# 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.

• Inflation is usually modelled using other models than Black-Scholes. Try looking up the Jarrow-Yildrim model for inflation derivatives (google.com/url?sa=t&source=web&rct=j&url=http://…). The option pricing formula for inflation will NOT be the B-S formula. So your root-finder might be failing because BS is not appropriate. Jun 22 '20 at 11:20
• @JanStuller I am aware of JY model. By calculating the B-S implied vol I am not assuming that the B-S model is appropriate. I have been told it is common practice to quote inflation caps/floors in terms of the B-S impled vol and that it should still produce a volatility smile. Jun 22 '20 at 11:26
• I see, ok. I haven't looked at Inflation options for a few years now: I just remember that we used the JY model to model inflation. Let me look at it properly tonight and try to answer later on. Jun 22 '20 at 11:47
• Thanks, I would really appreciate that. The purpose is to produce a volatility smile which can then be interpolated to provide estimates at all strike prices. I think it is quite common to try and interpolate the B-S impled vol smile rather than interpolate the price of the caps/floors directly. Jun 22 '20 at 11:56

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.

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.