Assume that S follows the geometric Brownian motion dynamics, dS = μSdt+σSdZ, with μ and σ constants. Find the stochastic differential equation satisfied by V (t,S) = er(T−t) S .
Added by Thusani R.
Step 1
Ito's lemma states that if we have a function V(t, S) and S follows the geometric Brownian motion dynamics, then the differential dV(t, S) can be found as follows: dV(t, S) = (∂V/∂t) dt + (∂V/∂S) dS + (1/2) (∂²V/∂S²) (dS)² Now, let's find the partial derivatives Show more…
Show all steps
Your feedback will help us improve your experience
Rachel Gore and 67 other Differential Equations educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
If S follows the geometric Brownian motion process dS = μSdt + σSdz, what is the process followed by (a) y = 2S, (b) y = S^2, (c) y = e^S, (d) y = e^(r(T-t))/S, and (e) y = S^α. In each case, express the coefficients of dt and dz in terms of y as well as in terms of S. (i.e. you will have a solution in terms of S, and a solution in terms of y)
Adi S.
Suppose that a stock price, S, follows geometric Brownian Motion with an expected return µ and volatility σ. dS(t) = µS(t)dt + σS(t)dW(t) Show that Sn, for n>0 constant, also follows a geometric Brownian Motion. Compute its expected return and volatility.
Sri K.
Recommended Textbooks
Differential Equations and Linear Algebra
Fundamentals of Differential Equations
A First Course in Differential Equations with Modeling Applications
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD