Dynamic Hedging with Replicating Portfolios

Introduction

In our last lesson, we mastered the 5-step method for pricing a derivative by constructing a replicating portfolio. We discovered that by combining a specific quantity of the underlying stock ( $Δ$ ) with a risk-free asset, we could create a portfolio that perfectly mimics the option's payoff in a single time step. The cost of setting up this portfolio, by the principle of no-arbitrage, had to be the option's fair price.

Today, we will build directly on that foundation to explore what the composition of this portfolio truly represents. The hedge ratio, $Δ$ , is not merely a static number for a one-off calculation. It is the key to a continuous risk management process. We will interpret the replicating portfolio as a dynamic hedging strategy, a method for continuously adjusting our holdings to remain insulated from risk as market conditions change. This lesson will bridge the gap between the theoretical concept of replication and its practical application in managing option risk over time.

From a Static Position to a Dynamic Strategy

In the one-period binomial model, our task was simple: build a portfolio at time $t = 0$ that would be worth the same as the option at time $t = 1$ , no matter what happened. This involved calculating a single hedge ratio, $Δ$ , and holding that position until expiry.

But what happens in a multi-period world, where the option has a long time until expiration and the underlying stock price can move through many states? The sensitivity of an option's price to the stock's price is not constant. It changes as the stock price moves and as time ticks away.

This means a "set it and forget it" or static hedge is insufficient. A portfolio that perfectly hedges an option today will become an imperfect hedge tomorrow. To maintain the risk-free nature of our combined position, we must continuously adjust the number of shares we hold in relation to our option position. This process of periodic rebalancing is the essence of dynamic hedging.

To begin, let's formalize the concept of our hedge ratio, $Δ$ , which in the world of derivatives is known as delta.

Binomial Trees — Econ 236 2016.03.28 documentation - Eric Aldrich

This document from Eric Aldrich's course notes at UC Santa Cruz provides a clear and concise definition of delta and dynamic hedging in the context of our binomial trees.

Please read the sections titled Delta and Call Deltas. As you read, focus on these key points: Delta (\Delta) is the ratio of the change in the option's value to the change in the stock's value. It tells us how many shares are needed to hedge one short option. The delta of an option is not constant; it changes as the stock price and time evolve. The act of rebalancing the portfolio to maintain the hedge as delta changes is called dynamic hedging.

Static vs. Dynamic Hedging

The distinction between a static and a dynamic hedge is crucial. The following video provides a very clear explanation of the difference and the trade-offs involved.

Delta Hedging Explained: Options Trading Strategies

This video by Ryan O'Connell clearly distinguishes between static and dynamic hedging approaches.

Please watch the segment from 9:02 to 10:22. Note the key differences: Static Hedge: Set up once and left alone. It's simpler but allows risk exposure to grow over time, leading to higher potential upside and downside. Dynamic Hedge: Continuously adjusted. The goal is to keep the portfolio's total delta as close to zero as possible (a "delta-neutral" position), minimizing exposure to price changes and thus resulting in lower upside and downside.

In the context of the replicating portfolio, we are implicitly performing a dynamic hedge. The "fair value" of the option at any point in time is the cost of the replicating portfolio. To ensure the replication holds, we must adjust the portfolio's composition at every step.

Dynamic Hedging in the Binomial Tree

Let's see how this works in the multi-period binomial model we've been using. Recall the formula for delta at any given node:

Δ = \frac{f _{u} - f _{d}}{S _{u} - S _{d}}

Here, $f_{u}$ and $f_{d}$ are the option values in the next period's up and down states, and $S_{u}$ and $S_{d}$ are the corresponding stock prices.

In a multi-period tree, we apply this logic recursively.

At time $t = 0$ , we calculate $Δ_{0}$ based on the option values at $t = 1$ . We buy $Δ_{0}$ shares to hedge our short option.
At $t = 1$ , the stock price has moved either up or down. Our option's delta has now changed.
- If the price went up, we are at the "up" node. We must calculate a new hedge ratio, $Δ_{u}$ , based on the option values at $t = 2$ . We then adjust our holding of shares from $Δ_{0}$ to $Δ_{u}$ .
- If the price went down, we are at the "down" node. We calculate $Δ_{d}$ and adjust our share holding accordingly.
We repeat this process at every time step until the option expires.

The document from the previous reading provides excellent numerical examples of this. Let's look at the put option deltas.

A one-step binomial model illustrating the calculation of hedge ratios for a call and put option. Dynamic hedging extends this single step into a continuous process.

In the LINK resource, under the heading "Put Deltas," the author calculates the following:

$Δ_{0} = - 0.4024$
$Δ_{u} = - 0.1667$
$Δ_{d} = - 1$

This shows numerically that the required hedge ratio changes at each stage. At time 0, we must short 0.4024 shares for every put option we own to create the hedge (since put deltas are negative, hedging a long put involves buying stock, while hedging a short put involves shorting stock). If the stock price moves up, our new delta is -0.1667, so we must adjust our position. If it moves down, the delta becomes -1, and we must rebalance again.

This rebalancing must be done in a self-financing way. This means that adjustments to the portfolio are funded by its existing components (e.g., selling some stock to raise cash, or using the cash component to buy more stock), not by injecting or withdrawing external funds. The value of the portfolio just before rebalancing is the same as its value just after.

For a more rigorous look at how this recursive, self-financing strategy works, you can review the following resource.

Hedging and portfolio replication in the binomial tree model.

This PDF from a Rutgers course provides the mathematical framework for dynamic, self-financing replication.

This is a more technical document, so focus on the conceptual flow rather than every detail of the equations. Start by reading the section on the multi-period model to understand the setup. Then, read the paragraph explaining the portfolio update equation (2) and the definition of a self-financing portfolio. Finally, skim the section that explains how to derive the replicating portfolio by applying the one-period delta hedging result at each step. The key idea is that by starting with the option's price (V_0) and always holding the correct delta (\delta_t = \Delta_t), the portfolio's value (\Pi_t) will always match the option's theoretical value (V_t) at every node.

A Practical Example of Dynamic Hedging

To solidify your understanding, let's watch a practical example where a trader maintains a delta-neutral portfolio over several weeks by dynamically hedging a short call option position.

Delta Hedging Explained: Options Trading Strategies

This section of the video walks through a week-by-week rebalancing example from John Hull's famous textbook.

Watch the dynamic hedging example from 10:22 to 13:53. Pay attention to how the trader's actions follow the principles we've discussed: Initial Hedge: They are short 100,000 call options with an initial delta of 0.522. To create a delta-neutral portfolio, they must buy 100,000 \times 0.522 = 52,200 shares. Rebalancing: In week 1, the stock price falls, causing the option's delta to fall to 0.458. The required share holding is now 100,000 \times 0.458 = 45,800. To rebalance, the trader must sell 52,200 - 45,800 = 6,400 shares. Continuous Adjustment: Notice how as the stock price fluctuates week by week, the delta changes, and the trader must continuously buy or sell shares to keep the portfolio's net delta close to zero.

This example perfectly illustrates that the replicating portfolio is not a static object but a continuous strategy. The goal of the hedger is to ensure that at any point, the value of their portfolio (option + shares + cash) is immune to small changes in the underlying stock price.

Conclusion

In this lesson, we elevated our understanding of the replicating portfolio from a one-time pricing tool to a continuous risk management strategy. We have seen that to perfectly replicate an option's payoffs over time, one cannot simply "set and forget" a hedge. Instead, one must dynamically adjust the portfolio to account for the ever-changing sensitivity (delta) of the option to the underlying asset.

Key Takeaways:

The hedge ratio $Δ$ is the option's delta, a measure of its price sensitivity to the underlying.
Because delta is not constant, a hedge must be continuously rebalanced. This process is called dynamic hedging.
A self-financing dynamic hedge involves adjusting the portfolio's composition (shares and cash) without adding or removing external funds.
In the binomial model, this means recalculating delta at each node and adjusting the number of shares held to maintain a risk-free, or delta-neutral, position.
Therefore, the replicating portfolio is best interpreted not as a fixed basket of assets, but as a dynamic hedging strategy.

In our next lesson, we will apply these discrete-time pricing concepts to more complex derivatives, specifically barrier options. We will see how the dynamic hedging strategy becomes particularly crucial—and challenging—when the option's very existence depends on the path taken by the underlying price.

Recommended resources