Solution of Derivative

Ba s 60 65 70 75 80 85 90 95 100 105 110 115 120 125

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Unformatted text preview: a 0.099 0.154 0.220 0.294 0.372 0.450 0.526 0.597 0.662 0.719 0.769 0.811 0.847 0.877 0.902 0.922 0.938 put delta -0.901 -0.846 -0.780 -0.706 -0.628 -0.550 -0.474 -0.403 -0.338 -0.281 -0.231 -0.189 -0.153 -0.123 -0.098 -0.078 -0.062 89 Part 3 Options We can clearly see that the entries for the one day expiration table are more extreme: There is only one day left for stock price changes, so a lot of uncertainty is resolved. For example, a deep out of the money call option (e.g. at a stock price of $ 60) is unlikely to change during one day to some price bigger than $ 100, so the option most likely does not pay off – therefore it’s delta is zero. On the other hand, with one year to maturity left, there is a decent chance of such a change, therefore the price of the option reacts to a one dollar increase in the stock price. ba) S 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 time to expiration: 1 day straddle delta -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 -0.999 0.018 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 straddle vega 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.042 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 straddle theta 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.019 -0.631 -0.027 -0.022 -0.022 -0.022 -0.022 -0.022 -0.022 -0.022 90 straddle rho -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 0.000 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 Chapter 12 The Black-Scholes Formula bb) S 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 bc) time to expiration: 1 year straddle delta -0.802 -0.692 -0.560 -0.412 -0.256 -0.100 0.052 0.194 0.323 0.438 0.537 0.623 0.694 0.754 0.803 0.844 0.876 Straddle vega 0.209 0.308 0.415 0.517 0.605 0.673 0.717 0.735 0.732 0.708 0.670 0.622 0.567 0.509 0.451 0.395 0.342 Straddle theta 0.009 0.004 -0.003 -0.009 -0.016 -0.021 -0.026 -0.030 -0.032 -0.034 -0.035 -0.035 -0.035 -0.034 -0.033 -0.032 -0.030 straddle rho -0.819 -0.750 -0.661 -0.554 -0.433 -0.304 -0.171 -0.040 0.086 0.203 0.310 0.406 0.490 0.564 0.626 0.679 0.724 Explanation of the one year greeks We need to keep in mind that we bought a call option and bought a put option, both with a strike of $100. Therefore, with a stock price smaller than $100, the put option is in the money, and the call option is out of the money. This pattern helps us when we look at the greeks: For small stock prices, delta is negative (the put dominates) and rho is negative (recall that since the put entitles the owner to receive cash, and the present value of this is lower with a higher interest rate, the rho of a put is negative). Deep in the money put options have a positive theta, therefore for very small stock prices, we expect (and see) a positive theta of the straddle. However, once we increase the stock price, the theta of a put becomes negative; the theta becomes progressively more negative as the negative theta effects of the call are integrated. Both put and call have the same vega, and we know that vega is highest for at the money options. 91 Part 3 Options c) The delta of the one day time to expiration graph is a lot steeper. However, delta changes only in a small area around the strike price. With only one day to expiration left, it becomes increasingly clear whether the call option ends out of the money (delta_c = 0) and the put option ends in the money (delta_p = -1) or the call option in the money (delta_c = 1) and the put option out of the money (delta_p = 0). Taken together, this yields a delta of the straddle of either –1 or 1. 92 Chapter 12 The Black-Scholes Formula da) Vega: The one-day time to expiration vega graph shows only a small hump around the strike price of the option position. With only one day time to expiration left, we do not have enough time to participate in the opportunities the one percentage point increase in volatility offers to our bought straddle. However, with one year left, we see that the volatility increase has a huge effect on our straddle. db) Theta: 93 Part 3 Options Remember, we bought a call and a put option on the same strike of $ 100. This figure is a nice demonstration that for bought at the money option positions, time decay is greatest for short time to maturity positions. Our long straddle will pay off if either the call or the put is in the money. If the current stock price is about 100 and we have only one day to expiration left, our option position will likely expire worthless. Therefore, there is a huge time decay. With longer time to maturities, chances of stock price movements away from a 100 are substantial. Therefore, the theta is much smoother and smaller. 3) rho: With one day to maturity left, a 100 basis point increase in the interest rate has no effect on the option position, because the time we could earn interest/lose interest on the strike is just too short. For the one year to maturity figure, we can see the following: If the stock price is higher than $100, it is the call option that is in the money, and the put expires worthless. Th...
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