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If a riskless security costs 100 today and will cost 120 at time T, a stock costs 50 today and will either be 70 or 30 at time T, and call options on the stock have strike price 50 expiring at time T, how would you construct a portfolio of stocks and call options such that the final value of the portfolio is 120 no matter the final stock price? Also how would you find the price of the call option?
Please explain, thanks.
If the riskless security cost $100$ at time $t=0$ and $120$ at time $T$ then the risk free rate, $r$, is $20\%$. So that, $r=0.2$. Denote the initial stock price as $S_0$ and price of the call option as $c$.
Suppose that at time $t=0$ you buy one stock and sell $\Delta$ options. Your portfolio value at time $t=0$ is $$P_0 = -\Delta\times c + S_0$$.
At time $t = T$ stock may cost either $S_u = 70$ or $S_d = 30$. At the maturity the price of a call option, $c$, can be calculated as $\max (S_T - K, 0)$. Hence at time $t = T$
when stock price goes up your portfolio cost $P_u = S_u - \Delta\times c = 70 - \Delta \times (70-50) = 70- \Delta \times20$;
when stock price goes down your portfolio costs $P_d = S_d - \Delta\times c = 30 - \Delta\times 0 = 30$.
Hence if you want to construct the risk free portfolio (one that does not depend on a stock price) you need to solve for $\Delta$: $$70 - 20\Delta = 30$$ $$\Delta = 2$$.
This means that if you buy at time $t=0$ one share for $50$ dollars and sell two options, then at time $t=T$ your porfolio value is $30$ no matter if the stock price goes up or down.
If you want your portfolio to cost $120$ at time $T$ you just need to multiply your securities by $4$ (buy $4$ shares and sell $8$ options at time $t=0$).
To find option price in your portfolio remember that it is a riskless portfolio. Hence it yields risk free interest rate $r=0.2$. Thus, $$P_0\times (1+r) = P_u = P_d$$ $$(50-2\times c)(1+0.2) = 30$$ $$c = 12.5$$
Put-call parity says that the difference between a call and a put is equivalent to the difference in the current stock price (adjusted down for dividends) and the strike price discounted at the risk-free rate.
$$Call - Put = S_0*e^{-div} - K*e^{-rt}$$
So, if you want to have 120 dollars in the future, you would need to need to have $120 worth of "K" or 2.4 shares worth. Accordingly, we adjust the equation as follows:
$$2.4*Put - 2.4*Call + 2.4*S_0*e^{-div} = 120/1.2 $$
So, you would need to long 2.4 puts, short 2.4 calls, and long 2.4 shares (or contracts) of the underlying. This is equivalent to holding the risk-free asset and earning the 20% interest.
To value the calls and puts, you would need to use a binomial lattice. See the attached page for details.
In short, you need to find the probability of an up and a down move in stock price, multiply the payoffs by those probabilities and discount the probability weighted payoff at the risk-free rate of 20%. The probability of an up move is:
$$\frac{rfr - r(down)}{r(up)-r(down)} or \frac{0.20 - -0.40}{0.40 - -0.40} = 75\%$$
So, the call is worth $\$20 *75\% /1.2 = 12.5$
The put is worth $\$20 *25\%/1.2 = 4.1667$
As a check, the outlay today is $2.4*50 = 120$ for the shares. You pay \$10 for the puts and collect \$30 for the calls. Total outlay is $100.
Also, check any scenario for the price of the shares, you will see that the payment will be \$120. The calls/puts create a synthetic short of the stock and only thing not canceled out is the 2.4 shares of strike at \$50 = \$120.
Let's call R the riskless security (100 today, 120 at time T).
And call S the stock = 50, and either 70 or 30 at time T.
One way to look at it is:
A] Consider: buy 2 call options (C), short the stock (S), invest 50 (proceeds from S) in R. At time T:
S=70: 2C=40, buy back S=-70, proceeds from R=60. net: 30
S=30: 2C=0, buy back S=-30, proceeds from R=60. net: 30
The value of the portfolio today must be the same as the value of the portfolio at time T, so the value of the 2 calls today must be worth the same as 30 at time T. Now, we can invest $\frac{30}{120}=\frac{1}{4}$ of R (100) today =(~\$25) which will be worth \$30 at time T. Thus each call today is worth \$25 / 2 ~= 12.5.
[Edit note: Thanks RF - helped me realise the error]
(Note: if the calls were cheaper we could make money by buying the above portfolio, or if they were more expensive we could sell calls and buy the stock, either way for more profit than the riskless security)
B] Also consider: buy the stock and 2 puts. Cost is 50 + 2xP. At time T:
S=30: 30+2x20=70; or
S=70: 70+2x0 =70
This is the same as if 58.33 is invested in R, as it will be worth .5833*120=70 at time T. As today's value of the portfolio must be also be 58.33, P must cost ~4.167
C] To have 120 at time T, you must start with 100, and either:
1. buy $\frac{120}{70}$ stock (cost=85.72) and $2\times\frac{120}{70}$ puts (cost=14.28).
OR
2. buy 8 call options, short 4 stock, invest S*4 in R. 8 call options @12.5 will cost 100.
OR
3. As RF pointed out, buy 2.4xS=120, buy 2.4xP=10, sell 2.4 calls at 12.5=30, for a cost of 100.
Either way, all portfolios will be worth 120 at time T, irrespective of what happens to S, just as if you had invested 100 in R.
