# Implied Volatility

We have seen that both the BSM model and Black model require the parameter, $$\sigma$$ which is the volatility of the underlying asset price. However, future volatility cannot be observed directly from the market but rather estimated.

One way of estimating volatility is by using an observed option price from the market and determining the volatility in line with this price. The values of other model parameters, including the underlying share price,  the risk-free rate of interest, and the dividend yield, can be observed. This makes it possible to solve for the volatility as it will be the only unknown parameter in the formula. The resulting estimate is known as the implied volatility.

The values of both European options are directly related to the volatility of the underlying asset. A call holder gains from the price increase but has limited downside risk. Moreover, the holder of put gains from the price decrease but has limited upside risk. Therefore, the value of options increases with an increase in volatility.

Lastly, implied volatility gives an understanding of the investor’s opinions on the volatility of the underlying asset. Higher implied volatility relative to the investor’s volatility expectations suggests that the option is overvalued. Additionally, implied volatility helps in revaluing existing positions over time.

## Question

An options dealer offers to sell a one-month in-the-money put on PayPal Holdings at 20% and a two-month at-the-money call on the SET index option at 15% implied volatility. Based on the current forecast, an options trader believes that PayPal volatility should be closer to 16%, and SET volatility should be closer to 22%. To benefit from his views, the trader should most likely:

1. Buy the PayPal put and the SET call
2. Sell the PayPal put and the SET call
3. Sell the PayPal put and buy the SET call

## Solution

The trader believes that the PayPal put is overvalued and that the SET call is undervalued. Thus, he expects the PayPal volatility to fall and that of SET to rise. Therefore, the PayPal put would be expected to decrease in value while the SET call would increase in value. The trader would then

As a result, the FTSE call would be expected to increase in value, and the VOD put would be expected to decrease in value. The trader would then Sell the PayPal put and buy the SET call.

Reading 38: Valuation of Contingent Claims

LOS (n): Define implied volatility and explain how it is used in options trading.

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