# Calculating risk free rates from risky options using put call parity

My questions relates to this post Implying risk-free rates using Put/Call parity , but I am using a different approach.

Given: ODAX (Options on "DAX") Settlement prices across different maturities and strikes. $$PCP: p_{i,t,T} - c_{i,t,T} = (P_{t,T} - S_t) + e^{-r_{t,T}(T-t)}K_i$$

Approach: Using the put call parity(=PCP) for same maturities but different Strikes, i.e. $$(1) \space p_{i,t,T} - c_{i,t,T} = S_t + e^{-r_{t,T}(T-t)}K_i \\ (2) \space p_{j,t,T} - c_{j,t,T} = S_t + e^{-r_{t,T}(T-t)}K_j$$ and solving (1) & (2) after S_t and equate them results in $$p_{i,t,T} - c_{i,t,T} - e^{-r_{t,T}(T-t)}K_i = p_{j,t,T} - c_{j,t,T} - e^{-r_{t,T}(T-t)}K_j$$ solving for the the rate r_t,T gives $$(*) \space r_{t,T} = \frac{1}{T-t} * log(\frac{c_{j,t,T}-p_{j,t,T}+p_{i,t,T}-c_{i,t,T}}{K_i-K_j})$$ Note I do not have to account for the present value of dividends paid out, since the DAX is a performance index.

Calculating the risk free rates on 16,17,18-Jan-2008 and plot the risk free rates through all available maturities on a day gives me these plots. Options with maturity in Jan-2008, would expire on 18-Jan-2008

16-Jan-2008: Range of risk free rates: ≈ (-40%,40%), these are the options pairs that expire on 18-Jan-2008 --> TAU = 2

17-Jan-2008: Range of risk free rates: ≈ (-75%,75%), these are the options pairs that expire on 18-Jan-2008 --> TAU = 1

18-Jan-2008: Range of risk free rates: ≈ (1%,8%), these are the options pairs that expire on 15-Feb-2008, getting a little more reasonable again

I only consider option pairs with a traded Volume > 0.

Apparently this phenomenon only happens for short dated option pairs close to maturity. In the post I've mentioned above they talk about liquidity mismatches. Is this really the reason. Could anybody please elaborate on that or give other reasons? Happy to discuss any ideas! $$Maturity\space in\space days = TAU\space in\space days = T-t$$

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After it was pointed out that I should try to take the most liquid ATM options I expected the following: For the rates that are "off" either 1) the traded volume is too low, or 2) one of the option pairs is too far OTM(to be precise this means looking at the formula above(*) either the call or the put is OTM for the respective option pair (p_i, c_i)), or 1) + 2).

But having a look a the rates from 17-Jan-2008 range (-75%,75%) I printed the following table, where you can see that for the calculated rate ≈73.3% nor the options pairs are far OTM and the traded volume is also decent!

On the other hand looking at a decent calculated rate, the traded volume is very low and also the option pairs which where used are OTM:

Somehow this doesn't support the statement. Do I miss something here?

• your bid/ask spreads plus liquidity effects pollute the estimator. Also, these options are on Futures, no? So you also have an effect from Futures / spot mismatch etc – Kermittfrog May 8 at 18:02
• 1) The options are not on the Future they are only on the index DAX, see: eurexchange.com/exchange-en/products/idx/dax/DAX-Options-139884 2) Since I am using the daily settlement price and not the mid price shouldn't the bid/ask spread be irrelevant? 3) About the liquidity, it is true if I choose a higher threshold, e.g. considering only call put pairs where the threshold is over a decent volume some of the high rates disappear. Though, not all – C0deP0ntage May 11 at 7:47
• Shouldn't the liquidity effect also the long dated options? For the options pairs that expire in > 1y the trading Volume on average is around 50 contracts, but there I do not get the outliers? – C0deP0ntage May 11 at 8:47