Jeffery and Smith (Equity Release Mortgages: Irish & UK Experience, p.30) discuss the apparent paradox that when we use a ‘real world’ model to project a forward price, then calculate the expected value of put and call options at different strikes, the internal rate of return of those options is considerably different from that obtained using the Black formula. See their table which I have copied below. Put options even have negative discount rates.
Taking the case of the put options, how can we rationalise these negative discount rates? Why would an investor even consider an asset that is expected to lose money, let alone one as risky as a put option which has a chance of expiring worthless, losing everything?
They continue.
The answer is that few rational investors hold a portfolio 100% in a put option. Rather, a put option is a form of insurance held in connection with other assets. An investor in shares can, sometimes with a modest outlay, acquire a put option that substantially mitigates losses in a market crash. The willingness to accept a negative expected return on the put option reflects the reduction of risk to the portfolio as a whole. This is the same reason that buyers of household or motor insurance would not expect (or hope) to make a profit on that insurance.
Are they right? Does the ‘willingness to accept a negative expected return’ really reflect the need to reduce the risk?
Their argument bears an uncanny resemblance to the one by Warren Buffett which I discussed earlier in September and again in November last year. Stock index prices are bound to go up considerably in the long run, so put options have no real value. But Black Scholes places a value on put options, therefore Black Scholes overvalues put options.
I appreciate Smith and Jeffery aren’t falling into Buffett’s trap, but the question is the same. Valuing put options using projection-discount methods yields dramatically lower values. Why so? Is it really that purchasers of said options want to reduce risk?
Now there is no puzzle about the put price. Consider a call and a put option both struck at 150, with the underlying at 100 (and by implication the forward at around 95, given interest at 2% and dividend at 3%). If we buy the call at the Black 76 price of 0.53, and sell the put at the Black 76 price of 50.18 then we have what is called a synthetic forward. The call is the right to buy at 150, but the put is the obligation to buy at the same price, which is contractually equivalent to a position in the forward contract. We have received a total of 49.65 for entering the position, because the forward contract is currently priced at 95, whereas we have agreed to purchase it at 150. The difference between 150 and 95 is 55, which discounted at 2% over 5 years gives 49.65.
So there isn’t even the whiff of a paradox here, at least concerning the option pricing. If the forward is trading at 95 and we buy forward at 150, i.e. at a considerable loss, we expect to be compensated. And since the put price represents almost all that expected loss, it follows that the Black 76 value correctly represents the option’s value to a market participant. Black 76 is not overvaluing anything.
Note also that a rational investor is indifferent to any forecast of growth in the underlying asset. The Smith and Jeffery expected values will change according to their risk premium assumption of 8%. The Black Scholes value will not change at all. If we buy an asset forward for 150 with its current forward price at 95, we expect the discounted value of the difference as compensation, and that value depends only on the risk free interest rate, not the growth rate.
Perhaps there is a paradox here, but it is nothing to do with option pricing, nor even forward pricing, but rather the pricing of the spot. Rephrasing Smith and Jeffery’s puzzle, why would an investor even consider selling an asset when its expected future price is many multiples of its current one? Let alone selling the asset short, where the seller has a good chance of losing an indefinite amount of money? The puzzle is about why assets are currently valued so low, given that their forecast future value, discounted at risk free, is significantly higher than their current market value. Why is the FTSE currently (June 3 2019) trading at the dismal level of around 7,100, when its true value should be 20,000 or 50,000?
That’s the real mystery. The FTSE is at its current value because of an equilibrium between buyers and sellers. If no one wished to sell (or sell short) at 7,100, or at any level short of 20,000 then happily and hey presto the price would rise to levels which would enrich Eumaeus. But sadly not.
| Strike | Expected | Market | Discount | Expected | Market | Discount |
| 50 | 91.907 | 40.8746 | 16.21% | 0.0003 | 0.0456 | -103.20% |
| 80 | 62.0729 | 16.9841 | 25.92% | 0.1661 | 3.3003 | -59.78% |
| 100 | 43.3396 | 7.3772 | 35.41% | 1.4328 | 11.7901 | -42.15% |
| 120 | 27.5496 | 2.7574 | 46.03% | 5.6429 | 25.2671 | -29.98% |
| 150 | 11.8601 | 0.5301 | 62.16% | 19.9533 | 50.1849 | -18.45% |
| 200 | 2.1392 | 0.028 | 86.72% | 60.2324 | 94.9247 | -9.10% |
| 250 | 0.3153 | 0.0014 | 107.70% | 108.4086 | 140.14 | -5.13% |
Table: theoretical option prices versus ‘expected’. Source: Smith and Jeffery 2019.