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Social ScienceThe prices of a certain security follow a geometric Brownian motion with parameters mu=.12 and sigma=.24. If the security's price is presently 40, what is the probability that a call option, having four months until its expiration time and with a strike price of K=42, will be exercised? (A security whose price at the time of expiration of a call option is above the strike price is said to finish in the money.)If the interest rate is 8%, what is the risk-neutral valuation of the call option?
The prices of a certain security follow a geometric Brownian motion with parameters mu=.12 and sigma=.24. If the security's price is presently 40, what is the probability that a call option, having four months until its expiration time and with a strike price of K=42, will be exercised? (A security whose price at the time of expiration of a call option is above the strike price is said to finish in the money.)
If the interest rate is 8%, what is the risk-neutral valuation of the call option?
Expert Answer
The question has two parts:
- The probability that a call option will get exercised in 4 months - this question has been addressed in the previous question posted by you
- The risk neutral valuation of the call option - I will address this part here.
We will evaluate the risk neutral valuation of the call option using the Black Scholes Model.
Inputs are:
- Current stock Price, S
- Strike Price, K
- Risk free rate, r
- Volatility, σ
- Time to maturity, t
The Black Schole formula to value the European call option on a non dividend paying stock is as shown on the white board.
Image Transcriptionclose
Value of call S N (d1) - Ke N(d2) where (1 d2 d1-o Vt
Since the entire calculation is intensive, i will resort to excel.
Please see the table on the white board. Please be guided by the last column titled “Formula used” to understand the mathematics. The last r...
Image Transcriptionclose
C D 7 Inputs 8 S 40 0.3333 = 4 months = 4 / 12 years 24% Sigma is = 0.24 9 t 10 a 11 K 42 8.0% Formula Used 12 r Values 13 Output -0.090380558 (LN(C8/C11)+(C12+C10^2/2)*C9)/(C10*SQRT(C9)) -0.228944623 =C14-C10*SQRT(C9) 14 d1 15 d2 0.463992403 =NORMSDIST(C14) 0.409455979 NORMSDIST(C15) 1.82 C8*C16-C11*EXP(-C12*C9)*c17 16 N(d1)= 17 N(d2)= 18 C
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