A trader in S&P500 options recently faced a choice. He needed to price a 3 month at the money SP500 calls for a client and was unsure as to what should be the correct volatility measure to put in the model? The implied vol of ATM calls were around 14.7%. But the problem was that he believed that the market will jump due a variety of factors. Therefore, he was quite certain that SP500 will either go up 30 points or down 30 points pretty rapidly in the next few days time.

He could take 14.7% and price the calls or he could use a volatility measure that incorporated this movement, a binomial movement of up or down with equal probability. He decided to do that latter.

Say, the SP500 are trading at 1260 (this example is adapted for illustration). The trader needs to price 3 month 1260 calls. But he believes that the index has a 50% chance of moving up by 30 points and 50% chance of moving down by 30 points in the next few days. Thus, inputting the relevant index prices in a trinomial option pricing model with a thousand steps the trader gets the following values:

Given the above the table the fair value of the option - as per trader's estimate - should be given by probability weighted average of the two call prices (under positive and negative jump):

Call Value = 50% * 61.52 + 50% * 27.61 = 44.56

Therefore, with the call price of 44.56 points the trader estimated the 3 month SP500 implied volatility to be **14.21% **(this is calculated using a Black-Scholes model with a risk free rate of 5.40% and a dividend yield of 1.4%). This is the volatility he input in his pricing model (a trinomial model with 1000 steps) to price the 3 month SP500 calls for his client.

**Note **: *The theoretical aspects of the above problem is also mentioned in John Hull's book *.

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