Homework 7 Possible Solutions

 

K Foster, Options & Futures, Eco 275, CCNY, Spring 2010

 

 

  1. A put option on a stock has strike of 75; the current price of the stock is 78.50.  The riskfree rate is 3%, time to expiry is 45 days, and the volatility is 17%. 
    1. What is the Black-Scholes implied value? 

The put is .5328.  (See hw7sol.xls)

    1. What is the value of a call with same characteristics? 

The call is 4.3135.

    1. Does put-call parity hold?

Check: c – p = 3.7807.

    1. If the price of the stock falls to 77.5, how much do the call and put prices change?  What does this imply about the delta of the call and put?  Using the formula, what is the delta and gamma for each?

The put changes by -.2286 for a dollar change in the underlier; the call changes by .77114 for a dollar change in the underlier.  The formula for infinitesimal changes would give the put delta equal to  and  , so for the put is -.1973 and for the call is .8028.  We could also use the spreadsheet to evaluate for small changes (I show an example for .001) to also get a numerical value of delta.

  1. In class we discussed hedging a portfolio of SPX contracts with options on that index.  Calculate the implied volatilities for those option prices, with riskfree rate from LIBOR (look it up!), assume q=2%, S0 = 1192.13 and T=35 days. 

I used r=0.02 and q=0.01 because LIBOR is crazy low and otherwise r-q would be negative (bad).  See hw7sol.xls for details (used GoalSeek):

K=

puts

Bl-S-M valuation

implied sig

925

0.75

0.7500

0.3933

950

0.95

0.9501

0.3702

975

1.1

1.1001

0.3417

1000

1.6

1.6008

0.3258

1025

2.1

2.1000

0.3033

1050

2.7

2.7002

0.2788

1075

3.5

3.5003

0.2536

1100

5

5.0000

0.2333

1125

7.5

7.5008

0.2154

1150

10.7

10.7002

0.1918

1175

17.4

17.4000

0.1801

1200

26

26.0000

0.1586

 

 

 

Find payoffs to the fund, if you bought different put options.  Which do you think offers the best set of risks?

We can graph the different payoff functions, for a portfolio of one stock and one put; this will put a lower bound on the possible losses.  Puts that are far out of the money are cheap.  With the index currently at 1192.13, for just 0.75 (6 bps) you can protect against the value falling below 975 (losing more than 23.5%).  For more money you can buy more protection.

  1. A set of options mature in 2 months.  The stock has volatility of 32%.  LIBOR is 3%.  The current market price of the underlier is 25.  It pays dividends at a continuous rate of 1%. 
    1. What is the price of a bull spread, with strikes of 24 and 26, using the B-S-M model?

The call with K=24 costs 1.883; a call with K=26 is 0.917.  So a long position in the call with K=24 and a short on K=26 costs net just 0.965.

    1. If the security paid no dividends, what would be the cost of a bull spread?  Is it more or less costly?  Why?

Without dividends, the calls are 1.910 and 0.935 so net 0.975.  The slight increase in cost reflects the change in 'earning power' of the stock as it accumulates dividends.

4.        Consider the prices of currency options on the "zing" (the currency of Zembla).  Zemblan interest rates are at 2% while US riskfree rates are 1%.  The zing trades at 10 zing/$ today; its volatility is 20%.  What is the cost of an ATM straddle, expiring in 3 months, as implied by B-S-M model?  Would the cost be more or less, for an ATM straddle expiring in 6 months?  If the zing suddenly became more volatile?

An ATM straddle would buy a call and a put; the Bl-S-M values of these are c= 0.1865 and p=0.1860 so the total cost is 0.3725.  For a six-month position, c=.2558 and p=.2543 so the straddle is .5101 – more expensive since it offers insurance over a longer time period.  (The payoff graph of the net profit on the position sinks lower.)  If the currency became more volatile then the straddles would be more expensive (since there is a higher chance of them expiring in the money).