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Simple Trading In this example, a derivates trader wishes to find the optimal trading strategy for a complex derivative of two publicly-traded biotech stocks (“BioHuge” and “BioSmall”). The future expected volatilities of BioHugeand the future risk-free rates will be modeled as certain but changing. The values will be found by matching the Treasury yield curve rates and the average of the current bid and ask price of American-style call options on BioHuge; data can be found in the Appendix. Building the Theoretical Model Matching the Yield Curve with Risk-Free Rates We matched the yield curve by calculating the forward discount factors. Matching the Option Prices with Expected Volatilities Option prices are matched by creating a modelfor each stock option with the riskfree rates found above. Beginning with the nearest exercise date, we find the expected volatility over that time period that matches the value of the option. This volatility is used to solve for the expected volatility over the next time period which matches the value of the next longest option. The implied volatilities can be found in the Appendix. Finding the Theoretical Price and Exercise Strategy of the Derivative The derivative is similar to an American-style put option of 3 months. It can be exercised early and the payoff is the difference between the current value and the strike price. The main difference for this example is that the value is calculated as the stock price of BioHuge (currently 130) times the stock price of BioSmall (currently 13). The strike price of this derivative is 1500. The derivates trader believes that BioSmall has a volatility of 30% and that the two stock prices have a future expected correlation of 40%, based on past data and our knowledge about the future (e.g., they are launching a joint product for the first time ever). The theoretical price is $64.66 and the optimal strategy is to only exercise at the expiration date. Building the Trading Model Mispricing and Transaction Costs The derivatives trader believes that this derivative could be mispriced by the market. The trader’s experience with illiquid complex derivatives markets tells him that the mispricing in percentage terms (where 100% represents no mispricing) is a mean-reverting process with a volatility of 15% and reversion rate of 450%. The mispricing has no relationship with the general market (i.e., the model property “Mispricing Market Correlation” is zero). This derivative is most-easily traded in blocks of 1000 units. The bid/ask spread is 2% (twice the size of the “Percentage Trading Cost” model property) and the fixed trading cost is $10. The trader believes that he will not be able to trade an entire block at the bid or ask price. His buy and sell orders will influence the price and he will end up getting an average of the starting and ending price. If he buys or sells 1 block, the price will change by 0.5% (the “Traded Block Percentage Change” model property). If he buys or sells 2 blocks, the price will change by 2% (the “Double Traded Block Percentage Change” model property). He believes that in such an illiquid market, he won’t be able to get a reasonable price by buying or selling any more than 2 blocks per month. The current mid-trading price of the derivative is $64.66. Optimal Trading Strategy The optimal strategy begins by buying 1 block. Then if next month the derivative is mispriced on the low side, buy 2 more blocks. If the derivative is priced on the high side and BioHuge and BioSmall both increased in price, hold onto the position. If the derivative is priced on the high side otherwise, sell the purchased block. After that, the strategy varies considerably depending on the circumstances. The True Shareholder Value of this trading strategy to the trader’s firm is $3,215. Analysis The NPV and optimal trading strategy are sensitive to the current situation. For example, the initial trading choice (wait, buy 1 block, buy 2 blocks) changes depending on the current derivative price. A break-even analysis tells the trader at what derivative prices he would choose various trades. Derivative Price Range <= $63.03 $63.04 - $65.04 >= $65.05 Optimal Initial Trading Choice Buy 2 Blocks Buy 1 Block Wait Changing the block size increases the NPV by about $3 per unit. Below are analyses of how the other model properties affect the NPV for model properties that are in terms of percentages and in terms of dollars. Model Property Percentage Trading Cost BioSmall Volatility Double Traded Block Percentage Change Mispricing Volatility Traded Block Percentage Change Correlation Mispricing Market Correlation Mispricing Reversion Rate BioSmall Growth BioHuge Growth d $NPV / d % -1283 1201 -475 356 -161 124 1 +0 0 0 Model Property BioSmall Initial Value BioHuge Initial Value Initial Derivative Price Strike Price Fixed Trading Cost d $NPV / d $ -15992 -1599 -442 158 -1 Discussion of Price-Matching Traded Assets By default, the Market and a single risk-free rate (see Market) are considered in the Provisdom Decision Platform. This is an approximation as the risk-free rate is not actually constant. For those decisions in which more accuracy is required, the market’s expected future risk-free rates can be determined from the yield curve. For models that rely heavily on the accuracy of the risk-free rates, an interest rate (IR) model can be used to capture the uncertainty in the future risk-free rates. Using one these IR models, the parameters of the IR model can be matched against traded IR derivatives. IR models add a level of complexity to the decision model. Similarly, a single or set of expected future volatilities of an asset price (e.g., stock) can be determined from the traded derivatives (e.g., options) on that asset. Again, a more complex model of the expected growth rates and volatilities can be used to more closely match the currently traded derivatives prices. In the example above, the model could be updated frequently with the current prices of stock options and the treasury yield curve rates. A more accurate model could be built to model the uncertainty or to “smooth out” the transitions in the future risk-free rates and the future expected volatilities. Appendix Treasury Yield Curve Rates The following rates represent the current Treasury yield curve: 1-month 3-month 6-month 1-yr 2.02% 1.91% 1.93% 2.17% BioHuge Traded Prices BioHugeis currently trading at 130. Call options with a strike price of 130 and varying lengths have the following averages between their ask and bid prices: 1-month 4-month 4.20 7.90 7-month 10.65 Implied Volatilities The implied volatilities were calculated as follows: 0-1 month 1-4 month 4-7 month 28.280% 23.193% 25.812%
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