In the problems following, use an equity risk premium of 5.5 percent if none is
ID: 2780947 • Letter: I
Question
In the problems following, use an equity risk premium of 5.5 percent if none is specified.
1. The following are prices of options traded on Microsoft Corporation, which ays no dividends Call Put One-month Three-month Six-month K=85 K=90 $2.75 $1.00 $4.00 $2.75 $7.75 $6.00 K=85 K=90 $4.50 $ 7.50 $5.75 9.00 $8.00 $12.00 The stock is trading at $83, and the annualized riskless rate is 3.8%. The stan- dard deviation in ln(stock prices) (based on historical data) is 30% a. Estimate the value of a three-month call with a strike price of $85 b. Using the inputs from the Black-Scholes model, specify how you would repli- cate this call. c. What is the implied standard deviation in this call? d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position. e. Using put-call parity, estimate the value of a three-month put with a strike price of $85.Explanation / Answer
1a) The Black Scholes model formula for a call option without dividend can be written as:
C=S0× N(d1)-Xe-rt×N(d2)…….(1)
Where N (d1) and N (d2) are the standard normal cumulative distribution function
The formula for d1 and d2:
d1= (ln(S0/X)+((r+(2/2))t)/(t)…..(2)
d2=d1-t……..(3)
Given information in the question:
S0=83 (Current Stock Price)
X=85 (Strike Price)
r=0.0380 (Risk free rate)
t=3 months=3/12=0.25 (time to expiry)
=0.30 (standard deviation)
First let’s find out d1 and d2 using equation 2 and 3:
d1= (ln(83/85)+((0.0380+(0.302/2))0.25)/(0.300.25)
= (-0.02381+0.02075)/ (0.15)
= -0.0204
Therefore, N(d1) in the Z table or excel function of normsdist(-0.0204)=0.49186
d2=-0.0204-0.300.25
= -0.1704
Therefore, N(d2) in the Z table or excel function of normsdist(-0.1704)= 0.432348
Using equation 1 the value of the 85 strike call option is:
C=83×0.49186-85e-0.0380(0.25) × 0.432348
=40.82441 -36.40209
= 4.422321 (rounded to $ 4.42)
1b) To replicate this call option using Black Scholes formula you need to buy delta N(d1) and borrow an amount equivalent to the term Xe-rt× N(d2).
Therefore, you need to buy 0.49186 shares (from problem 1a) , and borrow 85e-0.0380(0.25) × 0.432348=$ 36.40209 to replicate the same payoff as to that off the $ 85 strike price call.
1c) So this questions asks us to find the implied volatility of call option. We know the value of 85 strike call option from Black Scholes pricing model to be $ 4.42, while the market price of the call option $ 4.
So the call is undervalued. So using equation 1 (Black Scholes model) we have to keep on changing the value of standard deviation () unless Black Scholes model returns us a value of $ 4. Kindly note there are no closed form solutions to find the implied volatility of an option.
We know that as volatility (standard deviation) increases the call price will increase. In this case the model price ($ 4.42) is greater than the market price ($ 4), so we need to plug in values of standard deviation below 30 % (originally given) to match the market price.
So when standard deviation is changed from 30 % to 27.39 % the black Scholes model also returns a price of $ 4.
d1= (ln(83/85)+((0.0380+(0.27392/2))0.25)/(0.27390.25)
= (-0.02381+0.018878)/ (0.13695)
= -0.03602
N(d1)=N(-0.03602)= 0.485633
d2=-0.03602- 0.27390.25
=--0.17297
N(d2)=N(-0.17297)= 0.431337
Therefore, C=83×0.485633-85e-0.0380(0.25) × 0.431337
=$ 3.990524 (Rounded to $ 4)
Conclusion: The implied standard deviation of the call is 27.39%.
1d)
Pay off (Buy)
Payoff (Sell)
Stock Price
85
90
Net Payoff
0
-4
2.75
-1.25
20
-4
2.75
-1.25
50
-4
2.75
-1.25
70
-4
2.75
-1.25
80
-4
2.75
-1.25
85
-4
2.75
-1.25
86
-3
2.75
-0.25
87
-2
2.75
0.75
88
-1
2.75
1.75
89
0
2.75
2.75
90
1
2.75
3.75
100
11
-7.25
3.75
120
31
-27.25
3.75
200
111
-107.25
3.75
(Call premium for 85 strike is taken to be $ 4, while for 90 strike call is taken to be $ 2.75)
1e) Put call parity can be written as:
C+Xe-rt=S+P
Where C=Call Price ( $ 4.42 from 1a)
X= Strike Price= 85
S=Stock Price=83
r=risk free rate=0.0380
t=time to maturity=0.25
Rearranging the above equation in terms of put:
P= C+Xe-rt-S
=4.42+85e-0.0380×0.25-83
=4.42+ 84.19632-83
=5.61632 (Rounded to $ 5.62)
Pay off (Buy)
Payoff (Sell)
Stock Price
85
90
Net Payoff
0
-4
2.75
-1.25
20
-4
2.75
-1.25
50
-4
2.75
-1.25
70
-4
2.75
-1.25
80
-4
2.75
-1.25
85
-4
2.75
-1.25
86
-3
2.75
-0.25
87
-2
2.75
0.75
88
-1
2.75
1.75
89
0
2.75
2.75
90
1
2.75
3.75
100
11
-7.25
3.75
120
31
-27.25
3.75
200
111
-107.25
3.75
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.