Pricing Derivatives Without Probability: The Sequel
In the previous chapter, Wilmott introduced the Epstein-Wilmott model for interest rates: no probability, just bounds on where rates can go and how fast they move. We saw how to value bonds and generate the Yield Envelope. Chapter 69 takes this framework and applies it to real portfolios and more complex derivatives. Bond options, index amortizing rate swaps, convertible bonds. The nonlinear, non-probabilistic approach handles them all.
A Real Portfolio: Dresdner Kleinwort Benson
The chapter starts with something refreshing: a real-life example. Wilmott shows the actual sterling cashflows from a leasing portfolio at Dresdner Kleinwort Benson, with payments stretching out to 2008.
Using the yield curve from January 8th 1998, the present value of this portfolio is negative 3.5 million pounds. The traditional way to stress-test this is to shift the yield curve up and down in parallel. A 2% downward shift gives negative 4.4 million pounds. But is a 2% parallel shift realistic over a ten-year horizon? Probably not enough.
The Epstein-Wilmott model gives a much scarier number. Without any hedging, the worst-case value is negative 6.1 million pounds. That is 2.6 million worse than the simple present value calculation. The model is not being pessimistic for fun. It is accounting for all the ways rates could move within the constraints, not just parallel shifts.
Now, hedge the portfolio with benchmark bonds. The optimal static hedge uses 15 different bonds, with the three-year bond heavily shorted at 123,013 units. After hedging, the worst case improves to negative 4.0 million pounds. That is only 470,000 worse than the deterministic present value. The hedging has absorbed most of the interest rate risk.
The summary table tells the story:
- Deterministic PV: negative 3.5 million
- 2% downward shift: negative 4.4 million
- Worst case unhedged: negative 6.1 million
- Worst case hedged: negative 4.0 million
The hedged worst case is far more robust than the parallel shift analysis. It covers a much broader set of scenarios.
Bond Options: European Style
Now things get interesting. How do you price an option when there is no probability? The approach is clever. You consider two separate portfolios: one for the case where you exercise the option, and one where you do not. For each portfolio, you solve the worst-case equation. Then at the option expiry, you pick whichever case gives the holder more value.
For a European call on a zero-coupon bond, you exercise if the portfolio value (with bond delivery and hedging instruments) minus the strike price exceeds the portfolio value without exercising. The decision depends on the interest rate at expiry.
The example prices a one-year call option on a five-year zero-coupon bond with a strike of 0.5. Without hedging, the worst-case value is 0.009 and the best case is 0.372, a huge spread. With optimal static hedging using seven bonds, the spread collapses to 0.220 vs 0.221. That is practically a single price.
The static hedges include roughly one unit of the underlying bond (short for the call, long for the put), which makes sense. You are essentially decomposing the option into the underlying plus some adjustments from shorter-dated bonds.
American Options: Exercise Anytime
The American option version works the same way but with an additional constraint. At every moment before expiry, we check: is it better to exercise now or keep waiting? If exercising gives a higher portfolio value, we add a constraint that prevents the value from dropping below the exercise value.
This is the same free boundary problem as in the standard Black-Scholes world, but now within the non-probabilistic framework. The results show that American options have tighter spreads than European options because the extra exercise opportunities make the option more likely to be exercised. When exercise is likely, you can hedge more effectively with the underlying bond.
A result that carries over from standard theory: the American call on a zero-coupon bond has the same value as the European call. No early exercise advantage without coupons. But if the underlying were a coupon bond, the American call would be worth more.
The chapter also mentions Bermudan options (exercise allowed only on specified dates). Same framework, just restrict when the exercise constraint applies.
There is a computational catch, though. For each either/or decision in the portfolio, you double the number of scenarios. A portfolio with N vanilla options requires pricing 2^N separate portfolios. That gets expensive fast.
Contracts With Embedded Decisions
The flexiswap is a nice example. The holder has M possible cashflow dates and must take exactly m of them (where m is less than M). On each date, they choose whether to exchange floating for fixed. The trick is introducing m+1 functions, one for each number of remaining cashflows.
At each date, if you must take every remaining cashflow (no slack left), it is a forced jump condition. If you have more dates than remaining obligations, you only take the cashflow if it is optimal. This creates a constraint linking the functions, similar to the American option.
The example prices an eight-choice swap where the holder must take four of eight possible cashflows. The hedging is done with vanilla par swaps rather than zero-coupon bonds, because you should hedge with instruments similar in form to what you are pricing.
Without hedging, the worst-to-best spread is about 177,000 dollars. With optimal hedging (using only three of the nine available swaps), it drops to about 30,000. Most of the hedging value comes from the two-year, three-year, and four-year swaps, the ones closest in maturity to the contract.
The Index Amortizing Rate Swap
The IAR swap is a swap where the principal decreases over time based on an amortizing schedule that depends on interest rates. The lower the rates, the faster the amortization. This makes it path-dependent in a sense, because the current principal level matters.
The good news is there is a similarity reduction. The value can be written as the principal times a function of just the interest rate and time. This simplifies the computation significantly, and the nonlinearity does not mess up this reduction.
Hedging the IAR swap with vanilla swaps is natural since they share the same cashflow structure. The optimal hedge uses primarily the three-year and four-year swaps. Hedging improves the worst case from negative 30,820 to negative 11,753 on a million-dollar notional.
One comparison is telling. The hedged worst case (negative 11,753) is similar to the deterministic value with a 1% yield curve shift (negative 10,908). This pattern keeps showing up: the optimally hedged worst case is roughly equivalent to a 1% adverse shift in the yield curve.
Convertible Bonds
The convertible bond adds stock price dependence on top of interest rate uncertainty. The pricing equation now has terms for both the stock process (with its usual second-order diffusion term) and the interest rate (handled non-probabilistically as before). The value depends on S, r, and t.
The chapter compares three models:
- Constant interest rate (7%): bond worth 1.058
- Vasicek model fitted to 7% flat curve: bond worth 1.059
- Epstein-Wilmott worst case: 1.042, best case: 1.089
The first two models give nearly identical answers, which might make you feel confident. But try changing the Vasicek parameters slightly. Bump the mean reversion speed from 0.1 to 0.11 and the value changes. Bump the volatility from 0.02 to 0.25 and it changes more. The “precise” answer is quite sensitive to inputs you cannot measure well.
The Epstein-Wilmott spread of 1.042 to 1.089 is wide, but honest. Static hedging with three zero-coupon bonds narrows the worst case to 1.052 and the best case to 1.067. The spread shrinks from 0.047 to 0.015, a factor of three. And the hedged worst case of 1.052 is less sensitive to model assumptions than either the constant-rate or Vasicek prices.
The Big Picture
Wilmott clearly has a soft spot for this model, and the chapter shows why. The framework delivers something that traditional stochastic models cannot: robustness. You do not need to estimate volatilities, correlations, or mean reversion speeds. You set broad constraints and find the worst case.
The main computational challenge is the nonlinearity. You have to value the entire portfolio at once because the worst-case interest rate path depends on what instruments you hold. Adding a hedge changes the path, which changes the value, which changes the optimal hedge. It is a joint optimization problem.
But the PDE itself is not hard to solve numerically. It is first-order, not second-order, so it is actually simpler than Black-Scholes in some ways. The difficulty is in the optimization over hedge quantities, not in the equation solving.
The examples throughout this chapter reinforce one lesson: hedge with instruments that are similar in form to what you are pricing. Zero-coupon bonds hedge bonds well. Swaps hedge swaps well. Obvious in hindsight, but the model actually proves it rather than just assuming it.
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