SHOW and EXPLAIN in FULL DETAIL! 1. Assume the Black-Scholes framework. The cont
ID: 2710132 • Letter: S
Question
SHOW and EXPLAIN in FULL DETAIL!
1. Assume the Black-Scholes framework. The continuously compounded
risk-free interest rate is r is unknown, but for a non-dividend paying
stock S we know:
• The current stock price S0 = 10
• The stocks volatility is 10%.
• The price of a 6-month European gap call option on S, with a
strike price of K1 = 10 and a payment trigger of K2 = 9.90, is 1.
• The price of a 6-month European gap put option on S, with a
strike price of K1 = 10 and a payment trigger of K2 = 9.90, is
0.50.
The definition of the payoffs is then
GGapCall(S) = (S - K1)1{S > K2}
GGapPut(S) = (K1 - S)1{S < K2}
Calculate r.
Explanation / Answer
Continuously compounded risk-free rate = 3.58%
Current Stock Price S0 = 10
Stock Volatility = 10%
Price of 6-month European call option = 1
Strike price K1 = 10, payment trigger K2 = 9.90
Price of 6-month European Put Option = 0.50
Strike Price K1 = 10 and Payment Trigger K2 = 9.90
Value of the call option C(S,t) = N(d1) S – N(d2) Ke^-r(T-t)
Where d1 = 1/(T-t)^1/2 * [ln (S/K) + (r+ ^2/2) (T-t)]
d2 = d1 - (T-t)^1/2
Where T-t = time to maturity = 6 months
S is the spot price = 10
K strike price = 10
r is risk-free rate
is volatility = 10% or 0.10
d1 = 1/(0.10*(6)^1/2) *[ln(10/10) + (r + (0.10)^2/2)(6)]
d1 = 1/(0.10*2.449489743) *[ln(1) + (r+ (0.01/2)) * 6]
d1 = 1/0.24449489743 * [0 + (r+0.005)*6]
d1 = 4.08247 * (6r + 0.03)
=24.49482 * r + 0.1224741
d2 = d1 – 0.10 * 6^1/2
d2 = 24.49482 * r + 0.1224741 – 0.10 * 2.4494897428
d2 = 24.49482 * r + 0.1224741 -0.244948974
d2 = 24.49482 * r – 0.12247487
Substituting values of d1 and d2 in value of call
1 = 24.49482 * r + 0.1224741 * 10 - 24.49482 * r – 0.12247487 * 10 *e^-r*6
1 – [ 24.49482 * r + 0.1224741 * 10 - 24.49482 * r – 0.12247487 * 10 *e^-r*6] = 0
1- 24.49482 * r - 0.1224741 * 10 + 24.49482 * r + 0.12247487 * 10 *e^-r*6 = 0
1-0.01224741 + 1.2247487 * e^-6r = 0
0.98775259 + 1.2247487 * (2.71828)^-6r = 0
2.71828^-6r = 0.98775259/1.2247487
2.71828^-6r = 0.8064941
Taking Logarithms
log(2.71828^-6r) = log 0.8064941
-6r * log 2.71828 = log 0.8064941
-6r = log 0.8064941/log2.71828
-6r = - 0.09339881/0.43429242
6r = 0.215059728
r = 0.215059728/6 = 0.0358432 or 3.58% (rounded off)
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