS B.
[2] For PROCESS B, the statement in [1] is not true. Explain why, in PROCESS B, the variance
of the instantaneous return (per unit time) depends on the value S(t).
Let's manipulate PROCESS B using a change of variable (and Ito's Formula) to see what we
come up with. Worth a try. Let Y(t) = 1/Š(t).
[3] Apply Ito directly and show that we obtain:
dY = (-1/S²)dS + (1/2)(2/S³) (ds)²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85fafaa4-de34-47c7-98e7-8408756b8b80%2F82334abe-df53-48b4-b90f-229153d50d9d%2F8udnzop_processed.png&w=3840&q=75)
Transcribed Image Text:PROCESS A:
"Driftless" geometric Brownian motion (GBM). "Driftless" means no "dt" term. So it's our
familiar process: dS = o S dW with S(0) = 1. o is the volatility.
PROCESS B: dS = ∞ S² dW_ for some constant x, with S(0) = 1
the instantaneous return over [t, t+dt] is the random variable:
dS/S = (S(t + dt) - S(t))/S(t)
[1] Explain why, for PROCESS A, the variance of the instantaneous return is constant (per
unit time). Hint: What's the variance of dW?
The rest of this problem involves PROCESS B.
[2] For PROCESS B, the statement in [1] is not true. Explain why, in PROCESS B, the variance
of the instantaneous return (per unit time) depends on the value S(t).
Let's manipulate PROCESS B using a change of variable (and Ito's Formula) to see what we
come up with. Worth a try. Let Y(t) = 1/Š(t).
[3] Apply Ito directly and show that we obtain:
dY = (-1/S²)dS + (1/2)(2/S³) (ds)²
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