Time Varying Parameters

When I discussed the BS equation here, one of the assumptions was that r and \inline \sigma were constant parameters. In reality, neither of these will be constant: how much of a problem is this for us? In general, they will both be stochastic and hence unpredictable in the future. In this post however I’m going to stick to deterministic quantities and demonstrate that BS can be readily extended to time-varying rates and vols. This will enable us to price at-the-money options correctly, but still won’t help us with the vol smile effect that I discussed here.

If r and \inline \sigma are time varying, the stochastic differential equation describing the underlying spot price in BS is

{ dS \over S} = r(t)dt + \sigma(t)dW_t

Using Ito’s Lemma as discussed before, this can be re-written in terms of the log as

d(\ln S) = \Bigl( r(t) - {1 \over 2}\sigma^2(t)\Bigr) dt + \sigma(t) dW_t

And hence

S(t) = S(0).\exp{ \Bigr[ \Bigr( \bar{r} - {1 \over 2}\bar{\sigma}^2 \Bigl)t + \bar{\sigma} \sqrt{t} z \Bigl] }

where \inline z \sim {\mathbb N}(0,1). This is still lognormal as before, but we now have an effective rate and an implied volatility defined by (you can see these by comparing to the distribution coming from constant r and \inline \sigma):

\bar{r}(t) = \int_0^t r(t')dt'

\bar{\sigma}(t) = \sqrt{ {1 \over t} \int_0^t \sigma^2(t')dt'}

\inline \sigma(t) is called the instantaneous vol, but the relevant quantity for option pricing is always the implied vol \inline \bar{\sigma}(t). Assuming that we have a discount curve constructed from traded bonds we can calculate r(t) from that as I described here, and if we can see liquid at-the-money (ATM) options on the market at different times we can also calculate the \inline \sigma(t) function consistent with them from their implied volatilities, which we will need to simulate paths in Monte Carlo, via the following procedure

  • Take all available ATM options, label their times in order \inline \{ t_1, t_2, \cdots , t_n \}
  • Calculate their implied vols using a vol solver (eg. the one on my PRICERS page). These correspond to \inline \{ \bar{\sigma}(t_1), \bar{\sigma}(t_2), \cdots , \bar{\sigma}(t_n) \}
  • As a first approximation, we will assume \inline \sigma(t) is constant between time windows (although you could make approximations arbitrarily complicated). In this case, for \inline 0 < t < t_1 we have

\bar{\sigma}(t_1) = \sqrt{ {1 \over t_1} \int_0^{t_1} \sigma^2dt'} \qquad 0 < t' < t_1

  • Which works out to simply \inline \sigma(t) = \bar{\sigma}(t) in this region
  • For later vols, the procedure is a little more complicated:

\bar{\sigma}(t_2) = \sqrt{ {1 \over t_2} \int_0^{t_2} \sigma^2(t')dt'} = \sqrt{ {1 \over t_2} \int_{t_1}^{t_2} \sigma^2(t')dt' + \bar{\sigma}^2(t_1)}

t_2 \cdot\bar{\sigma}^2(t_2) - t_1 \cdot\bar{\sigma}^2(t_1) = \int_{t_1}^{t_2} \sigma^2(t')dt' = (t_2 - t_1) \sigma^2(t_1)

\sigma(t_1) = \sqrt{t_2 \cdot\bar{\sigma}^2(t_2) - t_1 \cdot\bar{\sigma}^2(t_1) \over (t_2 - t_1) }

  • And for general time window \inline t_n < t < t_{n+1}:

\sigma(t) = \sqrt{t_{n+1} \cdot\bar{\sigma}^2(t_{n+1}) - t_n \cdot\bar{\sigma}^2(t_n) \over (t_{n+1} - t_n) } \qquad t_n < t < t_{n+1}

  • And this final expression can be used to calculate the value of the instantaneous vol required between each time window to give the correct ATM implied vol

Note that this will fail if \inline t_{n+1} \cdot\bar{\sigma}^2(t_{n+1}) < t_n \cdot\bar{\sigma}^2(t_n) – this is because we expect the distribution variance \inline t \cdot\sigma^2(t) to be an increasing function of time. If it didn’t hold for any time windows we’d have an arbitrage opportunity, selling the first option and buying the second to lock in a risk-free profit.

As I mentioned, this extension allows BS to correctly price options and forwards at different times by matching to market observables, but still doesn’t predict any volatility smile. Can we take it any further? In fact we can: in a later post I will discuss the local vol model which extends \inline \sigma(t) to be \inline \sigma(S,t) and will allow us to match any set of arbitrage-free market prices.

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