 This is the part where we need to make assumptions about the dynamics of price returns under the statistical measure and apply econometric tools to estimate parameters of the model for returns under the statistical measure. As the SVJ model implies a higher P L volatility, the optimal maturity is shorter than the optimal maturity under the diffusion process. However, for trading strategies in vanilla options held to maturity, the risk of returns volatility under the statistical measure is the primary factor that we need to model, while the implied volatility serves only as a measure. As we expect, using the price and delta bands increases the realized Sharpe ratio as these strategies allow savings on the realized transaction costs. I assume that the trading strategy is based on a proprietary view on the volatility dynamics coming from either systematic strategies or discretionary views. I can apply my approach for a quantitative assessment of risk-reward for options with different maturities. I show that the total expected transaction costs are proportional to the square root of the hedging frequency times the BidAskSpread and some constant C: Expected Costs (HedgingFrequency) BidAskSpread *sqrt(HedgingFrequency C The total expected P. As a result, the total realized P L is the sum of single period P Ls: Total P L Sum of P L (n) over rebalancing times As a result, the total P L is a sum of squared. In that case, the IR equals.4. Information ratio Excel, finally, we implement the calculation of the information ratio in Excel. Volatility trading refers to investment and trading strategies that provide the exposure to the implied and realized volatilities of underlying asset, without assuming any exposure to the performance of the underlying asset. Since price and delta based rebalancing imply a random number of re-hedging times, the P L volatility as well as the realized transaction costs will be different when compared to the strategy with re-hedging at fixed time intervals.

We see that the result is again sensitive to the model assumption about the returns dynamics. By increasing the hedging frequency the P L volatility declines but the reduction becomes smaller and smaller as the hedging frequency increases. What is a good information ratio? The expected P L does little depend on the model assumptions since different models would imply both similar values of the expected returns volatility (if these models are estimated using the same data set) as well as similar values of the expected transaction costs. It is an easy and fast way to detect whether a security or price series is trending, mean reverting or following a random walk. Using the approximations for the price and delta bands, we can first find the optimal hedging frequency for the time-based re-hedging and then apply these approximations to get the equivalent variance ratio trading strategy price and delta bands.

(1999 When You Cannot Hedge Continuously: The Corrections of Black-Scholes, risk, 12(1 82-85. If you hedge 5,000 worth of the equity with a currency position, your hedge ratio.5 (5,000 / 10,000). For non-vanilla products, which require some vega/volatility hedging, and for strategies, when the option position can be terminated before its maturity, the risk of changes in the implied volatility can be significant. Matlab files with sample code for these computations are available at References Derman,. While this analysis can also be performed using Monte Carlo simulations, my analytic approach provide a fast and an accurate way to estimate the risk-reward characteristic of a delta-hedging strategy and my method can be implemented in a system for real time computations. Figure 5 illustrates the optimal Sharpe ratio as function of option maturity for a fixed spread between the implied and realized volatilities. Example: test1 VarianceRatio(close, 40, 200) 1; Check whether the security shows a tendency to form trends or not.

We ignore the higher order terms as their impact is much smaller than the impact from the first three terms (this observation is valid for near at-the-money options). Now we want to choose the maturity that provides us with the maximum Sharpe ratio. Intuitively, the diffusion process will imply that a larger part of the returns volatility can be delta-hedged with the residual volatility coming only from the spread between implied and realized volatilities. A positive information ratio, as is the case in the following figure, means that the manager is outperforming the benchmark. The analytic solution for the optimal hedging frequency can be found in my paper. #### Information ratio - Breaking Down Finance

Derman (1999) first makes the observation that, when the implied and realized volatilities are equal with zero spread between them, the part of the P L volatility that we can eliminate by delta-hedging is inversely proportional to the hedging frequency. In the abovementioned paper, I consider the four dynamics for the returns process under the statistical measure: the diffusion process as in the Black-Scholes-Merton model, the diffusion process with jumps as in the Merton model, the stochastic volatility. First, my method provides a very good approximation to the actual Sharpe ratio obtained by Monte Carlo simulations under the time-based re-hedging. The reason is that the volatility of the P L increases faster relative to the expected P L for OTM options, so that the trading strategies in OTM options have smaller optimal Sharpe ratios. Where the first term is the time decay measured by the option Theta, the second term is the delta term related to the change in price measured by the option Delta, and the third term is the option Gamma. El Karoui,., Jeanblanc,., Shreve,.

Figure 3 shows the expected P L volatility under the diffusion model by the red dotted line and under the SVJ model by the red dashed line. The hedge ratio compares the value of a position protected through the use of a hedge with the size of the entire position itself. Figure 2 illustrates the P L volatility by the red continuous line. Suppose we have a strategy that earned an average of.a. The other three quantities are computed using the Monte Carlo simulations. Clearly, the higher the IR the better. If option sellers seek for uniform Sharpe ratios across options strikes, option sellers would demand a higher level of the implied volatility for OTM options to compensate for the higher P L volatility of delta-hedged short volatility strategies in OTM options. I remind that, for simplicity, I assume that option implied volatility does not depend on price changes under the sticky strike volatility dynamics. Figure 1, we see that the P L is zero when the break-even price return is about plus/minus. As a result, we expect different Sharpe ratios and optimal hedging frequencies under different assumptions about the returns dynamics. The first course is the difference between implied and realized volatilities. I also assume that the option position is related to a single expiry and the position is held to maturity, so that the primary variance ratio trading strategy risk to the terminal P L depends only on the realized path. After calculating the optimal hedge ratio, the optimal number of contracts needed to hedge a position is calculated by dividing the product of the optimal hedge ratio and the units of the position being hedged by the size of one futures contract.

#### Variance, ratio, tests Archives - quantitative research AND

Share This - Share, download, you have to log in to bookmark this object. We can clearly see the humped shape of the Sharpe ratio. Taylor expansion in the time passed from the last hedge rebalancing, which is denoted TimePassed, and in the price change, which is denoted by PriceChange: Option Price change -Theta*TimePassed Delta*PriceChange.5*Gamma*PriceChange2. In particular, we show how to determine the benchmark return, the portfolio return, and the volatility of the difference between both (which is sometimes called the tracking error ). Impact of model assumptions To compute the expected Sharpe ratio and the optimal hedging frequency, we need to estimate the expected volatility of asset returns. I remind that, under the strategy with the price and delta bands, the actual number of re-hedging times is different across different realized price paths. Also, I have demonstrated that the Sharpe ratio of the delta-hedging strategy can be improved by incorporating the price and delta bands for the rebalancing of the delta-hedge and provided analytical approximations for computing the optimal bands to apply in my optimization approach.

On the other hand, the realized volatility is computed using available time series of price returns. The airline company expects to purchase 15 million gallons of jet fuel over the next year, and wishes to hedge its purchase price. Furthermore, the volatility of this difference was 5 annualized. We observe in option markets that typically longer-dated options imply a higher spread, however longer-dated options imply a higher P L volatility and transaction costs. On this page, we discuss the information ratio definition, we give some numerical examples, and finally we provide an Excel file at the bottom of the page that illustrates how to calculate the information ratio in Excel. What is volatility trading? For example, imagine you are holding 10,000 in foreign equity, which exposes you to currency risk. Whereas the benchmark in the case of Sharpe ratio is the risk-free rate r f, in the case of the IR it is the relevant expected benchmark return to which the manager is benchmarked. The appraisal ratio measures the level of outperformance of an active strategy over the benchmark. We see that in the absence of the volatility smile, which would assign higher volatility for out-of-the-money (OTM) puts and calls, the optimal Sharpe ratio declines for out-of-the-money options. 19-59, m/abstract1865998 Sepp,.

#### Variance ratio as a measure of backtest reliability Futures Magazine

That is why I would like to highlight some of my research and discuss my approach under the discrete time setting and the transaction costs to optimize the delta-hedging. (1998 Robustness of the Black and Scholes formula, Mathematical Finance, 8(2 93-126. This result is remarkable because it holds under different dynamics of price returns under the statistical measure. Futures contracts are essentially investment vehicles that let the investor lock in a price for a physical asset at some point in the future. Delta-hedging with transaction costs is first considered by Leland (1985). Outperforming the benchmark is not an easy task, of course. .

When we apply models with jumps and stochastic volatility, the residual part of the P L volatility increases because jumps and stochastic volatility cannot be hedged away by the delta hedging. Information ratio example, using the above definition, we can calculate the IR for any strategy, as long as we have a sufficiently long history of daily or monthly returns for both the strategy and the benchmark portfolio. In that case, the manager is underperforming the benchmark. We see that the stochastic volatility with jumps (SVJ) model implies a higher volatility of the P L which is also less sensitive to the hedging frequency. You could enter into a hedge to protect against losses in this position, which can be constructed through a variety of positions to take an offsetting position to the foreign equity investment.

#### Trading : Time-Consistent Mean-, variance

This is a significant topic in itself, which would deserve a separate post. The second source is the un-hedgeable delta risk due to jumps and gaps in the price. At the beginning, I assume that the re-hedging is applied at equidistant times with the time periods equal to the option maturity time divided by the hedging frequency. The residual volatility, which cannot be eliminated by delta-hedging, arises from the two sources. Optimal maturity for variance ratio trading strategy trading strategy Another interesting implication of the obtained results is the following. Figure 2 What we are interested in, is to find the optimal hedging frequency that maximizes the expected Sharpe ratio of the volatility trading strategy. Total P L I assume that we keep the short position until the option maturity. As an example, I consider an at-the-money (ATM) straddle, which constitutes a long position in an ATM call option and an ATM call option with strikes close to the current spot price at the contract inception. Rene Koch, the variance ratio indicator measures the degree of mean reversion or trendiness in a time series. BSM equation to represent the option Theta using the option Gamma as follows: as a result, the P L of the short delta-hedged position in the straddle becomes: P L (n) (ImpliedVol2 RealizedVol2) * where RealizedVol is the realized volatility computed. On one hand, the implied volatility is a forward looking estimate of the returns volatility implied from options market prices. Finally, we see that the primary risk driver of the P L is the realized volatility of price returns. An important consideration here is the assumption that we can compute the delta-hedge consistently with the statistical dynamics of price returns, so that we can eliminate the exposure to the realized drift. Suite 11, Second Floor, Sound Vision House, Francis Rachel Str.

This means that 50 of your foreign equity investment is sheltered from currency risk. While it is customary to assume a continuous-time hedging in most of the industrial applications and academic literature, the delta-hedging in practice is applied in the discrete time setting. By Brian Brown, 2711 days ago. Rene Koch, the variance ratio indicator measures the degree of mean reversion or trendiness in a time series. It is an easy and fast way to detect whether a security or price series is trending, mean reverting or following.