Black-Scholes Implied Volatilities in Practice

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Case Study: Black-Scholes Implied Volatilities in Practice

The topic for this case study is to apply the Black-Scholes model to calculate the strike price of the F.X. options and estimate the implied volatilities in practice, finally delta-hedged strategy will be described in detail in order to hedge F.X. option.

The below formulas for Black-Scholes pricing are applied to the case study problems:

Valuation of currency Europearn call option| Valuation of currency Europearn put option| C= S0*e^(-Rf*T)*N(d1) - Ke^(-R*T)*N(d2)| P=Ke^(-R*T)*N(-d2) - S0*e^(-Rf*T)*N(-d1)| d1 = (ln(S/K)+(R - Rf+ σ^2/2)*T)/(σ*sqrt(T))| d1 = (ln(S/K)+(R - Rf+ σ^2/2)*T)/(σ*sqrt(T))| d2 = d1 - σ*sqrt(T)| d2 = d1 - σ*sqrt(T)|

Δ= e^(−Rf *T)*N(d1)| Δ = e^(−Rf *T)*[N(d1) − 1]|

Q1. Complete the following table, by entering the strikes of the 50-delta options:

Answer:

Date| Option Strikes (measured in one GBP in terms of USD)|  | 1 week| 1 month| 3 months| 6 months| 1 year| 2 years| 14-Jan| USD 1.9578| USD 1.9556| USD 1.9496| USD 1.9397| USD 1.9185| USD 1.8717|

Detailed explanations:

Step 1: The below information is given in the questions as below:

14-Jan| 1 wk | 1 mth| 3 mths| 6 month| 1 yr| 2 yrs|
Delta| 0.5000| 0.5000| 0.5000| 0.5000| 0.5000| 0.5000| S0| 1.9584| 1.9584| 1.9584| 1.9584| 1.9584| 1.9584|
σ| 0.0890| 0.0918| 0.0918| 0.0897| 0.0894| 0.0881|
R (USA)| 0.0200| 0.0200| 0.0200| 0.0200| 0.0200| 0.0200| Rf (UK)| 0.0400| 0.0400| 0.0400| 0.0400| 0.0400| 0.0400| T | 0.0192| 0.0833| 0.2500| 0.5000| 1.0000| 2.0000|

Step 2: Using the formula Delta = e ^ (−Rf *T)*N (d1), N (d1) can be calculated with known delta, Rf and T. Thus we get N (d1) = delta / ( e ^ (−Rf *T) ). Once we calculate N (d1), d1 can be calculated using the NORMSINV function in excel, the result is shown as below:

14-Jan| 1 week | 1 month| 3 months| 6 month| 1 yr| 2 yrs| N(d1)| 0.5004| 0.5017| 0.5050| 0.5101| 0.5204| 0.5416| d1| 0.0010| 0.0042| 0.0126| 0.0253| 0.0512| 0.1046|
Step 3: Now we got d1, we can use d1= ((ln(S/K)+(R-Rf+ σ^2/2)*T))/(σ*sqrt(T)) to calculate back the Ln(S/K), then finally K can be obtained.

14-Jan| 1 week | 1 month| 3 months| 6 month| 1 year| 2 years| Ln(S/K)| 0.0003| 0.0014| 0.0045| 0.0096| 0.0206| 0.0453| K = S / e^(Ln(S/K))| 1.9578| 1.9556| 1.9496| 1.9397| 1.9185| 1.8717|

Q2. Complete the following table, by entering the strikes of the calls and puts underlying the 25-delta reversals and the 25-delta strangles:

Answer:

 | Put strikes| Call strikes|
 | 1 month| 3 months| 1 year| 1 month| 3 months| 1 year| 14-Jan| 1.9189| 1.8873| 1.8127| 1.9902| 2.0101| 2.0423|

Detailed explanations:

Step 1: To calculate the implied volatility for underlying call and put respectively. Given that:

Volatility of 25-delta reversal
= volatility of underlying call – volatility of the underlying put;

Volatility of 25-delta strangle
= average of the volatilities of the underlying call and put – the volatility of the corresponding 50-delta call

14-Jan| Implied Volatility|
 | 50 delta calls| 25 delta reversals| 25 delta strangles|  | A| B| C|
1 month| 0.0918| -0.0088| 0.0024|
3 months| 0.0918| -0.0094| 0.0031|
1 year| 0.0894| -0.0063| 0.0038|

Then we can get the relationships of underlying call and put as below:
14-Jan| Implied Volatility|
 | average volatility| volatility of call - volatility of put| volatility of call + volatility of put|  | D=C+A| B| E=D*2|
1 mth| 0.0942| -0.0088| 0.1884|
3 mths| 0.0949| -0.0094| 0.1898|
1 yr| 0.0932| -0.0063| 0.1864|
Then the volatility of underlying call and volatility of underlying can be obtained by solving simultaneously of the reversing formula:

14-Jan| Implied Volatility|
 | volatility of call| volatility of put|
 | F=(B+E)/2 |...
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