call put party

[bestmarcus]

Hi david

ABC Company stock sells for $5 per share. A six-
month call option with a $6.5 strike price sells for $0.1. The risk-
free rate is 0.5% per month. Please calculate the price of a six-
month put option with a $6.5 strike price?

please give me some ideas

 

[David Harper CFA FRM]

Hi bestmarcus,

One of the tips I give in this newsletter is to memorize the minimum value of a call option because it helps in three ways.

1. minimum value (lower bound): c > S - K*EXP(-rT)
2. Once memorized, then just subtract put on left side for put-call parity: c - p = S - K*EXP(-rT)
3. And wrap-in N()s for Black-Scholes: c = S*N(d1) - N(d)*K*EXP(-rt)

So, hopefully, as just a memory aid, this cluster is easier to remember:

c > S - K*EXP(-rT) [greater than]
c - p = S - K*EXP(-rT) [equal to]
c = S*N(d1) - N(d2)*K*EXP(-rt) [wrap in N()s]

Note how minimum value (lower bound) is common. The minimum value also may remind you of the value of a forward contract b/c without volatility, they are the same! If vol = 0, c = S - K*EXP(-rt). This little equation is quite the utility player.

Since c - p = S - discounted strike,
put = c - S + K*EXP(-rT) and
put = 0.1 - 5 + (6.5)*EXP(-3%) = 1.41

Let me add two observations:

1. This assumes a European option because if this were an American option, it must be worth at least it's intrinsic value; i.e., P >= $6.5 - $5 = $1.50. I can only get this counter-intuitive result (p < intrinsic value) because it's European
2. Related to this, your problem also illustrates a rate exception to "Theta is negative." The theta for this option is positive; it's value converges to $1.50 as it approaches maturity (value actually, counter-intuitively, increases with shorter term!)

David

 

[bestmarcus]

Dear David
please clearify
the time for six months should be set 6 or 0.5 ( since time unit is 1 for one year, the six months should be 0.5)
i don't sure that
if you say time for six months is 6, i don't understand how to express 6 years


C = $0.1
S = $5
K=6.5
R=0.005
T=0.5 OR 6 for 6 months

P=0.1-5+6.5×e(-0.005*6) or 0.1-5+6.5/(1+0.005)^0.5
which formula is correct for time?

 

[David Harper CFA FRM]

Hi bestmarcus,

Your first is correct. But this is why continuous compounding is used. With other compound frequencies, the difference would matter. But with CC the difference does not matter (Linda Allen: CC is "time consistent"). We can add rates without loss of accuracy, so here:

EXP(-0.5%*6) = EXP(-6%*0.5)

Your first is correct, because the rate is given in MONTHLY TERMS (.005) and six months to expire (.005 * 6). But CC is helpful because, when i looked at it, I just personally like to annualize so i mentally used the annual number (.5% * 12 = 6%) and then one-half of that for six months b/c I know the CC can be added without loss.

The key only is that rT must finally here match the maturity (6 months). It doesn't matter how you convert, under CC, to get there.

You could be given monthly rates of 1%, 2%, 3%, 3%, 2%,1% for each of six months, for example, and you only need to add them: EXP(1%)*EXP(2%)*EXP(3%)*EXP(3%)*EXP(2%)*EXP(1%) = EXP(1+2%+3%+3%+2%+1%).

David