# The Newton-Raphson method simply explained

Meet Alex, a derivatives trader at a major bank. Lately, the stock market has been fluctuating due to global events. Alex comes across a European put option on stock XYZ trading for \$15. The stock itself is currently priced at \$100, the option's strike price is \$100, and it matures in a year. The generally accepted risk-free rate is 5%.

Alex believes this put option might be overpriced. His theory is that even though the market is currently volatile, it might stabilize over the coming year. To verify his hunch, he decides to deduce the option's implied volatility. Knowing this can help him assess if the market is overestimating the stock's future volatility.

Steps:

1. Data Collection:

Alex gathers the following data:

- Stock price: \$100

- Option's strike price: \$100

- Time to maturity: 1 year

- Risk-free rate: 5%

- Observed market price for the put: \$15

2. Implied Volatility Estimation:

Alex uses the Newton-Raphson method combined with the Black-Scholes model for put options. His aim is to pinpoint the volatility that aligns with the observed market price. He initiates with an initial guess of 20% for implied volatility.

3. Analysis:

After running the calculations, Alex determines that the implied volatility is around 30%. When juxtaposing this with historical volatilities and other market insights, he concludes that the market is overestimating the stock's future volatility.

Given his insights, Alex decides to sell a bunch of these put options, surmising that he'll pocket more from the premiums now than he might lose later if the stock market stabilizes as he anticipates. However, to safeguard against unforeseen downturns, he also acquires some cheaper put options on the same stock as a hedge.

5. Ongoing Monitoring:

Alex remains vigilant, monitoring his trades in the subsequent weeks. If he discerns any pronounced price shifts in the options, he recalibrates the implied volatility to see if the market's perspective on future volatility adjusts.

In this adjusted scenario, Alex's strategy of selling put options aligns with his belief that the stock market might stabilize in the future.

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