How Can Calculus of Variations Optimize Stock Sales in a Bear Market?

[Liquidxlax]

This is for Classical mechanics 2

I'm not sure how to put partials in and the N with the dot beside it was supposed to be the derivative of N with respect to time

Suppose that you have N₀ shares of stock and you want to make as money as you can by selling all of them in a single day. If N₀ is large, and you sell your shares in small batches, the money you make can be written approximately as an integral

∫dt N'(t)P(N,N';t) t₁ to t₂

where t1 and t2 are the opening and closing times for the stock exchange, N(t) is a smooth function that is approximately the number of shares you have sold at time t (which satisfies N(t₁) = 0 and N(t₂) = N₀, N' is the rate at which you sell the stock (in shares per hour, say) and P(N, N';t) is the price per share as a function of time. The interesting thing is that the price depends on how you sell the shares. For example, if you sell them too fast, the price will drop. That is why P depends on N and N'.

a) Suppose that P(N,N';t) = P₀ - Bt -CN' for P₀, B, C all positive (This is a "bear market" because of the -Bt term as the stock price is going down with time). Find N(t) and N' that allow you to make the most money.

b) Discuss briefly what happens if B is too large.


2. Homework Equations

dP/dN−d/dt(dP/dN')=0



3. The Attempt at a Solution

I've really got nothing for this because I'm unsure of how to start minus checking the relevant equation

dP/dN=0=d/dt(dP/dN).

so

dP/dN.=−C

and i think there has to be an auxillary equation to do with the fact that N(t₁) = 0 and N(t₂) = N₀

 

[Ray Vickson]

This is for Classical mechanics 2

I'm not sure how to put partials in and the N with the dot beside it was supposed to be the derivative of N with respect to time

Suppose that you have N₀ shares of stock and you want to make as money as you can by selling all of them in a single day. If N₀ is large, and you sell your shares in small batches, the money you make can be written approximately as an integral

∫dt N'(t)P(N,N';t) t₁ to t₂

where t1 and t2 are the opening and closing times for the stock exchange, N(t) is a smooth function that is approximately the number of shares you have sold at time t (which satisfies N(t₁) = 0 and N(t₂) = N₀, N' is the rate at which you sell the stock (in shares per hour, say) and P(N, N';t) is the price per share as a function of time. The interesting thing is that the price depends on how you sell the shares. For example, if you sell them too fast, the price will drop. That is why P depends on N and N'.

a) Suppose that P(N,N';t) = P₀ - Bt -CN' for P₀, B, C all positive (This is a "bear market" because of the -Bt term as the stock price is going down with time). Find N(t) and N' that allow you to make the most money.

b) Discuss briefly what happens if B is too large.


2. Homework Equations

dP/dN−d/dt(dP/dN')=0



3. The Attempt at a Solution

I've really got nothing for this because I'm unsure of how to start minus checking the relevant equation

dP/dN=0=d/dt(dP/dN).

so

dP/dN.=−C

and i think there has to be an auxillary equation to do with the fact that N(t₁) = 0 and N(t₂) = N₀

The optimality (Euler) equation is dL/dN = (d/dt) (dL/dN'), where L is the integrand. For L = N'*(P0-bt-cN'), what do you get?

RGV

 

[Liquidxlax]

The optimality (Euler) equation is dL/dN = (d/dt) (dL/dN'), where L is the integrand. For L = N'*(P0-bt-cN'), what do you get?

RGV

dL/dN = d/dt(dL/dN') = (d/dt)(P₀ - Bt -2CN') = -b