Timeline for Black-Scholes Equation - Riskless portfolio derivation
Current License: CC BY-SA 3.0
24 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 14, 2014 at 11:50 | comment | added | user11800 | Thanks again for your help on this - I have updated the OP to show via stochastic calculus that we reach a contradiction if we suppose that we can hedge a claim using a self-financing portfolio consisting of one unit of the claim and -dv/ds units of stock - please let me know if you have any questions on it. | |
| Aug 14, 2014 at 11:17 | comment | added | emcor | @Mark ok I will post an edit for my original post on your issues later today. Self-fin. does not mean constant value, it only means that $V_t=a_tC_t+b_tS_t$, which holds here as I showed 2 times. The BS-portfolio can be declared as self-financing because the definition is fulfilled as I showed, but it does not mean that such function $C$ actually exists. That is then the great finding of Black-Scholes, they found the $C$. For American Puts e.g., $C$ does not exist. | |
| Aug 14, 2014 at 11:02 | comment | added | user11800 | I'm not sure what you mean by "accepting the definition" of self-financing - I understand the definition, but you can't just declare a portfolio to be self-financing, you need to deduce it by looking at the differential of the wealth process. Can you tell me what is wrong with the vector space argument laid out above if you believe that the hedging portfolio is self-financing? | |
| Aug 14, 2014 at 10:59 | comment | added | emcor | @Mark If you do not accept the definition of self-financing, I cannot help you. And AGAIN: For Ito, you must assume that $v$ satisfies the BS-PDE, then the Ito-representation changes. | |
| Aug 14, 2014 at 10:56 | comment | added | user11800 | ...and so we have a contradiction. Thus, the original claim that the hedging portfolio is self-financing must be incorrect. This is supported by p12 of the document I linked to earlier: math.nyu.edu/research/carrp/papers/pdf/faq2.pdf. | |
| Aug 14, 2014 at 10:54 | comment | added | user11800 | I've given several arguments in the thread supporting the claim that this portfolio isn't self-financing - I think the vector space one is particularly intuitive: If the hedging portfolio is self-financing, then since holding a single option to maturity is self-financing, then it follows that a portfolio consisting of $\frac{\partial v}{\partial s}(t,S_t)$ shares is self-financing too (since the space of self-financing strategies is closed under linear combinations). However, hopefully it's clear that if $\frac{\partial v}{\partial s}$ is non-constant, it can't be self-financing... | |
| Aug 14, 2014 at 10:50 | comment | added | user11800 | Exactly - so the fact that the value of the portfolio is constant (as stated in your post "You need to apply Ito") does not imply it is self-financing. Re the definition of self-financing, the point I have been making is that if we have a portfolio of 1 option and -dv/ds shares, then our wealth is v(t,S_t) - $\frac{\partial v}{\partial S}$ S_t, and applying Ito's lemma to this shows that the differential of the wealth process is [b]not[/b] dv - $\frac{\partial v}{\partial S}$ dS, and so the portfolio is not self-financing. | |
| Aug 14, 2014 at 10:33 | comment | added | emcor | @Mark Please see the definition of self-financing: $dV=adC+bdS\,\forall t$. Set $a=1,b=\partial_SC$ and you get the result (again). If you have a bond, you get:$ dV=dB\,\forall t$ is self-financing. Self-financing just means, that you keep holding the investment without adding/withdrawing money; if you reduce your holding in the bond that is withdrawing $100$ money, which is not self-financing. Then you would have at some point $dV=dB-100\neq dB$. | |
| Aug 14, 2014 at 10:17 | comment | added | user11800 | The portfolio value being constant does not imply that the strategy is self-financing - consider a holding of e^(-rt) in the risk-free asset at time t. This portfolio has a constant value of $B_0$, but we are continuously consuming wealth (by reducing our holding in the bond), and therefore the strategy is not self-financing. | |
| Aug 14, 2014 at 10:12 | comment | added | emcor | @Mark You need to apply Ito under the assumption that $C$ satisfies the Black-Scholes PDE, see my answer, then you come to the same result. Also I can only repeat that $dX_t=dC-\partial_SCdS=dC-dC/dS\cdot dS=dC-dC=0$ is self-financing, because you always hold the same total value short as you hold long. | |
| Aug 14, 2014 at 10:04 | comment | added | user11800 | The comment in the final paragraph of p12 of math.nyu.edu/research/carrp/papers/pdf/faq2.pdf agrees that the portfolio is not self-financing. The change in the value of the option is irrelevant as we are not changing our holding in the option. Mathematically: $X_t = v(t,S_t) - dv/ds . S_t$ Self-financing says: $dX_t = dv(t,S_t) - dv/ds . dS_t$ But Ito says: $dX_t = dv(t,S_t) - d(dv/ds) . S_t - dv/ds .dSt - d<dv/ds,S>$ Equating these two expressions for $dX$ gives $d(dv/ds) S_t = - d<dv/ds,S>$, which is not true (RHS represents a Leb integral, LHS is an Ito integral). | |
| Aug 14, 2014 at 9:56 | comment | added | emcor | @Mark I cannot do more than the mathematical proof above that it is self-financing. I think you misunderstood that not only the weights change, but the prices aswell. Your total positions always cancel off. You hold one "unit" of option constantly, but the option value changes, so you adjust your stock position aswell. The stock-weight is always exactly offsetting the tangential option-change after an underlying change. See the proof above, they exactly cancel. | |
| Aug 14, 2014 at 9:54 | comment | added | user11800 | To put it another way: self-financing strategies form a vector space. The strategy of holding one option until maturity is obviously self-financing, so if the hedging strategy described above is self-financing, then by closure under linear combinations of self-financing strategies, it should follow that the strategy that consists solely of holding $\frac{\partial V}{\partial S}(t, S_t)$ shares is self-financing too, but obviously this is not the case - we need external funds whenever we want to increase our holding in the shares. | |
| Aug 14, 2014 at 9:52 | comment | added | user11800 | The portfolio is not self-financing - this can be shown by looking at the derivative of the wealth process, but there is also an intuitive reason why; we always hold one unit of the option, but our holding in the stock changes (i.e. if dV/dS increases, then we respond by buying more shares) - this cannot be a self-financing strategy, as extra funds are needed to buy the extra shares. | |
| Aug 14, 2014 at 9:27 | comment | added | emcor | @Mark The portfolio is self-financing, because the stock holding changes always exactly opposite to the change in the option value, so you never add/withdraw money: $\partial_SC\cdot dS_t=dC/dS\cdot dS=dC$. | |
| Aug 14, 2014 at 1:06 | comment | added | user11800 | I can see that the holding in the stock needs to be dynamic to hedge the claim, but I'm saying that the portfolio of 1 claim and -dV/dS stocks isn't self-financing (the only self-financing portfolio consisting of 1 claim and some amount of stocks is when a constant amount of stocks are held). The fact that this portfolio is not self-financing then means that we can't argue that its rate of return must be the same as the risk-free asset. | |
| Aug 14, 2014 at 0:41 | comment | added | emcor | @Mark you can see from my derivation, that the self-fin portfolio exists with the above dynamic weights. You need to adjust your portfolio dynamically, because the option is a sublinear asymmetric payoff while the underlying is linear only, so the match changes with every move as $C$ reacts differently than $S$. At a single point in time though, you are in theory hedged because the values coincide, but they react differently to future up/downmoves. | |
| Aug 14, 2014 at 0:37 | comment | added | emcor | @Mark $dX_t=dC-\partial_SCdS_t$ is just the definition of any self-financing portfolio with the given weights. You can define it so as an Ansatz, and then check if you get a solution. It must not be shown, because it is just a definition/condition which is imposed. It does not mean that there will be a solution under such conditions, but for European BS-Model there is. So the price $C(t,S_t)$ you get satisfies a self-fin. portfolio in itself as a solution under this assumption. | |
| Aug 14, 2014 at 0:22 | comment | added | user11800 | , and Ito's formula gives $dX_t = dV(t,S_t) - d(\frac{\partial V}{\partial S}) S_t - \frac{\partial V}{\partial S} dS_t - d<\frac{\partial V}{\partial S}, S_t>$. Equating these two gives $d(\frac{\partial V}{\partial S}) S_t = - d<\frac{\partial V}{\partial S}, S_t>$, which leads to a contradiction. | |
| Aug 14, 2014 at 0:18 | comment | added | user11800 | Thanks again for your response. I don't see that the portfolio value being uniform shows that the portfolio is self-financing - consider a holding in the risk-free bond of $B_0^{-1}e^(-rt)$ at time $t$ - this portfolio always has value 1, but is not self-financing. For the portfolio suggested by wikipedia, the wealth process is $X_t = V(t, S_t) - \frac{\partial V}{\partial S} S_t$. The self-financing condition is $dX_t = dV(t, S_t) - \frac{\partial V}{\partial S} dS_t$... | |
| Aug 13, 2014 at 23:49 | comment | added | emcor | @Mark I see your point, the article uses Ansatz-technique which is not as straightforward. You can see that the portfolio $\pi$ is self-financing by calculating out the partial derivative of the black-scholes price, then you get: $dC=\partial_S CdS$ (where $C$ is the Black-Scholes-Formula), so $\Pi=0\forall t$, which is then obviously self-financing. | |
| Aug 13, 2014 at 23:29 | comment | added | user11800 | Thank you for your answer. I understand how a European call is replicated using a portfolio of stock and risk-free bond, but the wikipedia article I linked to has an argument involving a portfolio consisting of holdings in the claim and the stock. In your example, we specify that $\alpha_t = \frac{\partial V}{\partial S}$, and then $\beta_t$, the holding in the bond, is determined by the self-financing condition. However, in the wikipedia example, the holdings in the option and stock are fully specified, and I don't believe this leads to a self-financing portfolio. | |
| Aug 13, 2014 at 22:02 | history | edited | emcor | CC BY-SA 3.0 |
deleted 68 characters in body
|
| Aug 13, 2014 at 19:46 | history | answered | emcor | CC BY-SA 3.0 |