Question

You own one call option and one put option on Shell, both with a strike price of 80. The interest rate is 5% and the time to expiration is nine months. The standard deviation of Shell is 25 percent. Graph on the same graph the value of the call and the put as the price of Shell goes from 70 to 130. (So that is two lines on the same graph.) Note:at least 50 data points on the graphs

Answer 1

We use Black-Scholes Model to calculate the value of the call and put options.

As an example, we calculate the value of the call and put option when the price of Shell is 70

The value of a call and put option are:

C = (S₀ * N(d₁)) - (Ke^-rt * N(d₂))

P = (K * e^-rt)*N(-d₂) - (S₀)*N(-d₁)

where :

S₀ = current spot price

K = strike price

N(x) is the cumulative normal distribution function

r = risk-free interest rate

t is the time to maturity in years

d₁ = (ln(S₀ / K) + (r + σ²/2)*T) / σ√T

d₂ = d₁ - σ√T

σ = standard deviation of underlying stock returns

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(70 / 80). We input the same formula into Excel, i.e. =LN (70 / 80)
• (r + σ²/2)*T = (0.05 + (0.25²/2)*(9 / 12)
• σ√T = 0.25 * √(9 / 12)

d₁ = -0.3353

d₂ = -0.5518

N(d₁), N(-d₁), N(d₂),N(-d₂) are calculated in Excel using the NORMSDIST function and inputting the value of d₁ and d₂ into the function.

N(d₁) = 0.3687

N(d₂) = 0.2905

N(-d₁) = 0.6313

N(-d₂) = 0.7095

Now, we calculate the values of the call and put options as below:

C = (S₀ * N(d₁))   - (Ke^-rt * N(d₂)), which is (70 * 0.3687) - (80 * e^{(-0.05 * (9/12))})*(0.2905)    ==> $3.4212

P = (K * e^-rt)*N(-d₂) - (S₀)*N(-d₁), which is (80 * e^{(-0.05 * (9/12))})*(0.7095) - (70 * (0.6313) ==> $10.4768

Value of call option is $3.4212

Value of put option is $10.4768

In this way, we calculate the values of call and put options when the price of Shell goes from 70 to 130

Shell Price	Call Option Value	Put Option Value
70	3.4212	10.4768
71	3.8024	9.8579
72	4.2086	9.2641
73	4.6398	8.6953
74	5.0960	8.1516
75	5.5770	7.6326
76	6.0826	7.1382
77	6.6124	6.6680
78	7.1661	6.2216
79	7.7430	5.7985
80	8.3426	5.3982
81	8.9644	5.0199
82	9.6076	4.6632
83	10.2717	4.3272
84	10.9558	4.0113
85	11.6591	3.7147
86	12.3810	3.4366
87	13.1206	3.1762
88	13.8772	2.9327
89	14.6499	2.7054
90	15.4379	2.4935
91	16.2405	2.2961
92	17.0569	2.1125
93	17.8863	1.9419
94	18.7281	1.7836
95	19.5814	1.6369
96	20.4456	1.5011
97	21.3199	1.3755
98	22.2039	1.2595
99	23.0968	1.1524
100	23.9981	1.0537
101	24.9072	0.9627
102	25.8235	0.8790
103	26.7466	0.8021
104	27.6759	0.7314
105	28.6110	0.6666
106	29.5516	0.6071
107	30.4971	0.5526
108	31.4472	0.5028
109	32.4016	0.4571
110	33.3599	0.4154
111	34.3218	0.3773
112	35.2870	0.3425
113	36.2552	0.3108
114	37.2263	0.2819
115	38.2000	0.2555
116	39.1760	0.2315
117	40.1542	0.2097
118	41.1343	0.1899
119	42.1163	0.1718
120	43.0999	0.1554
121	44.0850	0.1405
122	45.0715	0.1270
123	46.0592	0.1148
124	47.0481	0.1037
125	48.0381	0.0936
126	49.0290	0.0845
127	50.0207	0.0763
128	51.0133	0.0688
129	52.0065	0.0621
130	53.0004	0.0560