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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! $$ Maturity\space in\space days = TAU\space in\space days = T-t $$

enter image description here 17-Jan-2008 enter image description here

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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! enter image description here

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: enter image description here

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

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  • $\begingroup$ 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 $\endgroup$ – Kermittfrog May 8 at 18:02
  • $\begingroup$ 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 $\endgroup$ – C0deP0ntage May 11 at 7:47
  • $\begingroup$ 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? $\endgroup$ – C0deP0ntage May 11 at 8:47
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See my post: Implied interest rate using put-call parity. Maybe it helps. Liquidity is an issue for OTM and results should be more consistent using most liquid points (ATM).

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  • $\begingroup$ Thank you, it helps! Though, having a look at my chart on the 17-Jan-2008, only the rates that are completely off so the range (-75%,75%), I would expect 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, or 1) + 2). Do I see this correctly? $\endgroup$ – C0deP0ntage May 11 at 11:46
  • $\begingroup$ I edited my original post, regarding your suggestion: "Liquidity is an issue for OTM and results should be more consistent using most liquid points (ATM)" $\endgroup$ – C0deP0ntage May 11 at 12:17
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    $\begingroup$ Note also that you are looking at very short tenors (here 1d). It is usually quite difficult to get estimates from such short term options. The time value is almost 0, so you can see that the effect of interest rates in the discounting elements is neglectable (really tiny). Also, your skew will be highly unstable due to the same. Also, you should use bids and asks to get your mid and not the last price. $\endgroup$ – raptor22 May 11 at 12:51
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    $\begingroup$ I would also not expect any liquidity for highly OTM options given the short tenor. Your bids should basically be equal to 0. $\endgroup$ – raptor22 May 11 at 12:55
  • $\begingroup$ Totally agree, that was very helpful! Thx $\endgroup$ – C0deP0ntage May 11 at 13:00

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