| European call option price | |||||||||||
| According to Black Scholes Model= | SN (d1)- Ke-rt N(d2) | ||||||||||
| d1= | Ln (S/X) + ( r + s2/2)t | ||||||||||
| s * sqrt (t) | |||||||||||
| d2= | d1-(s*sqrt(t)) | ||||||||||
| d1= | Ln (20/20) + (0.05+(0.20)2/2)*0.5 | ||||||||||
| 0.20* sqrt (0.5) | |||||||||||
| 0.035 | |||||||||||
| 0.14142 | |||||||||||
| d1= | 0.24749 | ||||||||||
| d2= | 0.10607 | ||||||||||
| Price of call option | 1.37552 | ||||||||||
| Price of call option | c + x | = | S | ||||||||
| (Using put call parity formula | (1+r)t | P | |||||||||
| 21.37552 | = | 20 | |||||||||
| (1+.005)0.5 | P | ||||||||||
| Price of call option | P= | 0.89 | |||||||||
| Part 2 | Cost of buying Protective Put option price | Long put + long stock | |||||||||
| 20.89 | |||||||||||
| Part 3 | Cost of writing covered call | short call option + long stock | |||||||||
| 18.62 | |||||||||||
| Part 4 | Price | Pay-off | Profit | ||||||||
| $ 16.00 | 0 | -0.89 | |||||||||
| $ 18.00 | 0 | -0.89 | |||||||||
| $ 20.00 | 0 | -0.89 | |||||||||
| $ 22.00 | 2 | 1.11 | |||||||||
| $ 24.00 | 4 | 3.11 | |||||||||
| q= | e-rt – d | ||||||||||
| u-d | |||||||||||
| 2.7180.06*6/12 | |||||||||||
| 1.0304-.90 | |||||||||||
| 1.10-.90 | |||||||||||
| 0.1304 | |||||||||||
| 0.2 | |||||||||||
| Probability of price increase (q)= | 0.65 | ||||||||||
| Probability of price decrease (1-q)= | 0.35 | ||||||||||
| Exercise Price | Actual Price | Option Exercise | Value of Premium | Probability | Expected Value | DCF @ 3.040% | PV of call | ||||
| 50 | 45 | Lapse | 0 | 0.35 | 0 | 0.492514 | 0 | ||||
| 50 | 55 | Exercise | 5 | 0.65 | 3.25 | 0.492514 | 1.60067 | ||||
| Part b | Upside | Downside | |||||||||
| % of fluctuation | 55-50 | 45-50 | |||||||||
| 50 | 50 | ||||||||||
| 10% | -10% | ||||||||||
| 60.5 | |||||||||||
| 55 | 10.5 | ||||||||||
| 6.623641 | |||||||||||
| 50 | 49.5 | ||||||||||
| 4.18 | 0 | ||||||||||
| 45 | |||||||||||
| 0 | 40.5 | ||||||||||
| 0 | |||||||||||
| P= | R-d | ||||||||||
| u-d | |||||||||||
| 1.0304-.90 | |||||||||||
| 1.10-.90 | |||||||||||
| 0.1304 | |||||||||||
| 0.2 | |||||||||||
| Probability of price increase= | 0.65 | ||||||||||
| Probability of price decrease= | 0.35 | ||||||||||
| Exercise Price | Actual Price | Option Exercise | Value of Premium | Probability | Expected Value | DCF @ 3.040% | PV of call | ||||
| 50 | 45 | Exercise | 5 | 0.35 | 1.75 | 0.492514 | 0.861899 | ||||
| 50 | 55 | Lapse | 0 | 0.65 | 0 | 0.492514 | 0 | ||||
| 0.861899 | |||||||||||
| p= | Cu p + Cd (1-p) | ||||||||||
| R | |||||||||||
| (10.5 *.65)+(0*.35) | |||||||||||
| 1.0304 | |||||||||||
| 6.825 | |||||||||||
| 1.0304 | |||||||||||
| 6.62 | |||||||||||
| 4.305367 | |||||||||||
| 1.0304 | |||||||||||
| At node A | 4.18 | ||||||||||
| Part 1 | Amount of deposit on investment = | No. of contracts * Initial Margin Deposit | |||||||||
| 7*2000 | |||||||||||
| 14000 | |||||||||||
| Initial Margin | 14000 | ||||||||||
| Maintenance Margin | 8750 | ||||||||||
| Changes in prices | No. of bushels | Changes in futures value | Margin Money | ||||||||
| 1st Day | 1.64 | 14000 | |||||||||
| 2nd Day | 1.6 | -0.04 | 5000 | -200 | 13800 | ||||||
| 3rd Day | 1.66 | 0.06 | 5000 | 300 | 14100 | ||||||
| 4th Day | 1.7 | 0.04 | 5000 | 200 | 14300 | ||||||
| 5th Day | 1.75 | 0.05 | 5000 | 250 | 14550 | ||||||
| 6th Day | 1.8 | 0.05 | 5000 | 250 | 14800 | ||||||
| Part 2 | Current balance at the end of 5th day = | 14550 | |||||||||
| Part 3 | Final gain over 5 days | 550 | |||||||||
| 14000 | |||||||||||
| 3.93% | |||||||||||
| Part4 | |||||||||||
| 5th Day balance in margin | 14550 | ||||||||||
| 6th day profit/loss | 250 | ||||||||||
| 6th Day balance in margin | 14800 | ||||||||||
| As the balance in margin account on 6th day is higher than maitenance margin hence investor will not face any margin call. | |||||||||||
| Spot | 1 Euro= | $1.3000 | |||||||||
| Forward | 1 Euro= | $1.2900 | |||||||||
| Interest rate parity | 1+ rh | = | f | ||||||||
| 1+ rf | s | ||||||||||
| 1+0.015 | = | f | |||||||||
| 1+.002 | $1.3000 | ||||||||||
| f= | $1.3195 | ||||||||||
| 1.02 | |||||||||||
| Ideal forward rate | f= | $1.2936 | |||||||||
| Actual forward rate | f= | $1.2900 | |||||||||
| Arbitrage strategy | |||||||||||
| borrow 1000 dollar @ 1.5% | |||||||||||
| convert them in euro @1.30 | 769.2308 | ||||||||||
| Invest euro @2% interest | |||||||||||
| Receive after one year | 784.6146 | ||||||||||
| convert them in dollar @1.29 | 1012.153 | ||||||||||
| Profit (dollar) | 12.15283 | ||||||||||
| Part 1 | Fund A | Fund B | Fund C | Market Portfolio | |||||||
| Average return | 18% | 12% | 30% | 15% | |||||||
| Standard deviation | 25% | 15% | 30% | 20% | |||||||
| Beta | 1.25 | 0.6 | 2.5 | ||||||||
| rf | 5% | ||||||||||
| Sharpe’s Ratio= | rp – rf | ||||||||||
| sdp | |||||||||||
| Ratio | Rank | ||||||||||
| Fund A | 52.00% | II | |||||||||
| Fund B | 46.67% | III | |||||||||
| Fund C | 83.33% | I | |||||||||
| Treynor’s ratio= | ri – rf | ||||||||||
| Beta i | |||||||||||
| Ratio | Rank | ||||||||||
| Fund A | 10.40% | II | |||||||||
| Fund B | 11.67% | I | |||||||||
| Fund C | 10.00% | III | |||||||||
| Jenson’s Ratio | rp -(rf +Beta *(rm – rf)) | ||||||||||
| Fund A | .18-(.05 + 1.25(.15-.05)) | ||||||||||
| 0.50% | |||||||||||
| Fund B | .12-(.05+0.6(.15-.05)) | ||||||||||
| 1.00% | |||||||||||
| Fund C | .30-(.05+2.5(.15-.05)) | ||||||||||
| 0.00% | |||||||||||
| Part 2 | As per sharpe ratio Fund C has outperformed Fund A and B | ||||||||||
| As per Treyno’s ratio Fund A has outperformed Fund B and C | |||||||||||
| Part 3 | No, the rankings as per sharpe ratio and treynor ratio are not consistent. | ||||||||||
| The reason for the inconsistency of ranking is that Sharpe measure evaluates | |||||||||||
| the portfolio performance on the basis of overall risk of portfolio. However, Treynor | |||||||||||
| measure evaluates portfolio performance on the basis of systematic risk of the | |||||||||||
| portfolio fund. | |||||||||||
| In Sharpe ratio the risk is identified as a degree of volatility in the return whereas in | |||||||||||
| Treynor ratio the degree of momentum built into portfolio to derive excess returns. | |||||||||||
| Part 1 | Benchmark Portfolio | Joe’s Portfolio | Kim’s Portfolio | ||||||||
| Weight | Return | Weight | Return | Weight | Return | ||||||
| Stocks | 0.6 | -5.00% | Stocks | 0.5 | -4.00% | Stocks | 0.3 | -5.00% | |||
| Bonds | 0.3 | 3.50% | Bonds | 0.2 | 2.50% | Bonds | 0.4 | 3.50% | |||
| T-Bills | 0.1 | 1.00% | Cash | 0.3 | 1.00% | Cash | 0.3 | 1.00% | |||
| SD | 3.500% | 5% | 3% | ||||||||
| Rf = | 1.00% | ||||||||||
| Portfolio Return | -1.85% | -1.20% | 0.20% | ||||||||
| Part 2 | a) as regards selection of stocks, Joe has been superior as the loss on stocks is less compared to Kim’s portfolio. As regards, Bonds, its Kim is superior as the profit on bonds is higher as compared to Joe. | ||||||||||
| b) Joe has allocated 50% to stocks and 20% to bonds while Kim allocated 30% to stocks and 40% to bonds. Bonds providing positive return while stocks incurring losses, hence, it could be inferred that Kim has been superior in asset allcoation. | |||||||||||
| Part 3 | Sharpe’s Ratio= | rp – rf | |||||||||
| sdp | |||||||||||
| -81.43% | -44.00% | -26.67% | |||||||||
| III | II | I | |||||||||
|
References |
|||||||||||
| Chance, D. M., & Brooks, R. (2015). Introduction to derivatives and risk management. Cengage Learning. | |||||||||||
| Rao, G. S. (2012). Derivatives in risk management. International Journal of Advanced Research in Management and Social Sciences, 1(4), 55-60. | |||||||||||
| Sharpe, W. F. (1994). The sharpe ratio. Journal of portfolio management, 21(1), 49-58. | |||||||||||
| Chinn, M. D. (2006). The (partial) rehabilitation of interest rate parity in the floating rate era: Longer horizons, alternative expectations, and emerging markets. Journal of International Money and Finance, 25(1), 7-21. |
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