The Adjusters approach lowers the pricing error that often comes from difficult model selection, poor calibration, or simply the lack of precision inherent to numerical methods.
In this Monthly Focus, we present LexiFi’s Smart Adjusters: a modification of the Adjusters method that was introduced in 2002 by Patrick S. Hagan, but rarely used due to its high computational cost. We suggest a modification of Hagan’s method implying another choice for the weights of the calibration instruments.
We have enhanced the previous classical approach to make it generic, fast, and adapted to equity/FX structured products.
Derivatives and structured products are often too complex to be priced with closed-form solutions. Numerical methods such as Monte Carlo or Partial Differential Equation are needed, and advanced pricing models are frequently adopted. But these methods are slower than closed-form solutions, particularly for sophisticated models! This issue is accentuated for risk metric computations such as the Value at Risk (VAR) or stress testing. Of course, simpler models such as standard Black-Scholes would be fast, but not satisfactory in terms of precision.
To remain fast while preserving accuracy, we suggest using LexiFi’s Smart Adjusters method.
Using the Adjusters method means decomposing a given financial contract in two parts as follows:
Where C1 would be priced based on conventional pricing models, while C2’s valuation has either a closed-form solution or an observed price on the market: it is the adjuster.
LexiFi’s research has improved the speed and genericity of the standard Adjusters method. Our Smart Adjusters approach is close to the control variates variance reduction method. We use it to provide an efficient and generic choice of control variables making it fully automatic in LexiFi Apropos.
"LexiFi Smart adjusters provide faster and more precise pricing results!"
Let’s price a structured product for which the Local Volatility (LV) model would be best suited. So we take the LV model as the Reference Model. In this example, we first price the product using the Black-Scholes (BS) model. Then, we use LexiFi’s Adjusters method to get the Adjusted BS price.
In the table below we compare both pricing results with the calculated reference price.
We find the adjusted BS price (96.45%) to be closer to the reference price (96.34%) with a 0.11% difference. The BS model without the Adjusters method has overpriced the product by 1.94%. Also, the Adjusted BS is significantly faster than the LV.
Let’s calculate the Value at Risk (VaR) on a portfolio of 94 equity and FX structured products. We first use the Local Volatility (LV) model and take it as a Reference Model. We then use the Black-Scholes (BS) model, and finally, we apply LexiFi’s Adjuster method to get the Adjusted BS results.
When comparing the BS with the Adjusted BS results, we find that the VaR computation time is faster and the computing error is twice lower with the Adjusted model.
We can improve product pricing precision even with the simplest models (Black-Scholes), by transferring model risk to some hedging instruments using LexiFi’s Smart Adjusters method. This approach dramatically reduces the computational cost and total pricing time.
Vivien Begot, LexiFi’s Head of Quantitative Research provides numerical results on a traded structured product portfolio in the Smart Adjusters white paper. These results validate the method of adjusting the price and reducing the Monte Carlo variance. The generic choice of the hedging instruments makes the Smart Adjusters approach fully generic and automatic in LexiFi Apropos.