Dr. P.V. Viswanath
Introduction to Mergers and Acquisitions: Options
© P.V. Viswanath, 2006
Consider the following security:
Note that this security has no risk. Assuming an equilibrium rate of return of 8% on other assets with zero risk, the appropriate discount rate in this case should yield a price/present value of 100/1.08 = 92.59.
Now, let's look at another case.
What is the present value in this case? Let us assume that the S&P 500 Index represents the market portfolio. Then, since the asset covaries perfectly with the market index, the discount rate is simply the equilibrium rate of return on the market portfolio. Let's assume that this is 12%. In this case, we take the expected value of the payoff on the security, and discount it back to period t at 12%, which in this case, works out to $85.71.
Now what about the following case:
The expected cash flow in period t+1 is $98. If we discount this at 8%, the present value is $90.74; if we discount it at 12%, the answer is $87.50. But which is the correct answer?
Security 3's cashflows are a simple average of the cash flows of securities 1 and 2 in the 2 scenarios (states of the world). Hence, the value of security 3 is also a simple average of the values of securities 1 and 2. The present value of security 3 is 0.5(92.59) + 0.5(85.71) = 89.15.
Note that we did not need to know the probabilities of the two states, once we had the prices of the riskfree asset and the S&P portfolio. This is because we are using the principle of no riskless arbitrage. For the application of this principle, we do not need to know probabilities. This principle states:
If the cashflows on an asset can be expressed as a weighted average of the cashflows on a given set of traded assets, the market price of the first asset must be equal to the same weighted average of the market prices of the given set of traded assets. If this is not so, then there will be opportunities for riskless arbitrage profits.
But once we have this idea of a security being reckoned in terms of an average of other securities, we can take this further. Take security 4.
Can security 4 be considered a weighted average of securities 1 and 2? Let's conjecture that this is indeed possible. Then, let the weights be l and s. In this case, it must be true that
l(Cashflow in state 1 for sec. 1) + s(Cashflow in state 1 for sec. 2) = 10;
l(Cashflow in state 2 for sec. 1) + s(Cashflow in state 2 for sec. 2) = 0, which gives us the two simultaneous equations:
100l + 100s = 10
100l + 80s = 0
Solving these equations simultaneously, we get s = 0.5, and l = -0.4. Reconstituting the cashflows, we see that 100 (-0.4) + 100 (0.5) = 10 in state 1, and 100 (-0.4) + 80 (0.5) = 0 in state 2, as required. But this implies that the value of security 4 is simply (-0.4) (92.59) + (0.5) (85.71) = $5.82.
A call option on an asset is the right (but not the obligation) to buy a given quantity of that asset at a fixed (exercise) price within a fixed time period.
Suppose we have a call option on the portfolio of S&P 500 index stocks with an exercise price of $490. If the index (portfolio) value at time t+1 is $500, then the value of the call option is exactly $10, since we can buy the index portfolio for $490 and sell it for $500. If the index value at time t+1 is $400, then the call option is clearly worthless. We can graph this relation, using a payoff diagram (which shows the payoff to holding a certain asset or following a certain strategy as a function of another underlying asset's value):
Assume that only the values $400 and $500 are possible for the index portfolio at time t+1. Then, if we draw the binomial tree for this call option, we see that the cashflows are indeed the same as for security 4, and hence its value is the same.
The replicating portfolio of the index and the riskfree asset can be computed by solving the equations:
100l + 500s = 10
100l + 400s = 0
The solution is l = -0.4, and s = 0.1. The option value, as we remarked above is $5.82.
The value of an option depends on several parameters and we need to know how changing these parameters, one at a time, affects the option value:
We will use the previous example as our base example: a call option on the portfolio of S&P 500 index stocks with an exercise price of $490.
The volatility in the return on an asset is usually measured by the standard deviation of the return on the asset. Let us increase the volatility of the return on the S&P 500 Index, keeping the mean return constant. For example, let us assume that the values of the index portfolio at time t+1 can be either 360 or 510.
The value of the option will go up because the value of the option in state 1 is now $20, where it was formerly $10. The value of the option in state 2 continues to be zero. The replicating portfolio will, of course, be different, but the net effect will still be to increase the value of the call option.
To confirm, we can compute the replicating portfolio. We must solve the equations:
100l + 510s = 20
100l + 360s = 0
Solving, we find l = -12/25, and s = 2/15. This gives us an option value of 12.70, which is greater than 5.82, as conjectured.
We have assumed a riskfree rate of 8%. If we increase the riskfree rate to 10%, the option (being equivalent to a portfolio long the stock and short the riskless asset) will increase in value, since the present value of the amount required at t+1 has dropped. The equations to be solved are the same as before, and hence the replicating portfolio stays the same.
100l + 500s = 10
However, the value of the option, now, is -0.4 (100/1.1) + 0.1 (428.57) = 6.49 > 5.82.
Effect of changes in the current asset price:
Let us now change the current price of the asset. Since we need to keep the return volatitliy constant, this can be achieved by changing the discount rate for the asset. Suppose the discount rate is 14% (instead of 12%). Then the stock price will drop, and the value of the option, which consists of a portfolio long in stocks will drop as well. The equations to be solved are the same as before, and hence the replicating portfolio stays the same.
100l + 500s = 10
However, the value of the option, now, is -0.4 (100/1.08) + 0.1 (480/1.14) = 5.07 < 5.82.
If we change the exercise price to 480 instead of 490, the equations to be solved change to:
100l + 500s = 20
The solution to these equations is: = -0.8; = 0.2; hence the option value is -0.8 (100/1.08) + 0.2 (480/1.12) = 11.64 > 5.82.
Intuitively, this is clear, because the higher the exercise price, the lower the upside potential of the option.
Let us assume that there is yet another period before the option matures. Now, from date t to t+1, the stock price has been assumed to either increase by 16.67% (from 428.57 to 500) or decrease by 6.67% (from 428.57 to 400). Let us assume that during the next period too, only the same kind of increase or decrease is possible. Hence the scenario can be shown as follows:
In this instance, we need to compute the value of the call option at time t in two stages. We first compute the value of the option at time t+1 conditional on the index value. In other words, we compute what the call option value would be at t+1 if the index value were 500 or 400. Then we compute the option value at t, given the option values at t+1. This can be done in exactly the same way as we did it before.
Using this technique, we solve the equations:
100l + 583.33s = 93.33 and 100l + 466.66s = 0
to obtain the value of the call option at t+1 if the index value is 500.
Ans: l = -3.73; s = 0.8; call value = -3.73(100/1.08) + (0.8)(500) = 54.63.
If the value of the index value at t+1 is 400, the equations to solve are:
100l + 466.66s = 0 and 100l + 373.33s = 0
This gives us l = 0 and s = 0, or a call option value of 0.
The solutions to solve, in this case, are:
100l + 500s = 54.63 and 100l + 400s = 0
Solving, we get l = -2.185; s = 0.5463, for an option value of 31.81 > 5.82.
The reason that a greater time to maturity increases the value of the option is simply that the upside potential is greater, while the downside potential is always fixed.
The Black-Scholes formula gives the value, C, of a call option with exercise price E and expiring in t years, written on an asset with variance of the continuous return equal to s2 per year and currently selling at S. (This formula assumes that there is continuous trading, and that the option can only be exercised at maturity):
C = S N(d1) - E e-rt N(d2); and d2 = d1 -
N(.) is the area under the standardized normal curve (i.e. the plot of probability against value for a normally distributed variable with mean zero and s.d. = 1), and can be obtained by consulting normal probability tables.
Consider a call option with exercise price 490 and time to maturity equal to 1 year, on a stock selling at $428.57, with a variance of 0.01245 per year. The riskless rate of return is 8% per year. What is the value of the option?
d1 = -0.4277; d2 = -0.5393; N(d1) = 0.3345 and N(d2) = 0.2946.
C = 428.57 (0.3345) - 490 (.9231) (.2946) = 10.10
A put option on an asset is the right (but not the obligation) to sell a given quantity of that asset at a fixed (exercise) price within a fixed time period.
Example: Suppose we are considering a put option on the portfolio of S&P 500 index stocks with an exercise price of $490. If the index (portfolio) value at time t+1 is $500, then the value of the put option is worthless. If the index value at time t+1 is $400, then the put option is worth exactly $90, since we can buy the index portfolio for $400 and sell it for $490. We can graph this relation:
A person who sells a call option (with an exercise price of $490) and a person who sells a put option with an exercise price of $490 will, however, have the following payoffs:
We can also graph the payoff to the holder of one unit of an asset as a function of the asset price at some time in the future.
Now consider combining a long position in the index portfolio along with a put on the index portfolio.
Finally, let us combine a call with an investment in riskless bonds maturing at the same time as the call.
The payoff at expiration of the two combinations is the same. Hence, the present value of the two combinations must also be the same. In general:
C + E e-rt = P + S, where P is the price of the put.
We can graph the value of the call option before maturity as a function of the underlying asset price. In this case, the graph line will no longer be linear.
We can explain some of the agency problems of debt by looking at the equity in the firm as a call option on the assets of the firm. Consider what happens to the equityholders of a firm. If the firm does well, they pay off the bondholders and they get the entire upside. If the firm is not doing well, they don't have to payoff the bondholders in full; they can just walk away from the firm. Hence, what they have is similar to an option.
Example: Consider a firm with a security currently worth $100. Next period, if the security succeeds, the firm will be worth $200; else the value of the firm will drop to $40. Suppose the firm currently has debt with a promised payment of $60 next period.
Assuming an interest rate of 5%, the equations to be solved are:
200s + 1.05 L = 140 and 40s + 1.05 L = 0
The solution is s = 0.875, and L = -33.33, for an option value (equity value) of $54.17.
Correspondingly, the value of the debt can be diagrammed as follows:
The EM.TV case
In March 2000, EM.TV bought 50% of SLEC for €1.88b. At the same time, EM.TV also got a call to buy another 25% of SLEC for €1.16b by Feb. 28, 2001 (in 4 quarters). Simultanesly, Ecclestone got a put option to sell 25% of SLEC to EM.TV for €1.16b by May 2001 (in 5 quarters). However, the put option details were not released to the mariket at this time.
See Excel spreadsheet for analysis. In March 2000, according to the market, the change in the value of EM.TV (that is, the value including 25% of SLEC plus the call less the €1.88b payment), conditioned on all the information available at this point is -€2b. Again, assuming market efficiency, the change in the value of EM.TV when the news of the put option came out, equals -€2.2b. This includes the value of the put plus any reevaluation in EM.TV's value.
What is the value of the call that EM.TV got, and what is the value of the put that EM.TV gave? We don't really have enough information here to estimate these values precisely since along with the actual transfers of money and options, there is also updating by the market on the value of the other assets that EM.TV has; however, we can evaluate certain hypotheses -- for example, did the change in value reflect on the value of the options or was there an information component, as well?