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
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
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![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%2Feadbed5d-21fd-4dc2-b262-8c56690527fe%2Fsih20jf_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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