Black–Scholes Implied Volatility for Inflation Floor Options

# Calculating Implied Volatility from a put option


# Calculating Implied Volatility from a put option












I am trying to find the Black-Scholes implied vol from a put option. I know how to do this in the case of a regular put option on an underlier $S(t)$ where $$ p(t, K) = e^{-r(T-t)}\mathbb{E}_Q\Big[ (K - S(T))_+ \vert \mathcal{F}_t \Big] $$ However, in my case I am working with an inflation floor (a put option on the annual inflaiton rate). In this case the price of the put option (when assuming constant short rate) is given by $$ p(t, K) = e^{-r(T-t)}\mathbb{E}_{Q}\Big[ \Big((1+k)^{T-t} - \frac{I(T)}{I(t)}\Big)_+ \vert \mathcal{F}_t \Big] $$ where $I(t)$ denotes a price index, and $k$ denotes the strike price of the floor

Now, to translate this problem into the case that I already know how to solve I take $$ K = (1+k)^{T-t} $$ and $$ S(T) = \frac{I(T)}{I(t)} $$ and then just calculate the implied vol like I usually would (using a root finder). However, my root finder doesn't yield any roots.

The data I am using is as follows:

$S(t) = \frac{I(t)}{I(t)} = 1$

Time to maturity $= 1$ year

$r = -0.1425\%$

$K = (1+0.025)^{1} = 1.025$

Price of the option $= 0.0156$

This is real data and I am confident it is correct. Therefore there is either a mistake in my methodology or in my interpretation of the data. Any help would be appreciated.

## Answer by ffbzona (score 2, accepted)

https://quant.stackexchange.com/a/55141

I’m not an expert on Inflation derivatives, so I will just give you an explanation on why your finder doesn’t yield any root.

In the Black & Scholes framework, it holds for the price of European Put options:

$$P_{B S}(\sigma=0, T, K, S)=\left(K e^{-r(T-t)}-S\right)^{+},$$ $$P_{B S}(\sigma=\infty, T, K, S)=K e^{-r(T-t)}.$$

Given the parameters you provided, the price of your Inflation Put option assuming zero volatility is roughly:

$$\left(K e^{-r(T-t)}-S\right)^{+}\approx0.02646.$$

The European Put option price is a monotone increasing and continuous function of volatility. Hence, because the price for 0 volatility is higher then your reference price, there exists no volatility that yields your reference price in the BS framework.

## Answer by ir7 (score 0)

https://quant.stackexchange.com/a/55133

At time $T$, standard floorlet pays:

$$ N\tau \left[\kappa - (I(T)I(S)^{-1} -1)\right]^+$$

with $N$ notional, $\kappa$ strike, $S < T$, and $\tau$ day count fraction.

Standard floor is simply a strip of floorlets sharing a common strike paying at each $T_i$, $i=1,...,M$:

$$ N\tau_i \left[\kappa - (I(T_i)I(T_{i-1})^{-1} -1)\right]^+$$

Your payoff is for a zero-coupon put option and pays at maturity $T$ (in years here):

$$ N\left[(1+\kappa)^T - I(T)I_0^{-1} \right]^+ $$

For a pricing framework see Brigo and Mercurio's book, Interest Rate Models - Theory and Practice With Smile, Inflation and Credit. There are two standard models introduced there:

- Jarrow-Yildirim model that needs volatility of real rates and

- a second market model that uses volatility of index and correlation of index and nominal rate.