Forums > Pricing & Modelling > Can risk free interest rate be different for each strike and maturity

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 npips Total Posts: 1 Joined: Jun 2016
 Posted: 2016-06-02 21:28 I'm trying to reverse engineer the risk free interest rate (RFIR) that the market uses to price options. I am using the Black Scholes Merton pricing formula. I downloaded SPX (monthly) data from Yahoo Finance and Interactive Brokers (to double check my findings). I'm using a bisection function to estimate the risk free interest rate. I was thinking that I would get a RFIR between 0.27% and 0.66% based on data from http://www.bloomberg.com/markets/rates-bonds/government-bonds/us , but I'm getting rates that are all over the place.The data I downloaded is from May 30 (16:43). I get the following interest rates based on the ask price:Price S&P was 2099.06Time left: 17/366SPX160617C02100000 - Ask price: 18.8 - IV (implied vol. as quoted by Yahoo): 10.08 - RFIR: 0.022099SPX160617C02110000 - Ask price: 13.4 - IV: 9.58 - RFIR: 0.024787SPX160617P02080000 - Ask price: 14.4 - IV: 11.92 - RFIR: -0.03235SPX160617P02075000 - Ask price: 13.0 - IV: 12.14 - RFIR: -0.03493Time left: 108/366SPX160916C02100000 - Ask price: 57.7 - IV: 12.65 - RFIR: 0.002074SPX160916C02125000 - Ask price: 43.4 - IV: 11.97 - RFIR: 0.002382SPX160916P02080000 - Ask price: 60.9 - IV: 15.31 - RFIR: -0.00242SPX160916P02075000 - Ask price: 59.2 - IV: 15.45 - RFIR: -0.00252So RFIR for calls is positive, for puts it is negative. RFIR changes per strike and becomes ten times smaller when the date is further away. Something is obviously wrong, but I can not figure it out. All input is welcomed. Thanks.
 pj Total Posts: 3388 Joined: Jun 2004
 Posted: 2016-06-02 21:47 > Can risk free interest rate be different for each strike and maturityMaturity yes, strike no.How does the put call parity hold?My suspicions would be on volatility.This should be in basics. I saw a dead fish on the pavement and thought 'what did you expect? There's no water 'round here stupid, shoulda stayed where it was wet.'
 TakeItAndRun Total Posts: 96 Joined: Apr 2010
 Posted: 2016-06-03 13:55 For stock index options, iv depends on option model, dividend model, repo rate,...Not really simple. What is simple is to compute the implied rate from conversion -buy put, sell call, buy underlying- when options are European.So, for a zero-coupon curve, it is preferable to use interest rate instruments. For a usd curve, one can start with the 30-day fed funds futures.
 KarlHungus Total Posts: 9 Joined: Jul 2009
 Posted: 2016-06-03 16:58 In option pricing there are two interest rates involved: a discounting rate which you use for discounting the expected payoff and the funding rate which goes into the forward price of the underlying and is underlying-specific. For exchange traded options the discounting rate is also kind of risk-free due to margining with the clearing house. The discounting rate enters into box trades, for instance long call with strike X, short put with strike X, long put with strike Y, short call with strike Y, all with the same expiry. This portfolio should be worth (Y-X) discounted with the discounting rate. See here for more on box trades.When you want to back out the discounting rate from option quotes keep in mind that there might be a substantial amount of noise especially for short expiries and strikes closely together (like in your example).
 topdrudge Total Posts: 1 Joined: Jun 2016
 Posted: 2016-06-04 23:56 The problem is that you can't really do what you're trying to do because there is too much going on in the raw price data (bid/ask jiggles, index basis versus futures jiggles, and uncertain PVDivs). At the very least you need very accurate PVDivs and an accurate way to peg the implied SPX price from the futures bid/ask ... then you need to process a lot of data and take averages ... and you should be working with put/call parity, not raw option prices and implied vols.Anyway the only reason I'm posting is to mention this in case you care: If you have access to Interactive Brokers then it is VERY easy to reverse engineer THEIR rates, to 8 or 9 decimal places ... but their rates have a spotty record; currently they look good to me but 2 months ago they were probably off a little ... they SHOULD be good because IB = Timber Hill, but you can never trust anything with IB.
 fomisha Total Posts: 31 Joined: Jul 2007
 Posted: 2016-10-27 17:50 discount rate cannot depend on strike. funding rate may depend on strike for american options but in practice does not. for europen, it also cannot depend on strike. you need to separately imply both rates in order to achieve what you are trying to get.
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