Consider a stock paying continuous dividends of 1%. Assume r = 0.06, = 0.32 and
ID: 2730162 • Letter: C
Question
Consider a stock paying continuous dividends of 1%. Assume r = 0.06, = 0.32 and today’s stock price of S0 = 33. a) You sell 100 Calls with strike K = 35 and expiration in 68 days (assume 365 days in a year). You also construct a delta-hedge to manage your risk.
• If tomorrow (1 day later) stock price rises to $34.50, find your net profit/loss from your hedged portfolio. Hint: don’t forget about dividends!
• Repeat this problem for the case of selling 100 Puts with strike K = 35 and all other parameters staying the same
Explanation / Answer
Ans;
first we use black sholes model to determine the price of the call options
calculations are as follows
d1 = ln(spot/strike)+((rf-y)+(stddev^2)/2)*t))/(stdev*sqr(t))
here spot = 33
strike = 35
rf = 0.06
y=0.01
stdev=0.32
t=68/365
d1 = -0.28951
N(d1) = normsdist(d1) = 0.386097
d2= d1-stdev*sqrt(t) = -0.42763
N(d2) = normsdist(d2) = 0.334461
Call price = Spot * exp^(-y*t)*N(d1) - strike*exp^(-rf*t)*N(d2)
This gives the call option price as $1.141459
100 such calls would fetch $114.1459. This is the amount we got as premium on selling the call options
Next day when the spot rises to 34.5 we calculate the price of the option again in a similar fashion as above with only change being spot as 34.5 and t = 67/365
call price now becomes $1.798381.
This results in a loss for us as far as premiums from the options is concerned and the loss amount is 179.8381-114.1459 = $65.69224
for hedging the positions we bought the underlying stock 100 units at 33 and the price went up to 34.5 which results in a gain of $150
So overall gain = 150-65.69224 = $84.3078
You can do the same for put options, the only change is in the formula for price of put option.
which is put option price = strike*exp^(-rf*t)*N(-d2)-spot*exp^(-rf*t)*N(-d1)
where N(-d1) = 1 - N(d1)