May 29

Put call parity - Practice question (Par 4 difficulty)

by David Harper, CFA, FRM, CIPM

FRM |

iStock_000002520146XSmall

Assume the following:

  • Stock price = $40
  • Strike price = $40
  • Volatility = 35.5%
  • Risk-free rate  4%
  • Term = 3 months
  • No dividend
  • Value of a European-style put option on this stock = $2.60

Question:

  • What is the value of a European-style call option on the same stock, also with strike of $40?
  • What is the minimum value (i.e., lower bound) of the call?
  • If the options are instead American-style (without dividends), when is it optimal to exercise them?
  • If the options are instead American-style (paying dividends), when is it optimal to exercise?
  • If the stock pays a dividend of 2%, how do we adjust the Black-Scholes-Merton to account for the dividends? What is the intuition?

 

(don't peek until you try)

 

 

 

 

 

 

 

 

 

 

 

 

Answers:

(i)

Put call parity (for European style options): c-p = S-(K)EXP[(-r)(T)].
I like to remember this as: c-p = minimum value.

In this case, call = $40 - ($40)EXP[(-4%)(0.25)] + $2.60 = $3.00

(ii)

Minimum value of a call option = Max[Stock - Discounted Strike,0]. In this case,
$0.40 = $40 - ($40)EXP[(-4%)(0.25)]

 

By the way, lower bound on the put = Max[Discounted Strike - Stock, 0]. In this case,
$0.0 because: ($40)EXP[(-4%)(0.25)] - $40 < 0.

(iii)

Never

(iv)

Only before the stock goes ex-dividend.

(v)

Primarily we reduce the stock price (d1 is also adjusted and, by extension d2). Instead of N(d1)(S), we use N(d1)(S)EXP[(-2%)(0.25)].

If you think about the cost of carry model, this is reducing the stock price by the present value of the dividend stream (the holder of the stock receives the dividends, the option holder forgoes them). Another way to think about this: if the total expected return on the stock = 12%, now 2% are paid in dividends, so the expected capital appreciation is 10%. The value of the option is therefore less; in the case, option goes from $3.00 to $2.80

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