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I am preparing for an interview on Monday and I came across a practice question which has me stumped.

"The implied volatility of a 1 year option is 20% and the implied volatility of a 2 year option on the same underlying is trading at 10%. Estimate the implied volatility of a one year option forward starting in 1 year."

Its basically asking, "what is the annualized volatility the market expects for the second year?" correct?

We can't just do $\frac{0.2 + x}{2} = 0.1$ because volatilities are not additive correct? So we have to convert to variances first. So now its $\frac{0.04 + x}{ 2} = 0.01$, which doesn't make any sense.

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    $\begingroup$ You're correct, total variance should be a monotonic increasing function. $\endgroup$ – will Nov 19 '16 at 17:22
  • $\begingroup$ The only thing I can think of is you have an interest rate curve that looks really strange, but I still don't think that will do it. $\endgroup$ – barrycarter Nov 21 '16 at 21:11
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The answer given above is probably the intended answer to the interview question. However it is not the whole story. Suppose the 1yr and 2yr options are both at the money and struck at usd100 (and that there are no dividends and interest rates are zero, to make things simple). Then if the options are priced at 20% and 10% respectively, the implied price of a usd100 call one year forward, is negative (an arbitrage, as stated). However, the price of a 'then at the money' 1yr option whose strike is determined 1yr from now, cannot be determined from the information we have. One cannot replicate this option using usd100 calls of 1yr and 2yr maturities. I would answer the question by saying that the forward option will be cheap, since we can replicate all the usd100 options we want for nothing, but there is a chance that the stock will be at usd200 by then , and we have no way to lock in the volatility of a 1yr usd 200 call at that time.

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Yup, the issue is that with your implied volatility structure, the options with longer maturity are cheaper than the ones with shorter maturity - try checking this with your Black Scholes formula. This implies calendar arbitrage, you should short the shorter one and go long with the longer maturity option. The implied forward variance is actually negative in your example, your calculations are correct.

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