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If I have volatility smile quoted with respect to the delta of an option on the forward, how can I convert this delta into the moneyness or strike of the option?

Is there any bult-in function of Matlab financial toolbox?

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    $\begingroup$ I can't help you with the MATLAB part of your question. For the general algorithm, I recommend you to have a look at Chapter 1 "FX Market Conventions" in Dimitri Reiswich's Ph.D. thesis "The Foreign Exchange Volatility Surface" or the related papers Reiswich and Wystup (2010) "A Guide to FX Options Quoting Conventions", Journal of Derivatives and Reiswich and Wystup (2012) "FX Volatility Smile Construction", Wilmott. $\endgroup$ – LocalVolatility Apr 10 '17 at 9:16
  • $\begingroup$ right, as LocalVolatility says - this sounds like options on FX (not futures but OTC)- is this correct? If yes, then adding this information to the question would help. $\endgroup$ – Ric Apr 10 '17 at 10:25
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The call delta in a Black framework is: $$\Delta = N(d_1)$$ with $d_1=\frac{\ln(F_t(T)/K)+(T-t)\frac{\sigma^2}{2}}{\sigma\sqrt{T-t}}$.

Then the strike of the option is: $$K=F_t(T) e^{-(N^{-1}(\Delta)+1/2) \sigma \sqrt{T-t}}$$

The same thing is done if the option is a put and we obtain:

$$K=F_t(T) e^{-(N^{-1}(\Delta+1)+1/2) \sigma \sqrt{T-t}}$$

In matlab it can be solved by doing:

fzero(@(Strike) blsdelta(Price,Strike,Rate,Time,Volatility,Yield)-Delta, K0)

where the initial guess can be K0 = Price

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    $\begingroup$ Two comments: 1) You are not clear about what type of delta you are dealing with. $\Delta = \mathcal{N} \left( d_1 \right)$ could either be the forward delta or the spot delta when the foreign rate is zero (which is unrealistic). 2) Your equations equally assume that the delta is not premium adjusted. In the premium adjusted case, you need to numerically invert the delta for the strike - see the reference I provided in my comment to your question. $\endgroup$ – LocalVolatility Apr 10 '17 at 10:14
  • $\begingroup$ I am dealing with the Black-like delta of an option on the forward. Given that it is a martingale, the drift and hence $r$ are not important. $\endgroup$ – NSZ Apr 10 '17 at 10:22
  • $\begingroup$ That's fine. My point was that should just be clear about it. The inversion from delta to strike usually arises in an FX context where there are many different conventions. Your answer just works in this particular context. $\endgroup$ – LocalVolatility Apr 10 '17 at 10:23
  • $\begingroup$ Sure, I slightly edited the question. Thanks! $\endgroup$ – NSZ Apr 10 '17 at 10:25

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