Saturday, February 28, 2015

volatility - Swaption valuation across time using vcub


On Bloomberg one has access to the rates vol cube with the VCUB function. For a given currency, today, one sees Black implied volatilities for swaptions of various expiries and strikes, for forward swaps of various tenors. Suppose I see 80% of implied volatility for a 1y10y payer physically settled swaption. This means that the corresponding market price is forward swap annuity×Black(s0,80%,)

where Black is the Black function and s0 is today's 1Y forward 10Y swap value and the s0 quote is available on the market (on Bloomberg).


Now imagine I come the next week (in 7 days), and I want to value the same swaption in the Black model again. How can I do this using the vol cube in 7 days ?



Answer



Forget for a moment that your option is delivering the immediate entrance in a swap (if the swaption is physically settled) or the cash amount of the swap (if the swaption is cash-settled), as your question doesn't depend on this fact, and take a "general" 1Y option.



Your today's (date t0) cube loses the "swap tenor dimension" and becomes a today's implied volatility surface, on which you read (through Black-Scholes function) the price of your option through implied volatity for 1Y expiry and given strike.


In 1W (date t0+1W) your option will be an option on the same underlying (work it out in the swaption case) with same strike K but expiry "1Y minus 1W" (date t1). So to value your option 1W after, you need to know the implied volatility at (t1,K). And it this one is note quoted, you'll have to resort probably to interpolation, or even extrapolation.


To make it simple, the time t price of the option is


πt(T,K)=Black(ˆσt(T,K),Tt,K,st)


where ˆσt(T,K) is the time t implied volatility for expiry T and strike K (and swap tenor 10Y) and where st is the forward swap rate (for the underlying forward swap of the swaption) at time t.


As I said the fact that ˆσt0(T,K) is quoted (i.e. is directly readable on VCUB) doesn't imply that ˆσt1(T,K) will be, hence you'll probably have to resort to some interpolation to get ˆσt1(T,K) from values observables in VCUB at t1.


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