When Delta Hedging Applies to Markov and Path-Dependent Options
Article Quant Q&A · Author: user54908
Summary
The document explains why the familiar hedge of holding the derivative’s sensitivity to the underlying price applies when the option value can be written as a function of time and current state variables. In a Markov model with a single state variable, that sensitivity is the derivative with respect to the asset price. For path-dependent claims, the price may depend on the history of the asset, so the asset price alone does not specify the option’s value and the simple delta in the question may not exist.
The cited discussion describes enlarging the state with additional processes that capture the payoff’s relevant path features, then deriving a pricing equation and matching Brownian risk terms to construct a hedge. It distinguishes this practical state-based method from the martingale representation theorem, which establishes that a hedge exists but does not itself supply a way to compute it. The document gives no worked exotic-option example and does not settle details for variable volatility or American exercise; those cases require an appropriate state description and model assumptions.
Key ideas
- A delta based only on the asset price requires the derivative value to be determined by time and current state.
- Path-dependent claims may need additional state variables to make their prices Markovian.
- A pricing equation can be derived from the discounted price dynamics after specifying the state processes.
- Hedge exposures are found by matching the Brownian risk terms in the derivative and portfolio dynamics.
- Martingale representation guarantees a hedge under the stated framework but does not provide a practical construction by itself.
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# For what options does the "delta hedging rule" apply?
# For what options does the "delta hedging rule" apply?
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?
## Answer by ir7 (score 1, accepted)
https://quant.stackexchange.com/a/63702
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. 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. 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. 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. 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.