Case Study: Black-Scholes Implied Volatilities in Practice
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
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