Why don't we take the differential to the Delta in the Delta hedge-portfolio

Question

For option $V(S,t)$ with underlying asset $S$, we have a hedge portfolio $$\Pi = V(S,t) - \Delta(S,t)S$$ I always confuse here, when we take the differential of $\Pi$ $$d\Pi = dV -\Delta dS$$ why needn't we take the differential for $\Delta(S,t),$ I think it should be $$d\Pi = dV -d(\Delta S).$$ We call the delta-hedge self-financing, can any one show me a clearly reason?

Answer 1

Note: I have edited the answer so as to give a clearer interpretation of the self-financing condition.

Let us specify the notation: $C_t$ is the derivative price; $S_t$ the stock's; $B_t$ the bond's; $w_C(t)$, $w_S(t)$ and $w_B(t)$ the option's, stock's and bond's holdings respectively; $C_T=h(S_T)$ the option's payoff, which we assume is European, hence $T$ is the derivative's maturity.

First, the equation you wrote is not exactly correct, because as it has been explained a few times in this site you cannot get a self-financing portfolio by holding an option and simply adjusting the stock's holding: indeed, you cannot dynamically hedge one option only with $w_S(t)$ stock shares, because when time passes and $w_S(t)$ changes, the only way you can change your allocation $-$ your holding $w_S(t)$ $-$ is to inject or withdraw cash from your portfolio, hence you need more degrees of freedom $-$ more weights $-$ in your equation to avoid cash injections/withdrawals. Now, there are a few ways to view the situation:

• You can form a self-financing portfolio made up of options and stocks, where the weights are selected so as to cancel all random terms. Because the portfolio is self-financing and also riskless, it must earn the risk-free rate, hence:

$$ B_t = w_C(t)C_t+w_S(t)S_t $$

• You can form a portfolio with a holding of stocks and bonds which replicate the option. Hence your portfolio value must be equal to the option value at any time:

$$ C_t = w_S(t)S_t+w_B(t)B_t $$

Let us consider the second perspective, namely let us form a portfolio consisting of stocks and bonds that replicate the option:

$$ C_t = w_S(t)S_t+w_B(t)B_t $$

Now, why must the portfolio be self-financing? Think of yourself as the option writer $-$ i.e. seller $-$ and assume the option is sold at $t=0$ and that you are charging its fair value: your client is then paying you the option's premium $\pi$ today in exchange for which he is expecting you to settle the liability $h(S_T)$ at maturity $T$. Because you are charging the fair value of the option $-$ you can rationalize this assumption by considering that, in a liquid, competitive market environment, market participants will force option premiums to converge to their fair value, i.e. no arbitrage $-$ it implies that somehow you should be able to meet your financial obligation at $T$ by appropriately allocating the cash $\pi$ and only that sum: you shouldn't add or retrieve any cash throughout the life of the contract. In other words, the premium $\pi$ should be the required quantity to finance your liability $h(S_T)$ at $T$, hence it must be the case that this sum can be allocated dynamically to stocks and bonds throughout the period $[0,T]$ so as to have the value $C_T=h(S_T)$ at $T$ at your disposal to pay your client. Consider the alternatives to this situation:

• You need to add cash to your position at some time $t \in [0,T]$: this would be a loss to you, something you want to avoid so you wouldn't do it. You will set $\pi$ so as to avoid this;
• You can retrieve cash from your position at some time $t \in [0,T]$: you would be extracting value from your client. Because we have assumed we are in a competitive market environment, other option writers will charge lower prices and hence you will lose your client. You will be forced to set $\pi$ so as to avoid this $-$ if you don't want to lose your clients.

We have established that there shouldn't be neither injections nor retrievals of cash throughout the life of the trade. Therefore, this implies that the net impact of rebalancing the portfolio on the portfolio value must be null. Applying Ito's Lemma on $dC_t$, this is equivalent to verifying the following self-financing condition:

$$ S_tdw_S(t) + dw_S(t)dS_t + B_tdw_B(t) + dw_B(t)dB_t = 0 \qquad (1) $$

$(1)$ implies that the replicating portfolio evolves according to:

$$ dC_t = w_S(t)dS_t+w_B(t)dB(t) \qquad (2) $$

Namely, the portfolio value should only change due to fluctuations in the option's, stocks' and bonds' prices.

Now, this does not mean that $w_S$ and $w_B$ cannot change throughout the life of the trade: these are functions that tell us how many stocks and bonds we must hold at $t$ to be able to pay for $h(S_T)$ at $T$, and because $S_t$ and $B_t$ evolve through time so must their holdings. Therefore we should be continuously rebalancing our holding of stocks and bonds but the purchase of additional stocks (bonds) must always be financed by the sale of bonds (stocks), without any injection or subtraction of cash.

Finally, please note that equation $(2)$ defines a replicating, self-financing portfolio, but we always need to ensure condition $(1)$ is fulfilled: in other words, not only should we posit equation $(2)$, but we also have to make sure that $w_S(t)$ and $w_B(t)$ are chosen in such a way that $(1)$ is verified.

I recommend taking a look at exercise 4.10, "Self-financing trading", in Shreve's Stochastic Calculus for Finance II, where discrete time and continuous time cases are discussed and connected.

Answer 2

Delta is intended to be a locally-constant holding to the stock (to create an instantaneously riskless portfolio) and determined by the current state of the system, hence passes through the derivative operator.

I visualise it similar to a tangent to a curve - it's different as you move along the curve (i.e. as the state of the system changes) but it's locally-constant and every time we pass through a given point, I recover the same tangent.

Same as under Black-Scholes, my option delta is uniquely determined by the state of my option economy (stock price, moneyness, time to expiry, etc)

Answer 3

I guess a discrete time approximation would be helpful here. You can follow these steps:

1. At time t choose $\alpha_t$ and $\Delta_t$ to set up your portfolio: $$\Pi_t=\alpha_t B_t +\Delta_tS_t$$
2. At time $t+dt$ the value of your portfolio is (pay attention to $\alpha_t$ and $\Delta_t$) $$\Pi_{t+dt}=\alpha_t B_{t+dt} +\Delta_t S_{t+dt}$$

It seems clear that for a small change in time, the change in the value of the portfolio is: $$d\Pi_t = \Pi_{t+dt}-\Pi_t=\alpha_t(B_{t+dt}-B_t)+\Delta_t(S_{t+dt}-S_t)=\alpha_t dB_t+\Delta_t dS_t$$ And this is all you need to compute the delta of the portfolio, given that it has been fixed at time t. The next step is to rebalance the portfolio (here is where the selfinancing part kicks in):

3. At time $t+dt$ modify your asset holdings mainting the value of your portfolio (pay attention to $\alpha_{t+dt}$ and $\Delta_{t+dt}$): $$\Pi_{t+dt}=\alpha_t B_{t+dt} +\Delta_t S_{t+dt}=\alpha_{t+dt} B_{t+dt} +\Delta_{t+dt} S_{t+dt}$$

As you can see, this third step is not needed to describe the sensitivity of your portfolio with respect to $S_t$. Only the first two steps suffice, meaning that you do not have to worry about $d(\Delta S)$ exactly because of the second equality in the last equation.