# University Problem about interpolation Implied volatility BS Model (volatility smile)

Good morning, this is my first question on this forum, I'm writing from Milan (Italy) and I have a question about a University Problem. The problem is about entering in a Long Range Forward (buy a call option and sell a put option). Now, the goal is to calculate the Strike of the option and get the volatility smile with this data:

Notional = 100.000,00 S(0) = 100,00 Delta Call = 20% Call Volatility = 10% Put Volatility = 12%

I tried to solve the problem and I get that in the black model I have:

Delta Put = 1 - Delta Call = 0,8 (80%)

also in the Black Model it is true -> Delta Call = 0,2 = N(d1) so I caculated d1 = -0,8405

at this point I don't know how I can solve it ca you help me? I've the final test on 1st December is the problem impossible to solve because of missing data?

Thank you

• Just to clarify: The put and call have different but unknown strikes and you want to calculate the strike of each of the two options, given their implied volatility, value of the underlying and delta under a 76 Black model? [btw: good luck with your exam!] Commented Nov 22, 2022 at 18:36
• yes my problem's that and I've to provide the volatility smile plot Commented Nov 23, 2022 at 8:44
• Thanks for the clarification Ivan. Are you sure that the value of the spot is given, not the futures price (i.e, no information about interest rates or dividends)? Do you know the maturity of these options? Commented Nov 23, 2022 at 10:46
• As you say, $\Delta_c=N(d_1)$, where $d_1=\frac{\ln(F/K)+\frac{1}{2}\sigma^2T}{\sigma\sqrt{T}}$. Thus, given $d_1$ (which you calculated correctly), we can solve easily for the strike price: $K=F\exp\left(\frac{1}{2}\sigma^2T-d_1\sigma\sqrt{T}\right)$. However, we do need to know the maturity of the option ($T$) and the futures price ($F$). Commented Nov 23, 2022 at 10:57
• Yes my problem is that I haven't maturity and interest rate, is it possible to solve it without this information? however is the strike price for Put option like K with -(-d1) so +d1? Commented Nov 23, 2022 at 15:37