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Homework 6
K Foster, Options & Futures, Eco 275, CCNY, Spring 2010 |
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1. A certain security is currently worth $23. Its annualized volatility is 25%. LIBOR is 0.045.
a. To value a call option that matures in 5 months with a strike price of 25,what price would be implied by the Black-Scholes-Merton model?
The call would be 0.8813 (see hw6sol.xls).
b. A put option with the same strike?
The put is 2.4170.
c. Is put-call parity satisfied?
Yes; c – p = -1.5356.
d. What if you used a one-step tree model -- how much different are the valuations? (Remember that Hull suggested a link between volatility and step size.)
Hull suggests and . So u=1.1751 and d = .8510 (growth rates) or Su=u*23 = 27.03 and Sd=d*23 = 19.57. So find the amount, ∆, such that S - ∆c has the same value in state U or D. S0 is 23; the call has value in u of (27.03 – 25) = 2.03; in d the call has no value. So set 27.03 – ∆2.03 = 19.57 and solve to get ∆ = 3.68. The current value of this portfolio is 23 - ∆c = 23 – 3.68c. The present value of the riskless payoff is = 19.21. Set 23 – 3.68c = 19.21 and find c=1.03.
2. You are overseeing a new portfolio manager, who currently has a $50,000 in shares of stock in Mega Corp (price of 37, 40% volatility). The new portfolio manager plans to write (i.e. take a short position in) 1300 calls with a strike of 38 expiring in 9 months and to write (i.e. take a short position in) 13000 puts with a strike of 35 expiring in 9 months. (Your company reports annual results in 10 months, thus the managers are focused on getting good returns over that horizon.) Assume LIBOR is 3%. The new portfolio manager says that the positions are an excellent hedge. Investigate whether this is correct.
a. What are the Black-Scholes-Merton (BSM) prices for the call and put (how much is the bank getting for writing these)?
The value of the call is 5.022; the value of the put is 3.63. So selling 1300 calls gets 1300*5.022 = $6528; selling 13000 puts gets $47,190. Assume these funds are invested into riskless bonds so the net gain on the transaction (as of the date of entry) is zero – the portfolio has a liability (from the call) of $6528 but assets (from the call) of $6528 also; same for the put.
b. If, after 4 months, the stock price has gone up to 40, what are now the BSM prices?
With now 5 months until expiration, the value of the call rose to 5.338 so the portfolio has lost $411 on that short position. The value of the put fell by 1.704 so the portfolio has gained $25,033 on that position. See the table below for the summary of position changes.
c. If, instead, after 4 months, the stock price has gone down to 35, what are now the BSM prices?
If instead the stock had fallen, then the value of the call is 2.60, so the portfolio lost $3154 here. The put fell to 3.36 so the portfolio gains here.
d. Evaluate the claims that this portfolio is hedged. (Assume that Mega Corp's beta with market returns is 80%.)
It does not appear that the portfolio is well hedged: when the stock position loses $100,000 the portfolio loses $92,792; when the stock position gains $150,000 the portfolio gains $175,161.
Table from hw6sol.xls
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begin |
portfolio |
if S=40 |
portfolio |
if S=35 |
portfolio |
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shares |
$ 1,850,000.00 |
shares |
$ 2,000,000.00 |
shares |
$ 1,750,000.00 |
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call (as liability) |
$ (6,528.05) |
call (as liability) |
$ (6,939.53) |
call (as liability) |
$ (3,373.93) |
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bond from call sale |
$ 6,528.05 |
bond from call sale |
$ 6,593.66 |
bond from call sale |
$ 6,593.66 |
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put (as liability) |
$ (47,189.74) |
put (as liability) |
$ (22,157.20) |
put (as liability) |
$ (43,675.85) |
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bond from put sale |
$ 47,189.74 |
bond from put sale |
$ 47,664.01 |
bond from put sale |
$ 47,664.01 |
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all: |
$ 1,850,000.00 |
all: |
$ 2,025,160.94 |
all: |
$ 1,757,207.89 |
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change |
$ 175,160.94 |
change |
$ (92,792.11) |