19. Options

Let’s consider the example of a call option on a bushel of wheat, with an exercise price of $10. Figure 19.2 shows the payoff structure at maturity of this call option.

A line graph of the pay-off versus the underlying value of the price of a bushel of wheat. The line is flat till (10, 0) and follows an increasing trend up to (20, 10). Data are estimated.

The underlying value is the price of a bushel of wheat. The exercise price of $10 means that the payoff is $0 for every price (of a bushel of wheat) below $10. For every price above $10, the payoff is the price minus the exercise price. So, at $10.70, the payoff is $10.70–$10 = $0.70.

The formula for the value of a long position in a call at expiration is:

With S = the underlying value, and K = strike or exercise price.

At the time of writing in October 2022, wheat was traded for $9.66 per bushel. And it was about $12 at the start of the Russian invasion of Ukraine in February 2022. We can use these prices to see what the payoff would be. At a price of $9.66 per bushel, C = max(S−K,0) = max($9.66–$10,$0) = 0, if the option were to expire at that point in time. As the price of the underlying value is below the strike price, the call is said to be out-of-the-money. However, that does not mean that the value of the call is 0, unless it is at expiration. The longer the maturity of the call, ceteris paribus, the higher the probability that the underlying value will at some time exceed the strike price (and be in-the-money), and the higher its value. And since it has value, investors will be willing to pay a price for it, called a premium (which in a well-functioning market equals the consensus view).

Let’s suppose the premium that the owner has paid is $0.40 (we’ll get back to the drivers of that premium later on). Then the profit net of the premium is as follows: C-premium = max(S-K-premium,-premium) = max($9.66–$10-$0.4,-$0.4) = −$0.4. The payoff structure in Fig. 19.2 does not include the premium paid, but the profit diagram in Fig. 19.3 does include the premium.

A line graph of the profit versus the underlying value of wheat price. The line is flat till (10, minus 0.5) and follows an increasing trend up to (20, 9.5). Data are estimated.

Note that the payoff line has shifted below 0, and the break-even point is now at $10.40, where the underlying value equals the sum of the exercise price and the premium. Note that this does not mean that premiums are fixed: they move with the price of the underlying security. However, the buyer pays the premium to the seller at the time both parties enter into the contract, at which time the premium payment is fixed.

All of the above represent the perspective of the owner of the call, who has a long position. The seller, who writes the call, has the exact opposite payoff profile, as shown in Fig. 19.4. The payoff diagram shows that the losses can become very large when the wheat price increases.

A line graph of the pay-off versus the underlying value of the wheat price. The line is flat till (10, 0) and follows a decreasing trend up to (20, minus 10). Data are estimated.

The payoff of the written call at expiration can be expressed as follows:

At a price per bushel of $9.66, there is no payoff, since the price is below the strike price: -C = -max($9.66–$10,$0) = $0. But at a price of $12, the call is in-the-money, and negative value for the writer of the option: -C = -max($12–$10,$0) = −$2. Again, this is without the premium paid. Since this is the same contract as the long call example, we should use the same $0.40 premium. But now it works the other way: the writer of the call receives the premium as a compensation for writing the call. Hence, the line that is flat below the exercise price moves from $0 to $0.4 in Fig. 19.5.

A line graph of the profit versus the underlying value of wheat price. The line is flat till (10, 0.5) and follows a decreasing trend up to (20, minus 9.5). Data are estimated.

Example 19.1 provides a problem to get familiar with calculating the value of a call option with premium.

As mentioned above, a put option gives the owner the right to sell a security. Let’s again consider the example of an option on a bushel of wheat, again with an exercise price of $10. But this time it’s a put instead of a call. Figure 19.6 shows the payoff structure of this wheat put option for its owner.

A line graph of the pay-off versus the underlying value of wheat price. The line follows a decreasing trend from (0, 10) to (10, 0) and remains flat till (20, 0). Data are estimated.

The formula for the value of a long position in a put at expiration is:

The exercise price of $10 means that the payoff is $0 for every price (of a bushel of wheat) above $10. For every price below $10, the payoff is the exercise price minus the price of wheat. So, at a wheat price of $4, the payoff is $10–$4 = $6.

As in the case of a call, a premium is paid. But we will skip discussion of the put premium here.

The seller, who writes the put, has the exact opposite payoff profile versus that of the buyer. This is shown in Fig. 19.7. At outcomes below the exercise price, the seller makes a loss, which can become quite significant at high volumes (i.e., many puts written) and at lower wheat prices (i.e., a bigger gap between price and exercise price to cover). Such losses may be neutralised by offsetting underlying positions, as we will see in the combined exposures below. But the losses can be dramatic in case of uncovered puts, i.e. without offsetting underlying exposures. The losses can be even worse for short calls. Figure 19.5 shows that these losses are not limited. For short put options, the downside is limited to the exercise price (which the writer has to pay to the owner of the put option).

A line graph of the pay-off versus the underlying value of wheat price. The line follows an increasing trend from (0, minus 10) to (10, 0) and remains flat till (20, 0). Data are estimated.

The payoff of the written put at expiration can be expressed as follows:

At a price per bushel of $9.66, the payoff is −$0.34 at maturity: -P = -max($10–$9.66,$0) = −$0.34. But at a price of $12, the put is out-of-money, and there is no payoff for the writer of the option: -P = -max($10–$12, $0) = $0. Example 19.2 provides an exercise for the valuation of a short put option.

The above payoffs were on a stand-alone basis, but in practice people can have composite exposures. And they may use options to hedge their exposures. Hedging is a strategy that tries to limit financial risk. For example, a farmer who has an upcoming wheat harvest will have a profit that depends on the wheat price, such as shown on the left-hand side of Fig. 19.8: since his costs are $5 per bushel, his profit per bushel will equal the wheat price minus $5. At any price below $5 he will be making a loss. However, by buying the put option described in Fig. 19.6 with an exercise price of $10, the farmer gets protection against potential losses. His payoff is his profitability before hedging (S-$5) plus the put’s payoff (max[K-S,0]) minus the put’s premium, so: (S-$5) + max($10-S,0)-$0.4. This means he obtains a profit of $4.6 at all wheat prices below $10. You can check the profit of $4.6: the farmer receives $10 as exercise price, has a cost of $5, and pays a premium of $0.4. Only if the wheat price goes above $10, does his profit rise proportionately with the wheat price, but he will still be making $0.4 per bushel less than in the case without hedging. This looks like a great deal: the put offers full protection on the downside against limited cost on the upside. So perhaps, the option is too cheap at $0.40? That depends on the chance of the wheat price ending up below $10.

3 line graphs of the pay-off versus the underlying value of wheat price. The graphs illustrate the profit of farmers before hedging plus the put option equals the profit of farmers after hedging. The lines of the profit of farmers before and after hedging follow an increasing trend.

Whether to buy the put, and with what strike price, is not the only decision to make. The farmer also needs to decide on the volume: how many bushels does he expect to harvest? And for how many does he need protection?

A producer of packaged food faces an (almost) opposite exposure. The price per bushel hurts its profits since the food company needs to buy wheat for its production: profit per bushel is $17-S. Therefore, it might want to buy a call to protect against price rises. Figure 19.9 describes the company’s exposures before and after adding the call. As a result of this protection, the company’s payoff is now its original profit formula ($17-S) plus the call including its premium (max[S-$10,$0]-$0.4). So, for example, at a wheat price of $4 per bushel, its payoff is ($17–$4) + (max[$4–$10,$0]-$0.4) = $13 + $0–$0.4 = $12.6. And at wheat prices of $10 and higher, its profits are locked in at $6.60: all the losses on the original profit formula are offset by equal gains on the call. Note however that just like for the farmer, the company’s full exposure is not this simple, since it will also be determined by its production volumes—which are probably more predictable than the farmer’s harvest. Hence, for a full analysis, one should not only look at payoffs per unit (the price component), but at total payoffs including units (price times volume).

3 line graphs of the pay-off versus the underlying value of wheat price. The graphs illustrate the co-profit of food before hedging plus the call option equals the co-profit of food after hedging. The lines of the co-profit of food before and after hedging follow a decreasing trend.

Example 19.3 provides an example of a steelmaker who hedges the price of iron. Iron is an important input to the steelmaker’s production process and the iron price can fluctuate.

If you take another look at Figs. 19.8 and 19.9, you might notice that the farmer’s resulting exposure (third graph in Fig. 19.8) looks a lot like the call option used by the producer (second graph in Fig. 19.9), but simply at a higher level. And likewise, the producer’s resulting exposure (third graph in Fig. 19.9) looks very much like the put option used by the farmer (second graph in Fig. 19.8), but again at a higher level. In fact, one could argue that the farmer’s payoff on the portfolio containing the underlying value (in this case wheat) plus a put is essentially a call plus a bond (riskless debt with a fixed payoff) and that the company’s payoff on the portfolio containing a negative exposure to the underlying value and a call is a put plus a bond. Please note that the size of the bond equals the present value of the exercise price: B = PV(K). This was $10 in Figs. 19.8 and 19.9. One can make different combinations to arrive at the same result. This can be expressed in put-call parity:

That is, the combined payoffs of the underlying value S and the put P are equal to the combined payoffs of a bond B and a call C. Figure 19.10 visualises these relationships.

5 line graphs of the pay-off versus the underlying value. They illustrate graph S plus graph P and graph B plus graph C equals graph S plus P equals B plus C. The line of graph S plus P equals B plus C follows an increasing trend.

One can view capital structure (Chap. 15) in terms of options as well. As we will show, equity can be seen as a call option on the company’s assets (Merton, 1974); and corporate debt is effectively riskless debt minus a put on the company’s assets.

Options are most commonly traded on stocks. 3M is a US conglomerate company that is well known for its Post-It and Scotch Tape products. Table 19.2 gives quotations for options on 3M stock with expiration in January 2025, priced as of Mid-October 2022, when the price of the stock was $113.49. Table 19.2 shows three strike prices (in reality, many more are available): one much below the stock price, one very close to the stock price, and one much above the stock price. We need some terminology to understand the option quotations in Table 19.2:

With a strike price of $60, the call is very much in-the-money. The payoff would be: C = max [S − K, $0] = $113.45 − $60 = $53.45, if the option on 3M stock were exercised today. This is called the intrinsic value of the call option (see Fig. 19.14). The option premium—the average of bid and ask—is $54.52, which is $1.07 higher. This is called the time value of the call option. It is the amount an investor is willing to pay for an option above its intrinsic value and reflects hope that the option’s value increases before expiration due to a favourable change in 3M’s price (see Fig. 19.14).

A dual-line graph of value versus stock price plots the option values and the intrinsic values. Both the lines follow an increasing trend. The distance between the lines is the time value.

Conversely, at that strike price of $60, the put is very much out-of-the-money if 3M stock hits $110 at maturity, since max(K − S, 0) = max ($60 − $110, $0) = $0. Hence, the much lower option premium of $3.44 (=($2.97 + $3.90)/2) on the put than on the call. The put premium reflects only the time value, since the intrinsic value is zero for this particular put option.

At a strike price of $160, the reverse holds: the put is very much in-the-money (max[K − S, 0] = max [$160 − $110, $0) = $50), with a price around $50; and the call is out-of-the-money (max[$110 − $160, $0] = $0), and a price (or premium) around $6, reflecting a small but serious chance that 3M’s stock price goes over $160 in the next 3.25 years.

At a strike price of $110, there is only a small gap between the strike price and the share price of $3.45. However, the prices of the puts and calls are much higher, reflecting the possibility that the share price will go much higher or much lower than the strike price during the lifetimes of the options.

As discussed, Fig. 19.14 illustrates the option value (also called option premium or option price) as the sum of the intrinsic value and the time value:

Example 19.5 gives an exercise to calculate the value of stock options.

The binomial option pricing model (Cox et al., 1979; Rendleman, 1979) prices options by making the simplistic assumption that at the end of the next period, the underlying value has only two possible values. The two-state single period model values a call option by building a replicating portfolio, which is a portfolio of other securities that has the same value as the option in one period. As they have the same payoffs, the law of one price (see Box 4.1) tells us that the call and the replicating portfolio of stocks and bonds should have the same value.

We can also check the formula in an analytical way. Filling in the put and call option formulas of Eqs. 19.19 and 19.16, we get: S + Δ_p ∙ S + B_p = PV(K) + Δ_c ∙ S − B_c. Rearranging, we obtain: S + B_p + B_c = (Δ_c − Δ_p) ∙ S + PV(K). This means for the stocks that 1 = (Δ_c − Δ_p). This holds in our example: 0.5 − − 0.5 = 1. Next, for the bonds: B_p + B_c = PV(K). Again, this holds $5.8252 + $3.8835 = $10/1.03 = $9.7087.

The formula for the option delta Δ can be interpreted as the sensitivity of the option’s value to changes in the stock price at each point in time. Let’s expand our one period binomial tree in Fig. 19.15 for a call to a two-period binomial tree. Figure 19.17 shows that in the up state at $12 at t = 1, the stock can go up again to $14 in t = 2. It can also go down to $10. In the down state at $8 at t = 1, the stock can go up to $10 at t − 2 (the same as the down movement of the up state) or go down to $6.

A 2-period binomial tree illustration of stock of $10 is classified into up and down of $12 and $8 respectively. The up value is classified into up and down of $14 and $10. The down value is classified into up and down of $10 and $6.

We can value the call option in the multiperiod binomial tree by working backwards from the end at t = 2. Let’s start with the situation that the stock price has gone up to $12 at t = 1. Figure 19.18 shows the payoffs for the stock and the call. Remember that the exercise price of the call is $10. This results in values for the call of C_u = $4 in the up state with a stock price of S_u = $14, and C_d = $0 in the down state with a stock price of S_d = $10. Using eq. 19.14, we can calculate the delta Δ:

A binomial tree illustration of a stock of $12 is classified as up and down. The up values are stock $14 and call $4. The down values are stock $10 and call $0.

The next step is to calculate the number of bonds needed to finance the stock position. Using Eq. 19.15, we can derive the number of bonds B:

The final step is to derive the value of the call option at t = 1. Using Eq. 19.16, the price of the call option C is:

So, the value of the call option in the up state at t = 1 is $2.291.

We can repeat the same exercise for the down state. Figure 19.19 shows that the payoff on the call is $0 in the up and down state at t = 2. A call with zero payoffs in the future also has zero value today. So, the value of the call option in the down state at t = 1 is $0. To check this, we can also calculate the delta

A binomial tree illustration of a stock of $8 is classified as up and down. The up values are stock $10 and call $0. The down values are stock $6 and call $0.

The delta is also zero, so we need zero stocks to hedge this position.

The final step is calculating the value of the call option at t = 0 in Fig. 19.20. As discussed before, the option delta—which is the number of stocks needed to hedge the call option—has to be recalculated at each point in time when the stock or call data change. Although Fig. 19.20 and Fig. 19.15 look almost the same, there is a small difference. The payoff on the original call option in the one-period model in Fig. 19.15 is $2 at t = 1 and that on the new call option is $2.291 in the two-period model in Fig. 19.20 at t = 1. Using Eq. 19.14, the delta Δ is:

A binomial tree illustration of a stock of $10 is classified as up and down. The up values are stock $12 and call $2.291. The down values are stock $8 and call $0.

Using Eq. 19.15, the number of bonds B needed to finance the stock position is:

Using Eq. 19.16, the price of the call option C at t = 0 is:

So, the value of the call option at t = 0 is $1.279, which is slightly higher than the one-period call with a value of $1.1165.

We can refine the tree by cutting the maturity of the call option in ever smaller increments. As a result, the return distribution at t = T (maturity) starts to approximate the real returns distribution ever better. There is also the limiting case in which the time to maturity is cut off into infinitely many increments that are each infinitely small. This gives the Black-Scholes option pricing model, which is in continuous time.

The Black-Scholes option pricing model (Black & Scholes, 1973) was developed independently and before the binomial model. But it is related to the binomial option pricing model and can be derived from it (see, for example, Hull, 2014). The Black-Scholes formula for the price of a European call on a non-dividend paying stock follows the set-up of the binominal model in Eq. 19.16:

where

Let’s discuss the features of the Black-Scholes formula. First, the stock price S plays an important role in option pricing. As stock prices cannot fall below zero, the left side of the stock price is limited. But stock prices can increase to very high numbers, though the chance of that happening is small. The lognormal function (denoted by ln in Eq. 19.20) is therefore used for the stock price.

The next step is to interpret the cumulative normal distribution N(d), which is illustrated in Fig. 19.21. This is the probability that a randomly distributed variable will be less than d: the shaded area left of d in Fig. 19.21. N(d₁) is the option delta, which we introduced earlier. If d₁ is large, then N(d₁) is close to one. In economic terms, this means that the stock price S is large relative to the present value of the exercise price PV(K). The call option is then very much in-the-money and one needs almost a full stock to hedge the fluctuations of the call option (i.e. the option delta is close to one). If d₁ is zero, then N(d₁) is 0.5. Figure 19.21 shows that half of the probabilities are in that case to the left.

A frequency graph of the cumulative normal distribution. The line follows a bell-shaped trend with a probability peak of 0.40 approximately.

N(d₂) is the normal distribution corresponding to the probability that the call option will be exercised at expiration (remember the Black-Scholes formulas are used for European call and put options, which can only be exercised at expiration). An easy way to calculate N(d) is to use the excel function NORMSDIST(d).

Finally, the volatility of the underlying stock σ is an essential variable of the Black-Scholes formula. Options are useful to protect or ‘hedge’ the owner of stocks against volatile stock prices, as discussed in Sect. 19.1. Without volatility in the underlying value, there is little use for options.

Example 19.7 calculates the value of the 3M call option with the Black-Scholes model.

Using the put-call parity from Eq. 19.5, we can show the put as follows:

We can now insert the Black-Scholes formula of Eq. 19.20 for the call C and rearrange. This gives us the Black-Scholes formula for the price of a European put on a non-dividend paying stock:

Interestingly, only a very limited number of input parameters are needed to price the Black-Scholes call and put options. For example, we don’t need to know the expected return on the stock (which is already in the current stock price). In Sect. 19.2.4, we review the drivers on option prices.

The Black-Scholes formulas reveal the drivers of option prices, which are summarised in Fig. 19.22.

A radial illustration lists 5 drivers of option prices. The drivers are the time to expire, risk-free interest rate, strike price, underlying value, and volatility.

Two out of these five drivers have the same sign for (long positions in) calls and puts: the time to expiration (+) and volatility (+). After all, the longer the option runs, and the higher its volatility, the bigger the chance that it will become in-the-money. The probability of really large swings increases, since there is little downside to a swing to one side and a lot of upside to the swing to the other side. Just imagine the reverse: an out-of-the-money call (or put) option with a very short time to expiration and a very low volatility—its price will likely be low.

The other three drivers have opposing signs for (long positions in) calls and puts, of which one has a negative sign for calls and a positive sign for puts: the strike price. The intuition on the strike price already followed from the discussion in Sect. 19.1: the higher the strike price, the lower the expected payoff on a call. This brings us to the two drivers that have positive signs for calls and negative signs for puts: the risk-free rate and the underlying value. The underlying value was already discussed in Sect. 19.1: the higher the underlying value, the higher the expected payoff on a call. And a higher risk-free rate lowers the PV of the strike (or exercise) price, which is positive for the call price and negative for the put price. Table 19.4 summarises the effect of drivers on call and put options.

Many corporate assets, particularly growth opportunities, can be viewed as call options (Myers, 1977). The value of such ‘real options’ depends on discretionary future investment by the firm. These options can be used in valuation and (corporate) investment decisions, including M&A. For example, suppose a company has developed a technology to make battery materials for electric vehicles that replaces polluting metals with lignin (wood). As almost always, the initial problem is the higher cost of the new technology versus existing alternatives. The company now effectively has a put option on the production costs of lignin-based battery materials (which need to go down to be competitive), where the exercise price is the production costs of traditional metals-based battery materials products (which are lower, but may rise). When the production costs of lignin-based battery materials go down, the put option comes in-the-money.

This is just one example, but there are many more. See Fig. 19.23 for a classification of more often occurring types. These options are very important for companies. In fact, one could argue that corporate strategy is all about options. As Luehrman (1998) puts it: ‘In financial terms, a business strategy is much more like a series of options than a series of static cash flows’. One could also view high risk companies as options, for example in a venture capital portfolio, where two-thirds of investments tend to lose money and most of the returns come from just a few blockbusters (see Chap. 10). Typical examples include biotech R&D investments, where most don’t pay off, but some pay off massively; and mining exploration, which gives the option to build a mine, and which is exercised if the price of the mined material is sufficiently high.

3 table illustrations. The real call options list option to defer, expand, extend, and increase scope. The real put options list abandonment option, the option to shrink and shorten. The combination lists follow switching options.

To stress the strategy angle, Box 19.3 shows how Shell created a put option to cancel refinery construction ahead of the oil crisis in the 1970s. This put option appeared to be very valuable when the oil crisis hit.

Koller et al. (2020) provide a classification of real options (see Fig. 19.23):

The drivers of real options follow from the drivers of financial options and are summarised in Fig. 19.24. A key point is that uncertainty—measured as volatility of cash flows—enhances the flexibility value (though it may also reduce future cash flows). Real options are about creating flexibility to change course in future circumstances. In this future, the cash flows of the existing course of action (e.g. the incumbent product or technology) and/or the alternative course of action (e.g. a new product or technology) may change.

A radial illustration lists the factors that enhance flexibility value. The components are the time to expire, risk-free interest rate, investment costs, cash flows existing business, lost cash flows, underlying cash flows alternative, and uncertainty about C F.

But these real options are typically not costless. Investment is often necessary to create the option and can thus be seen as a premium paid for the option. There may also be lost cash flows from an option to defer; the company could, for example, have increased sales today if it had not deferred the launch of a new product.

Companies can use real option analysis when faced with fundamental uncertainty, if not only the outcome, but also the probability distribution governing the outcome, is unknown. If the probability distribution is known, scenario analysis can be used (see Sect. 12.​8 in Chap. 12). If the probability distribution is unknown, real options can be used to structure the challenge for the company and prepare decisions in the future.

Let’s illustrate this with an example. In the energy transition, it is not (yet) clear which renewable energy source (wind, solar or hydrogen) is the most profitable and will emerge as the ‘winning’ technology. A company can prepare itself by making (small) investments in the various technologies. The company is then ready for expansion when it becomes clear which type of renewable is the most profitable. The option premium is then the upfront investment and the real options are the option to expand (in the winning technology) and to switch (from the non-winning technologies). As shown below, real option analysis works backwards. It starts with possible outcomes (in our example, possible renewable technologies) and then defines the options in the intermediate period (choice to expand/switch) and at the start (decision to do initial investment).

Real options can be analysed using decision trees that are reminiscent of the multiperiod option pricing model. A decision tree is a graphical representation of future decisions under uncertainty. Figure 19.25 gives the example of a decision tree regarding the investment in a mine. A decision tree is different from a binomial tree, in which the branches of the tree represent uncertainty that cannot be controlled: the nodes are only information nodes (uncertainty out of control of the decision maker; in red in Fig. 19.25). A decision tree also has decision nodes (in blue in Fig. 19.25).

A decision tree illustration of cash classified into to invest and do not invest in a mine. Invest in a mine is followed by minus $2500 which is classified into metal prices rise of $5500 million and fall of $800 million.

The investment in the mine (a cash flow of -$2500 million) at t = 0 effectively buys the company an exposure to metal prices, with a payoff of $5500 million if prices rise; and a $800 million payoff if prices fall at t = 2. Note that the investment in the mine is NOT a call anymore at t = 1 once the mine has been built at t = 0. The call lies before that at t = 0: the choice to build it or not; and before that at t = − 1 (outside Fig. 19.25): the choice to do the exploration or not. Table 19.5 shows the (expected) payoffs and how they add up. Please note that Table 19.5 starts from the right side of the decision tree and then works backwards. Table 19.5 shows a total expected payoff of $650 million, suggesting that the company should invest in the mine. Example 19.8 shows how one can calculate the expected value with a decision tree.

In reality it is of course more complicated, with a continuum of possible prices, and significant differences between small and large price rises. One can construct decision trees that better reflect that complexity by adding more branches. Up to a point, that can improve the quality of decision-making information.

In corporate practice, the use of real options is not as widespread as academics had imagined. Managers tend to favour DCF analysis in capex decisions, or simpler but flawed alternatives, such as the payback criterion—see Chaps. 6 and 7 for a discussion. Triantis and Borison (2001) identify three main corporate uses of real options:

Often, managers are not even aware of the options they have, since these options are not explicitly presented as such (‘opaque framing’) and are thus not identified in the first place. Moreover, they may suffer from behavioural biases, such as excessive optimism or overconfidence, which cause them to refrain from using real option techniques—they simply don’t see the need. That is a pity, because real option techniques can mitigate managers’ tendencies to invest fully in projects which could turn out to be unsuccessful. They can, for example, do a small upfront investment creating an option to expand. At the next stage, they can scale up when the project is successful or abandon when the project is unsuccessful.

A different and more cynical perspective is taken by Nassim Taleb (2012) in his book Antifragile. He argues that there are many cases in life, not just in corporate life, where people chase the upside while shifting the risk to others—they receive the option premium and leave the downside to fall on others. Outside of corporate life, examples include politicians who make dangerous claims and decisions that help them win elections, but which come at a high price to the health and wealth of the ordinary public. In corporate life, it can be managers who take cost-cutting measures at the expense of client safety, which helps them to boost EPS in the short run, while costing lives in the medium term. See the Boeing example in Chap. 3. Such shortcuts in safety are essentially written puts, in which every year the premium is earned (by means of cost savings and higher EPS) until one year a massive bill comes in.

How to think about real call options driven by E and S? On the long side, such a call results from grasping E and S opportunities. On the short side are the incumbents that are currently destroying value on E or S, whereas some competitors grasp E and S opportunities. Table 19.6 provides an overview of these real call options on E, long and short. To be concrete, we discuss Table 19.6 in terms of introducing a new low-carbon technology (or product) versus keeping the incumbent high-carbon technology. Think of the advent of electric vehicles in the car industry versus the incumbent internal combustion engine. But it applies more generally: a new course of action versus an existing course of action.

Let’s now consider a short put option. As mentioned in Sect. 19.3, the Boeing example from Chap. 3 was essentially a written (short) put on safety. By taking shortcuts in safety, every year Boeing earned the put premium in terms of cost savings and higher EPS, until one year a massive bill came in. Let’s start the analysis from the short perspective, i.e. Boeing’s. The underlying value is the safety level of the aircraft, which is the probability of no accident(s) happening that may result in multi-billion dollar fines. Boeing can avoid paying these potential fines by keeping passengers safe and alive via sufficient investment in safety. Safe arrival is the key threshold and hence the strike price (Table 19.7).

Another example of a short put option is BP’s Deepwater Horizon oil spill in 2010 (see Box 1.1 in Chap. 1). BP’s underinvestment in safety to increase short-term earnings contributed to the massive oil spill, resulting in a multi-billion-dollar settlement for the oil company.

As shown by the Merton model (Merton, 1974), any liability can be seen as a portfolio consisting of (among other positions) a short put. Such liabilities are called contingent liabilities or contingent claims. Contingent liabilities can also arise from the governance dimension (the G dimension from ESG). An example is accounting fraud, which can create large liabilities. The size of the liabilities can sometimes lead to the collapse of the company as discussed in Chaps. 3 and 17.