Estimating Implied Rates from Option Prices Across Strikes
Summary
The document examines an attempt to infer risk-free rates from put-call parity using DAX index option settlement prices. The proposed calculation compares put and call prices at two strikes with the same maturity, eliminating the underlying index level and solving for the discount rate. Because the index is described as a performance index, the author omits a dividend adjustment. The author reports implausibly wide rate estimates for options expiring very soon and asks whether liquidity mismatches explain them.
The question includes observations about trading volume and moneyness that challenge a simple explanation based on low volume or far out-of-the-money options. The accepted response points to a related discussion and suggests that estimates may be more consistent when based on the most liquid, near-the-money options. This is a limited answer, not a full diagnosis: it does not quantify bid-ask effects, settlement-price noise, parity violations, or the reliability of the inferred rates. The algebra also depends on consistent put-call parity sign conventions and matched option inputs.
Key ideas
- Comparing put-call parity across two strikes with a common maturity can eliminate the underlying price when estimating a discount rate.
- The author reports unstable estimates for options close to expiration.
- The response recommends focusing on the most liquid near-the-money option points.
- Trading volume and moneyness alone may not explain anomalous rate estimates.
- The discussion does not provide a quantitative error analysis or a complete treatment of market frictions.
Tags
Full text
# Calculating risk free rates from risky options using put call parity
# 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! $$ Maturity\space in\space days = TAU\space in\space days = T-t $$
-------------------------------------------Edit-------------------------------------
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?
## Answer by raptor22 (score 0, accepted)
https://quant.stackexchange.com/a/54033
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.