The original derivation of Black-Scholes-Merton Formula and its variants (Black′s Formula) somewhat obscured this dynamic. In a 1979 paper,² Cox, Ross, and Rubenstein (CRR) distilled the replication argument to a simple binomial tree model, and showed that the Black-Scholes-Merton Formula can be obtained as the number of time steps in the tree tends to infinity. This constructive algorithm did away with the stochastic calculus machinery and highlighted the dynamic replicating portfolio.

Extensions of the CRR binomial tree model to more general settings were swift. While the original CRR paper was for single tradeable assets (stocks, FX, commodities) under deterministic interest rates, Harrison and Kreps³ (in discrete-time setting) followed by Harrison and Pliska⁴ (continuous-time) generalized the CRR insights to cover multiple assets. Since interest rate options depend on multiple underlyings (zero coupon bonds, or discount factors as their prices), this allowed for a consistent framework for their pricing and risk management. Moreover, Harrison and colleagues′ papers formalized and generalized the principles in CRR, and showed the price of an option—which is the price of its dynamically replicated portfolio—can also be obtained by taking expected discounted value of the option payoff in a risk-neutral world. This valuation framework has become known as Risk-Neutral Valuation, and has introduced terms as martingales, numeraires, market completeness, change-of-measure, . . . into option pricing. Their result can be tersely summarized as follows: In an arbitrage-free market, there exists an equivalent market measure where assets′ prices relative to some numeraire are martingales. If the market is complete, this martingale measure is unique, and option prices are expected relative (to the numeraire) prices! While a mouthful, we will show that each component of this statement has a direct counterpart in CRR′s simple binomial model.

The works of Harrison et al. were further extended by Geman and colleagues⁵ who highlighted the flexibility in choosing the numeraire. This gave rise to a slew of new insights and extensions, the most famous being risk-neutral valuation under forward measures, and the ability to interchange discounting and taking expectation for interest-rate options. This interchange ability somewhat validated the long-term market practice of misapplying Black′s formula for interest-rate options.

Risk-neutral valuation is the modern framework for contingent claim valuation. As most of the core concepts of risk-neutral valuation have a direct counterpart in the CRR binomial model, we will present their model with a slight adaptation to allow for nondeterministic (random) interest rates.

Our goal is to construct a portfolio today (t = t₀) so that its value at expiration, t_e, replicates the value of the contingent claim. Therefore if we are the seller of the option, we can replicate the option′s contingent cash flows at expiration via this replicating portfolio. The fair price of the option today, C₀, would be the cost of setting up the portfolio.

Our portfolio consists of taking a position in the asset, Δ units of it, by financing it via a risk-free loan of size L at the prevalent risk-free rate r until expiration date t_e, so the value of the loan at expiration would be L(1 + r × (t_e - t₀)) = L/ D(t₀, t_e ) regardless of the state of the world.

At expiry, t_e, if we are in A_u state of the world, we want this portfolio to be worth C_u:

We have two equations and two unknowns (Δ, L). Solving for these, we get:

Therefore, today′s value of the contingent claim is:

The seller of the option can charge C₀ as above, make a loan of size L as above, and use these proceeds to buy Δ units of the asset at spot price A₀. At expiration, in either state of the world, A_u or A_d, the value of his holdings (Δ units of the asset) exactly offsets his liabilities: repayment of loan plus interest, and cash-settlement value of the option (C_u or C_d).

Note that in the preceding setup, we did not have to consider the probability of either state happening! As long as A_u, A_d can happen and are the only two possibilities, we are golden!

A bit of algebra allows us to rewrite the formula for C₀ as follows: where

Similarly, if p_u < 0, then A₀/D(t₀, t_e ) < A_d < A_u. In this case, we take a loan of A₀ to buy the asset today. At expiration, we owe A₀/ D(t₀, t_e) while we own an asset that is worth A_d or A_u. Regardless, we can sell the asset and pay off the loan for positive profit and no risk! Therefore, if there is no arbitrage in the above simple economy, p_u can be considered as a probability, and today′s value of the option is simply the expected discounted value of the option payoff under this probability.

Most people are risk-averse: between a guaranteed return and a risky investment with identical expected returns, they would opt for the former. That is why risky investments (stocks, real-estate, . . . ) need to have higher-than-average expected returns. Otherwise, one could simply put one′s money in the bank and have the same return with no volatility.

On the other hand, most of us have bought a lottery ticket or played in casinos, investments whose expected gain is less than what we paid for. These types of investing are examples of risk-taking, where although risky, we are batting for the fences.

In between, there is an investment behavior that considers any investments with the same expected return as equivalent, and does not require a risk premium for risky bets. Consider such an investor given a choice between two investments: (1) invest A₀ at the bank, and get A₀/D(t₀, t_e) at t_e, or (2) buy an asset at A₀ and either get A_u or A_d at t_e. For a risk-neutral investor, these two investments would be equivalent if or equivalently, when

Similarly, we observe that in our simple risk-neutral world, one-step expected future prices must equal forward prices:

Consider the two-step model shown in Figure 5.3. If we have moved to the up state (A_u, r_u) by time t₁, we have: where D(t₁, t₂, u) is derived from the financing rate r_u of the asset in the up-state for [t₁, t₂]:

Similar equations hold if we have moved to the down state (A_d, r_d) by time t₁: or expressed via risk-neutral probabilities: where

Once we have computed risk-neutral probabilities satisfying the preceding, today′s and future′s price of any contingent claim is

Consider the up state ( A_u, r_u). As we enter it, we hold a portfolio that consists of Δ₀ units of the asset (now worth A_u at t₁), and a loan of size L₀ plus its interest (worth L₀/D(t₀, t₁) at time t₁). Therefore the value of the portfolio value is:

The seller of the option can replicate the option cash flows as follows:

This characterization evokes fair games from probability theory. For example, if we continually toss a fair coin with payoff of ±1 if heads/tails, then at any point, our expected future stake is whatever our current stake is, since the expected gain from each coin toss is 0. In probability theory, this is called the martingale property. In general, martingales are an abstraction of fair games, that is, processes that have zero expected change. In this language, relative prices are martingales under risk-neutral probabilities.

Focusing on the underlying tradeable assets in fixed income, that is, zero-coupon bonds, and their prices expressed as discount factors, since a t_e -expiry zero-coupon bond has unit value at t_e , the risk-neutral probabilities must satisfy

This is in sharp contrast to pricing options on single stocks, FX, commodities, where the underlying is a single asset (1-dimensional), and since most of the risk is in the underlying, interest rates are conveniently assumed to be deterministic. For deterministic (nonrandom) interest rates, the stochastic discount factor can be replaced by its nonstochastic counterpart:

For deterministic interest rates, contingent claim valuation then reduces to computing discounted expected payoffs, an easier feat than calculating expected discounted payoffs, which would require co-evolution of assets and interest rates. As we will in later chapters, by assuming certain terminal distributions (Normal, Log-Normal) for the underlying asset, we can get closed-form formulae (Black) for simple European options such as calls, puts, digitals. Due to their relatively simple structure and ease of quoting—much like flat yield-to-maturity for quoting bonds—these Black formulae and their variants are commonly used even for interest-rate options, despite the inconsistency of simultaneously assuming deterministic discounting and nondeterministic interest rates at expiration.

Let us try to see if we can now parse the above and see their primordial counterparts in our simple CRR binomial model:

″Arbitrage-Free″: We showed that lack of arbitrage is equivalent to p_u being a probability, 0 ≤ p_u ≤ 1.

″Equivalent″: The asset possibility-space A_u, A_d was fixed. We ignored the real probabilities, and focused on new (risk-neutral) probabilities. Note that we can only do this as long as we are interested in pricing contingent claims. The risk-neutral probabilities are not true-world probabilities, and should not be used to forecast future expected outcomes (although many people do!).

″Complete″: The assumption that option payoffs can be replicated via underlyings.

″Unique″: We set up two equations and two unknowns, and we could solve for Δ, L. If we had more unknowns than equations, the solution would be indeterminate, and hence not unique.

″Numeraire″: The currency relative to which asset prices are expressed. We selected the rolled-over money-market account, M(0, t, ω), as our choice of numeraire.

″Relative prices are martingales″: E_t [C(T, ω)/M(T, ω)] = C(t)/M(t). This compact formula completely characterizes the risk-neutral probabilities, relating them to the assumed underlying asset value evolution, and also is the pricing operator for contingent claims.

Indeed, the two-step binomial model captures all the key ingredients of option pricing, and is all one really needs to understand the concept of risk-neutral valuation.

A final note is that the money-market account choice of numeraire, although common, is not required, and any positive-valued process can serve as the numeraire. For interest rate options, we chose the money-market account as this is the most common and intuitive numeraire. Chapter 11 picks up the discussion on the choice of other numeraires.