# Implied interest rate using put-call parity

In the process of asking this question, I acutally found the solution. I still let this post open if it can be interesting to someone else and have added a related question at the end.

I want to check if the interest rates that I assume for option pricing are consistent with the market-implied/assumed interest rates.

To do so, I assume that given "good enough" bid and ask prices for call and puts at different strikes $$k_i$$ for a fixed tenor $$T$$, I would expect to derive from the put-call parity a relatively horizontal line.

To be precise, I compute the following:

$$F^{ask}(k,T) := k + e^{r_T T}(C^{bid}(k,T) - P^{ask}(k,T)),$$ $$F^{bid}(k,T) := k + e^{r_T T}(C^{ask}(k,T) - P^{bid}(k,T)),$$

where $$r_T$$ is the assumed zero rate with continuous compounding for the period $$[0, T]$$. My expectation is that $$r_T$$ should be such that $$F^{bid}(k_1,T) \approx F^{bid}(k_2, T)$$ for different strikes $$k_1, k_2$$, same for $$F^{ask}$$.

Here I do not have any additional information about forward value, I just know that $$F = S_0 \exp((r_T-b-\delta)T)$$ where $$\delta$$ and $$b$$ are my dividend and borrowing rates. My goal is to extract $$r_T$$ and $$b + \delta$$.

Here is an example to illustrate based on S&P500 options. Here I assumed a swap zero rate curve to use in the put-call parity formula. I interpolated this curve using cubic splines.

In red is $$F^{ask}$$, in black $$F^{bid}$$ and in blue the average of the two.

Here seems to work great expect for spikes, they must be less liquid points?

Here a little less.

Increasing the rate seems to rotate the line clockwise and diminishing it counter-clockwise. This is expected because of the collar being linear. (Call - Put is a linear decreasing function of the strike with coefficient close to -1). Here is an illustration of the value of Put - Call for a fixed tenor:

From this I can fit a linear regression and obtain slop $$\hat{\beta}$$. The put - call parity being constant accross strikes rewrites to:

$$\exp(r_T\cdot T)(\hat{\beta} k + \alpha) + k = cst,$$

which is satisfied for $$r_T = -\frac{\ln(-\hat{\beta})}{T}.$$

Resulting "horizontalized" forward:

Question: is this "risk-free" rate $$r_T$$ usually consistent accross the index, i.e. can I use the same rate for an single-stock equity option constituent of the index?

• Thank you! Staying in the European option setup: why should the ease of borrowing have an impact on $r_T$ and not only on $b+\delta$ in the put call parity formula? I agree that borrowing rate $b$ and ultimately forward rate $r_T-b-\delta$ will be different for different stocks, however why should "risk-free" discounting change depending on ease of borrowing? In practice, since single equity options are American type, can I find box spreads? and in this case I guess that they are exposed to the underlying due to dividend payments? – raptor22 Sep 19 '19 at 23:05
• if there are no borrow fees, then put-call parity can be written as: $C-P + ke^{-rt} = S_0 e^{-qt}$, which we can rewrite as $(C-P)e^{rt} +k = S_0e^{(r-q)T}$. In this case the procedure gives us $r$ as there is no assumption made on the value of $F = S_0 e^{(r-q)t}$, the only assumption is that $F$ should be constant across strikes. Adding borrow cost, my intuition is that put-call parity can rewrite $(C-P)e^{rt} + k = S_0e^{(r-b-q)T}$ in which case it hsould hold, am I mistaken? – raptor22 Sep 20 '19 at 18:39