Delta hedging in fx options
While it is useful to get comfortable with the concept of Delta Hedging, most academic finance specialization programs provide cursory treatment of option price sensitivities and Greeks. Delta hedging as a concept is covered within the foundation of Black Scholes pricing at a theoretical level single step or two step binomial trees however actual implementation of a live delta hedging program is something an instructor rarely has time for in the first course on derivative products. Figure 1 Delta Hedging using Monte Carlo Simulation. Both Mark Broadie and John C Hull have put together illustrative sheets that simulate the actual process of Delta hedging for a call option. This session will help us walk through the basic model and then extend the model in later posts options answer questions around profitability and model behavior. Figure 2 Delta Hedging options Tracking Error between Hedging portfolio and options. We will also track the error in the model and discuss its sources. There is a mismatch and the tracking error shows that. Since then the essential model has remained hedging. The Hull approach is similar to the Broadie method and if you are comfortable with either of the two you will be fine. For back ground information on Greeks, see the Option Greeks Quick Reference Guide as well as the Derivatives Greeks Interview Guide. Figure 3 Delta Hedging — The baseline model and simulated values. If you are unfamiliar with the basics of Monte Carlo Simulation please see the Monte Caro Simulation Training Guide below as well as our complete collection of posts on Monte Carlo simulation before proceeding further. For the purpose of our simulation we will start off with Barclays Bank I know, I know and assume that the bank will pay no dividends over the life of the option. Figure 4 Delta Hedging — Barclays bank price chart. We will assume that the spot price is As discussed above, the stock will pay no dividends. Figure 5 Delta Hedging — Key Assumptions. Using the above assumptions we go ahead and simulate a path of Barclays share price over the next one year using a single month interval. We are assuming that the recurring options period before the option delta is evaluated and the replicating portfolio is delta is 1 month. The approach will give us 12 prices at monthly options, and 12 rebalancing points. Using our old friend the discrete edition of the Black Scholes equation we go ahead and simulate Barclays share price for the next 12 months. Figure 6 Simulating equity prices. The result should look as under. However our numbers will not match since the random seed used by the Excel RAND function are going to be different for everyone. Figure 7 Delta Hedging — Simulated price series. While the first eleven time steps are equidistant, the 12th one is a little short. We have done this by design because we want to examine the option the day before it expires, hence the value. The actual stock price simulation with the original discrete formula and the Excel implementation is shown below. Figure 8 Delta Hedging — Monte Carlo simulation — Excel implementation. The next step is to calculate d1, d2 and delta values based on the simulated stock prices at each step. The value of d1 and d2 are given by the standard Black Scholes implementation. Option Delta is simply N d1 or NORMSDIST d1. Now that we have option delta for each simulated stock price at each time step, it takes a simple multiplication step to calculate Dollars in stock Delta x S. However total borrowing requires a more involved calculation. At time zero when the option is written total borrowing is given as the difference between Dollars in Stock and the premium received for the option. This is given in the first cell at time zero. It is the second cell at time one where the calculation gets a little messy. Figure 10 Delta Hedging — Replicating portfolio components. One way of dissecting this calculation is to take a quick and close look at the top four rows of our delta hedge table and just do a simple step by step calculation that shows us how the total borrowing figure changes from one rebalancing period to the next. Its an essential step without which you cannot decode the delta hedging sheet. Figure 11 Delta hedging hedging dissecting total borrowing. How does this balance change at time step 1? At time step 1, the underlying stock has moved to In addition option delta has moved from. The combined position after the new purchase is So what is the incremental amount that was borrowed to finance the hedge? How did we hedging the exact number? That is the new net incremental borrowing. When delta and underlying prices fall, the formula will release funds. When they rise it will require funds. But there is one more step delta the total borrowing calculation is complete. What about the previous balance? Balance that was borrowed at step zero. We owe accrued interest on it for one period at the one period time step rate. When you put all of this together you end up with the formula used for calculating total borrowing balance at time step 1 in the delta hedge sheet. The same process is used to calculate the total borrowing balance at step 2, step 3 and onwards. Figure 12 Delta hedging — Marginal borrowing at each time step. You can clearly see now that while the replicating portfolio is doing a reasonable job of tracking the option value, there is a clear error in tracking, which moves up and down depending on how much in or out of money the option is. Figure 13 Delta hedging — the final picture. To calculate option value we use the standard Black Scholes formula for a non-dividend paying stock. Figure 14 Delta hedging — Option value — excel implementation. Now that the underlying simulation model is ready for delta hedging, here is a list of questions that we would like to answer. If our call is hedged by Delta x S — B, what would be required to hedge a put? Do you actually end up making money in this business? What are the sources of income and expense? How is that calculated? How do other drivers impact profitability? We will try and answer some of them in our posts later this month. Privacy Policy Site Map. ALM, Risk and Simulation Models — Training, Study Guides, Templates. Understanding Delta Hedging options using Monte Carlo Simulation in Excel Oct 23, by Jawwad Farid in Computational Finance Building a Monte Carlo Simulation model for Delta Hedging Delta in Excel While it is useful to delta comfortable with the concept of Delta Hedging, most academic finance specialization programs provide cursory treatment of delta price sensitivities and Greeks. Figure 1 Delta Hedging using Monte Carlo Simulation Both Mark Broadie and John C Hull have put together illustrative sheets that simulate the actual process of Delta hedging for a call option. Figure 2 Delta Hedging — Tracking Error between Replicating portfolio and options We will also track the error in the model and discuss its sources. Figure 3 Delta Hedging — The baseline model and simulated values The primary model simulates: Delta Hedging Model using Monte Carlo Simulations — Assumptions Figure 4 Delta Hedging — Barclays bank price chart We will assume that the spot price is Figure 5 Delta Hedging — Key Assumptions Using the above assumptions we go ahead and simulate a path of Barclays share price over the next one year using a single month interval. Delta Hedging Model — Monte Carlo — Simulating the stock price Using our old friend the discrete edition of the Black Scholes equation we go ahead and simulate Barclays share price for the next 12 months. Figure 6 Simulating options prices The result should look as under. Figure 7 Delta Hedging — Simulated price series While hedging first eleven time steps are equidistant, the 12th one is a little short. Figure 8 Delta Hedging — Monte Carlo simulation — Excel implementation Delta Hedging Model — Calculating Delta for our Simulation model The next step is to calculate delta, d2 and delta values based on the simulated stock prices at each step. Figure 10 Delta Hedging — Replicating portfolio components One way of dissecting this calculation is to take a quick and close look at the top four rows of our delta hedge table and just do a simple step by step calculation that shows us how the total borrowing figure changes from one hedging period to the next. Option delta is 0. Figure 13 Delta hedging — the final picture To calculate option value we use the standard Black Scholes formula for a non-dividend paying stock. The Excel implementation is shared below. How is that factored here? Can options also hedge them using a similar approach? Call OptionsCase StudiesDelta HedgingDerivative tradingDerivatives trainingEditors ChoiceMonte Carlo simulationOption hedge rebalancing. Jawwad Farid has been building and implementing risk models and back office systems since August Working with clients on four continents he helps bankers, board members hedging regulators take a market relevant approach to risk management. He is the author of Models at Work and Option Greeks Primer, both published by Palgrave Macmillan. Jawwad is a Fellow Society of Actuaries, FSA, Schaumburg, ILhe holds an MBA from Columbia Business School and is a computer science graduate from NUCES FAST. He is an adjunct faculty member at the SP Jain Global School of Management in Dubai and Singapore where he teaches Risk Management, Derivative Pricing and Entrepreneurship. Popular Posts 1 Capital Allocation Calculating Economic Capital — A Case Study. Commodities The knives are out in the Oil market. Computational Finance Implied and Local Volatility Surfaces in Excel — Final steps. 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