**More and more investors insist on guarantees on the investments. Theses investments are often created using options or dynamic strategies like CPPI. Recently, these strategies were made available in secularized form: Leveraged Exchange Traded Funds (LETF) and Exchange Traded Notes (ETN). Also, life insurance instruments often include guarantees on funds like Variable Annuities. In this article, we will look at the change in stock price dynamics due to hedging activities.**

## Overview

We will look at hedging of a popular put option. In a previous article about pricing and hedging of options, we saw that the issuer can replicate an option sold using a hedging strategy. For replicating a put option the issuer has to buy the underlying if the price rises and sell if the price is falling. (By the way: This is also true for call options, but here, we focus on puts to keep it simple.)

## Case Study: Hedging a Put Option and Market Feedback Effect

In this case, we use Theta Suite and ThetaML for analyzing the market feedback. First, we need the trading strategy, which is available in closed form:

model EuropeanPutAnalytic import S "Stock price" import K "Strike price" import Sigma "Volatility" import R "Interest rate" import T "Maturity time" export V "Black-Scholes price" export Delta d1 = ( log(S/K) + (R + (Sigma^2)/2)*T ) / (Sigma * sqrt(T)); d2 = d1 - Sigma*sqrt(T); V = -S * normcdf(-d1,0,1) + K * exp(-R*T)*normcdf(-d2) Delta = normcdf(d1,0,1) - 1 end

Now, we assume a linear dependency between a large trade and the change of the market price. That means, we use the following stock price model with Geometric Brownian motion:

model p_GBM export GBM_with_feedback export Delta TimeGrid = [1/250:1/250:1] % Daily time steps r = 0.02 % Risk free rate InitialValue = 100 % Initial Stock price Sigma = 0.2 % Volatility without feedback effect Multiplier = 30 % Size of price change GBM = InitialValue % GBM is the stock price process Delta_old = 0 % initial position in option hedge % loop through all time steps loop t:TimeGrid theta t-@time GBM = GBM * exp((r-0.5*Sigma^2)*@dt + Sigma*sqrt(@dt)*randn()) % Import Delta of Put option with Strike K=100 and Maturity T = 1.5 call EuropeanPutAnalytic export GBM to S, 80 to K, Sigma, r to R, 1.5-@time to T import Delta from Delta ChangeInDelta = Delta-Delta_old Delta_old = Delta % Change GBM according to change in hedge portfolio GBM = GBM + Multiplier * ChangeInDelta GBM_with_feedback = GBM end end

Using Theta Suite, we can now evaluate the p_GBM model by clicking on “Generate and Run”. Then, we see the Theta Suite Result Explorer and select “Probability Density”:

And we see the probability density distribution of the stock price in time: A bi-modal distribution.

Looking at the individual paths, we see that the prices rise and fall quicker if they move towards the options strike at 80. In the following plot, we see this region high-lighted in red.

## Conclusion

Using Theta Suite, it is simple to analyze the feedback effect of hedging an option: The hedging of capital market guarantees can significantly change the price process of the underlying. Especially, in illiquid markets, this effect can dominate the behavior of the stock prices. Looking at the volumes of traded options, the rising volatility in the markets might also be explained by increased hedging activities.

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