I'm reading Shreve's Stochastic Calculus for Finance, Volume II. In chapter 4, he derives the "delta hedging rule":

$$\Delta(t) = c_x(t, S(t)) \text{ for all } t \in [0, T)\text{.}\tag{1}$$

This says that a self-financing portfolio $X(t)$ that needs to replicate an option with random price $c(t, S(t))$ at time $t$,

$$X(t) = c(t, S(t))\text{,}\tag{2}$$

needs to long $c_x(t, S(t))$ of the underlying asset, assuming that the value of the asset follows a log-normal distribution.

This appears (to me) to be a very general solution to the hedging problem that applies to many classes of exotic options.

However, in chapter 5, Shreve requires the Martingale representation theorem to compute $\Delta(t)$. On page 223, we are given

$$\Delta(t)=\frac{\tilde{\Gamma}}{\sigma(t)D(t)S(t)},\ \ 0 \leq t \leq T\text{.}\tag{3}$$

About this formula, he says

The Martingale Representation Theorem argument of this section justifies the risk-neutral pricing formula (5.2.30) and (5.2.31), but it does not provide a practical method of finding the hedging portfolio $\Delta(t)$. The final formula [(3)] for $\Delta(t)$ involves the integrand $\tilde{\Gamma}(t)$ in the martingale representation (5.3.4) of the discounted derivative security price. While the Martingale Representation Theorem guarantees that such a process $\tilde{\Gamma}$ exists and hence a hedge $\Delta(t)$ exists, it does not provide a method for finding $\tilde{\Gamma}(t)$. We return to this point in Chapter 6.

I found this paragraph confusing because it does not seem to acknowledge that the author has already found such a $\Delta(t)$ in (1). I tried to read ahead to chapter 6, but I have found myself unable to find the answer to this question:

For what kinds of options should a replicating portfolio long/short $\Delta(t)$ of the underlying according to $(1)$? For what kinds of options is this hedge in any way "incorrect"?

Some examples: (1) obviously applies to European options with fixed volatility and rate of return. What if your model assumes variable volatility? What about American options?

  • 2
    $\begingroup$ The martingale representation theorem can be used to more formally show that a hedging strategy replicates the option payoff. If there are many sources of uncertainty the issue with a delta hedging strategy is that you are not hedged against all sources of risk, e.g. volatility of stock changing. $\endgroup$ – fesman Apr 30 at 7:21
  • 1
    $\begingroup$ Another issue may be if the underlying can experience jumps (e.g.. S(t) is a Jump Diffusion process) in that case also Delta hedging cannot remove all risk. The proof of delta hedging requires continuous paths for S(t). $\endgroup$ – noob2 Apr 30 at 17:31
  • $\begingroup$ @AlRacoon the delta does not change sign, but it sounds like binary/digital options may be a contradiction to the generality of (1). I don't see why that would be because binary options should satisfy the derivation of (1) (since they also adhere to the Black-Scholes PDE). $\endgroup$ – user54908 Apr 30 at 19:55
  • $\begingroup$ @user54908 I stand corrected. The change in delta changes sign at the barrier. $\endgroup$ – AlRacoon Apr 30 at 19:58

Chapter 5.5.2 (Hedging with One Stock) paragraph that includes (3) does not assume that the value of the payoff at any $t$ is Markovian, that is, it is a function of $S(t)$ only, so there is no "delta" as in (1) to use.

Summary 6.7 has the answer to what Shreve does want to say about formula (3) (in particular the three paragraphs I highlighted; note that Shreve defines and works with a Markov diffusion SDE as underlying).

6.7 Summary

When the underlying price of an asset is given by a stochastic differential equation, the asset price is Markov and the price of any non-path-dependent derivative security based on that asset is given by a partial differential equation. In order to price path-dependent securities, one first seeks to determine the variables on which the path-dependent payoff depends and then intro­ duce one or more additional stochastic differential equations in order to have a system of such equations that describes the relevant variables. If this can be done, then again the price of the derivative security is given by a partial differential equation. This leads to the following four-step procedure for finding the pricing differential equation and for constructing a hedge for a derivative security.

  1. Determine the variables on which the derivative security price depends. In addition to time t, these are the underlying asset price S(t) and possibly other stochastic processes. We call these stochastic processes the state processes. One must be able to represent the derivative security payoff in terms of these state processes.

  2. Write down a system of stochastic differential equations for the state pro­ cesses. Be sure that, except for the driving Brownian motions, the only random processes appearing on the right-hand sides of these equations are the state processes themselves. This ensures that the vector of state processes is Markov.

  3. The Markov property guarantees that the derivative security price at each time is a function of time and the state processes at that time. The discounted option price is a martingale under the risk-neutral measure. Compute the differential of the discounted option price, set the dt term equal to zero, and obtain thereby a partial differential equation.

  4. The terms multiplying the Brownian motion differentials in the discounted derivative security price differential must be matched by the terms multiplying the Brownian motion differentials in the evolution of the hedging portfolio value; see (5.4.27). Matching these terms determines the hedge for a short position in the derivative security.


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