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frolloos
Topic Author
Posts: 1621
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### self-financing for the clueless

I am embarassed to admit that I still don't quite understand the self-financing argument. I think I can see why self-financing implies dV = xdB + ydS, but I cannot find explicit demonstration that for the Black Scholes framework we indeed have dx(B + dB) + dy(S+dS) = 0.

emac
Posts: 89
Joined: July 7th, 2009, 7:15 pm

### self-financing for the clueless

frolloos
Topic Author
Posts: 1621
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### self-financing for the clueless

Thanks, have read that paper before. but I don't believe it really answers the question.
Last edited by frolloos on December 3rd, 2015, 11:00 pm, edited 1 time in total.

emac
Posts: 89
Joined: July 7th, 2009, 7:15 pm

### self-financing for the clueless

Maybe you could explain your question a bit better then? To me: dx(B + dB) + dy(S+dS) = 0 doesn't make sense Edit: do you mean: "when we apply Ito's formula to V, why do the other terms need to vanish?"
Last edited by emac on December 3rd, 2015, 11:00 pm, edited 1 time in total.

frolloos
Topic Author
Posts: 1621
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### self-financing for the clueless

The other terms need to vanish, and the / my question is how to show they really do vanish using the BS replicating / self-financng portfolio. That The other terms have to vanish can be seen from V(t+dt) = x(t+dt)B(t+dt) + y(t+dt)S(t+dt) = x(t)B(t+dt) + y(t)S(t+dt) accrding to the self-financing assumption.

emac
Posts: 89
Joined: July 7th, 2009, 7:15 pm

### self-financing for the clueless

You cannot show that the other terms vanish. I think most people agree that the argument you are trying to make is incorrect.That is what peter carr is saying when he says we don't want to calculate the total derivative of the portfolio value process, V. Instead, he says you should calculate what he calls the "gain" and his self-financing condition is that the gain should be zero.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### self-financing for the clueless

In 73 paper The Pricing of Options and Corporate Liabilities on page 643 is a remark. In particular it states that "Since there is no market risk in the hedged position , all of the risk due to the fact that the hedge is not continuously adjusted must be risk that can be diversify away."I do not know whether or not the program "must be risk that can be diversify away" has been performed
Last edited by list1 on December 3rd, 2015, 11:00 pm, edited 1 time in total.

frolloos
Topic Author
Posts: 1621
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### self-financing for the clueless

QuoteOriginally posted by: emacYou cannot show that the other terms vanish. I think most people agree that the argument you are trying to make is incorrect.That is what peter carr is saying when he says we don't want to calculate the total derivative of the portfolio value process, V. Instead, he says you should calculate what he calls the "gain" and his self-financing condition is that the gain should be zero.Actually you can show the other terms vanish if BS pde is satisfied, which makes sense. I did the algebra last night and it's only a few lines - not sure why I never managed to do it before. Don't understand what Peter Carr is saying. I am sure he is right, but formulates it in a way that adds to the confusion IMHO.

emac
Posts: 89
Joined: July 7th, 2009, 7:15 pm

### self-financing for the clueless

QuoteOriginally posted by: frolloosQuoteOriginally posted by: emacYou cannot show that the other terms vanish. I think most people agree that the argument you are trying to make is incorrect.That is what peter carr is saying when he says we don't want to calculate the total derivative of the portfolio value process, V. Instead, he says you should calculate what he calls the "gain" and his self-financing condition is that the gain should be zero.Actually you can show the other terms vanish if BS pde is satisfied, which makes sense. I did the algebra last night and it's only a few lines - not sure why I never managed to do it before. Don't understand what Peter Carr is saying. I am sure he is right, but formulates it in a way that adds to the confusion IMHO.Yes, but if you use the self-financing argument to derive the B-S PDE, then using the PDE in the derivation is circular, no?

frolloos
Topic Author
Posts: 1621
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### self-financing for the clueless

QuoteOriginally posted by: emacQuoteOriginally posted by: frolloosQuoteOriginally posted by: emacYou cannot show that the other terms vanish. I think most people agree that the argument you are trying to make is incorrect.That is what peter carr is saying when he says we don't want to calculate the total derivative of the portfolio value process, V. Instead, he says you should calculate what he calls the "gain" and his self-financing condition is that the gain should be zero.Actually you can show the other terms vanish if BS pde is satisfied, which makes sense. I did the algebra last night and it's only a few lines - not sure why I never managed to do it before. Don't understand what Peter Carr is saying. I am sure he is right, but formulates it in a way that adds to the confusion IMHO.Yes, but if you use the self-financing argument to derive the B-S PDE, then using the PDE in the derivation is circular, no?No, I don't believe it is circular. The argument is as follow: suppose $C(t,S)$ satisfies the BS pde $C_t + rSC_S + \frac{1}{2} \sigma^2 S^2 C_{SS} = rC$ then the portfolio $V = xB + yS$, with $x = (C - SC_S)/B, y = C_S$ is self-financing, i.e. satisfies $dx(B + dB) + dy (S+dS) = 0$. So you use the BS pde in showing that the budget condition holds, which a few lines of algebra will show is indeed the case.Conversely you can also show that if $V = xB + yS$ is self-financing with $x$ and $y$ as above, then $C$ must satisfy the BS pde.
Last edited by frolloos on December 4th, 2015, 11:00 pm, edited 1 time in total.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### self-financing for the clueless

QuoteOriginally posted by: list1In 73 paper The Pricing of Options and Corporate Liabilities on page 643 is a remark. In particular it states that "Since there is no market risk in the hedged position , all of the risk due to the fact that the hedge is not continuously adjusted must be risk that can be diversify away."I do not know whether or not the program "must be risk that can be diversify away" has been performedfrolos, your question is similar to my problem which forced me to think that BS derivation is formally incorrect. The hedged portfolio that is commonly used in derivation is written in the form$\Pi ( t ) = C ( t , S ( t ) ) - C_x^{\prime} ( t , S ( t ) ) S ( t )\qquad \qquad \qquad (1)$ then they take differential in t or they call change in the value at t and arrive at the formula$\Pi ( t + \Delta t ) = C ( t + \Delta t , S ( t + \Delta t ) ) - C_x^{\prime} ( t , S ( t ) ) S ( t + \Delta t )\qquad \qquad \qquad (2)$ Notation used to present transformation of the portfolio value from (1) to (2) is formally incorrect. WE cannot use the same letter t in the same formula assuming that one t is fixed and other is variable on small interval $[ t , t + \Delta t ]$ . To write transformation (1) - (2) correctly we should introduce portfolio value by the formula $\Pi ( t , u ) = C ( u , S ( u ) ) - C_x^{\prime} ( t , S ( t ) ) S ( u ) \qquad \qquad \qquad (3)$ where t is a fixed parameter taking values from [ 0 , T ] and variable $u \in [ t , t + \Delta t ]$We need to use two notations t and u to distinct fixed parameter t and variable which admits the same value t in (1). Thus using correct representation of the portfolio we conclude that $\Pi ( t ) = \Pi ( t , t )$ and $\Pi ( t + \Delta t ) = \Pi ( t , t + \Delta t )$ and therefore we did not lost anything by using new representation of the hedged portfolio. On the other hand it is clear that single variable t in $\Pi ( t )$ does not correctly describes dynamics portfolio.The next step is to present the adjustment of the portfolio at the next moment $t_1 = t + \Delta t$. The value of the adjustment at $t_1 = t + \Delta t$ is equal to$\Pi ( t_1 , t_1 ) - \Pi ( t_0 , t_1 ) = - [ C_x^{\prime} ( t_1 , S ( t_1 ) ) - C_x^{\prime} ( t_0 , S ( t_0 ) ) ] S ( t_1 ) \qquad \qquad \qquad (4)$ where $t_0 = t$. The adjustment is a 'risky' random variable can be either positive or negative. This remark highlight the sense of self financing. The full analysis should include the full cash flow (4) on [0 , T ] and taking limit when $\Delta t \rightarrow 0$ of the EPV of this cash flow. Here expected value E is the real world expectation.
Last edited by list1 on December 4th, 2015, 11:00 pm, edited 1 time in total.

emac
Posts: 89
Joined: July 7th, 2009, 7:15 pm

### self-financing for the clueless

Ok, yes. But usually the hedging argument is used to derive the B-S PDE.
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