I would like to calibrate my model to the current call option prices (with 17 different maturity times) but I don't know how to choose a risk-free rate in this case.
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To do this, you need to find some securities that depend only on the risk-free rate, and calibrate your risk-free rate curve to them, and then use that rate to price your options. In this way, your model will exactly reprice at lease two types of security.
There are many choices, but the easiest is government bonds in the currency of interest to you. You need to create a Zero Curve using 'govvies', and that gives you the risk-free rate. That is roughly explained here, and if you have some prices of bonds then packages like QuantLib can do the computations for you.
Note that other choices exist - you could bootstrap your Zero Curve from swaps, for example - and in fact, practitioners typically take a blended approach where cash instruments are used for short-dated parts of the curve and swaps for longer dated tenors
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$\begingroup$ What if my model assumes a constant risk free rate? Because I see that in one book the author used one risk-free rate to calibration, but I cannot find on what basis he chose this risk free rate $\endgroup$– Mr.PriceAug 6, 2020 at 12:23
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2$\begingroup$ If you want a single risk-free rate, you can only match a single govvie. Say the 1Y govvie is 99, which implies about 1% rate, but the 2Y govvie is 97, which implies about a 1.5% rate over two years, what do you do? You can either have a single constant rate (1% or 1.5%) and match one govvie but disagree with the market on the other, or you can let it be piecewise continuous (in this case, roughly 1% for the first year and roughly 2% for the second year) and match both. It just depends what is most important to you. $\endgroup$– StackGAug 6, 2020 at 12:27
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