Pricing: Trees, Hedges and Black-Scholes

The one-step tree, and the hedge that replaces the forecast

Take the simplest possible world: a $100 stock, one period, a 10% up move or a 10% down move, and a 5% risk-free rate. The option is a $100 call, so it pays $10 in the up state and $0 in the down state. The risk-neutral probability is (1 + r − d) ÷ (u − d) = (1.05 − 0.90) ÷ (1.10 − 0.90) = 0.75, and the option is worth (0.75 × 10 + 0.25 × 0) ÷ 1.05 = $7.14. Now the same option priced the intuitive way: if the true probability of the up move is 60%, the expected payoff is $6.00 and discounting that at 5% gives $5.71. The difference is $1.43, and it is not a rounding error — it is the difference between two theories of what a price is. The first answer is the right one, and what makes it right is that 0.75 is not a probability anybody has to believe — it is the weight that makes the hedge work. Build the replicating portfolio explicitly: hold Δ shares and borrow B, where Δ is the payoff spread divided by the price spread, (10 − 0) ÷ (110 − 90) = 0.50 of a share, and B is chosen so that the portfolio pays the same in both states. Holding half a share gives $55 if the stock goes up and $45 if it goes down, so the borrowing has to be the amount that turns those into the call’s payoffs — $10 and $0 — which means 1.05B must equal $45, so B = $42.86. The cost of the position is half a share less that loan: 0.50 × $100 − $42.86 = $7.14. That is the same number the risk-neutral calculation produced, and the agreement is not a coincidence; it is the definition. The price of the option is the cost of the portfolio an arbitrageur would build to reproduce it, and because the portfolio pays the call’s payoff in both states, the real odds of the up move never enter — the hedge does not care what the trader believes. That is the argument, and Black-Scholes is its continuous limit. Instead of one period with two outcomes, take a continuum of tiny moves, build the replicating portfolio exactly as above at every instant, and the option’s value solves a differential equation whose answer is the famous formula. The structure of the answer is what matters here: five inputs, and each one is a hedge parameter rather than a belief. The stock price and the strike set the moneyness and therefore the delta. The time sets how much can still happen. The interest rate sets the cost of the financing embedded in the hedge. And the volatility sets the width of the distribution of future prices, which is the only input that cannot be observed directly — which is why the formula, in practice, is used backwards: given a market price, solve for the volatility, and treat that number as the market’s opinion rather than as a fact. One option, two theories — Risk-neutral probability: (1.05 − 0.90) ÷ (1.10 − 0.90) = 0.75 · Option value: (0.75 × $10) ÷ 1.05 = $7.14 ← · Real probability 60%: Expected payoff $6.00, discounted = $5.71 — the wrong answer · Cost of the replicating portfolio: 0.50 × $100 − $42.86 = $7.14 — the same number, built from the hedge ← The five inputs — stock, strike, time, rate, volatility — are all hedge parameters except the last, which is the only one that cannot be looked up. That is why the formula is used backwards to produce implied volatility, which is O7 and O19 territory.

What the formula is good for, and what it is not

The formula is best understood as a relative-value tool rather than a valuation oracle. It does not tell you what an option is worth in any absolute sense, because two of its inputs — the volatility and, for long-dated contracts, the rate over the life — are forecasts. What it does extremely well is impose consistency: given the prices of a few liquid instruments and a volatility, every other strike and expiry has an implied price, and any deviation is either a quoting error, a liquidity difference, or a genuine disagreement about the future. That is why market makers quote an entire chain from a surface of volatilities, which O19 covers, rather than pricing each contract from first principles. It is also worth being precise about what the model assumes, because the assumptions are where it breaks. Constant volatility, log-normal returns, continuous trading, no transaction costs, no dividend, and a constant interest rate — none of which is true, and each of which is traded around rather than ignored. Real distributions have fatter tails, so far out-of-the-money options trade at volatilities well above near-money ones, which is the skew. Real trading is discrete, so hedging has error and gaps matter. And real rates move, which is why long-dated options carry more rate sensitivity than the model’s simplicity suggests. A learner who understands that the model is a consistency engine will read a deviation as information; a learner who treats it as the truth will be puzzled every time the market disagrees with it. • Use the formula for consistency between instruments, not for absolute value. • Volatility is the only unobservable input, so the model is used backwards to imply it. • Its assumptions — constant volatility, log-normal returns, continuous hedging, no costs — are where it fails. • Fatter tails show up as skew: far out-of-the-money options trade at higher implied volatilities. • Discrete hedging and gaps mean the replication is approximate, which is why market makers charge a spread. The mistake that costs money is confusing a cheap option with an underpriced one. Cheapness has a definition only relative to a volatility: a $0.20 option on a $20 stock is expensive if the implied volatility is 25% and realised turns out to be 15%, and a $12 option on the same stock is cheap if the reverse is true. Price in dollars on its own says nothing, and the whole of the next lesson — implied against realised — is about making the comparison that does.

Three ways to price the same thing

The one-step tree in this lesson is the argument, not the tool. Real pricing is done by three families of numerical method, and knowing which is used for what explains why two systems can report slightly different values for a contract neither of them has mispriced. The **lattice** family extends the tree: more steps, more branches, and a backward induction that converges to the continuous-time answer as the steps get small. It handles American exercise directly, because at every node the calculation compares holding with exercising, which is the property that made it useful for equity options. Its cost grows with the number of steps, and its accuracy in early-exercise problems converges slowly, needing many steps to be precise — so it is fast for a single contract and awkward for a large book. The **finite-difference** family solves the underlying partial differential equation on a grid rather than by simulating paths. It is efficient for American options, natural for a dividend schedule because dividends become boundary conditions, and good at producing a whole surface of values and sensitivities at once since the grid solves for all prices and times in one pass. Its weakness is that the grid must be built around the problem, so a book containing many maturities and strikes needs a grid per slice or a cleverer scheme. The **Monte Carlo** family simulates many paths and averages the discounted payoffs. Its strength is generality: anything whose payoff can be written as a function of a path can be valued, including structures whose path dependence defeats a lattice. Its two difficulties are well known. The first is that accuracy improves only with the square root of the number of paths, so precision is expensive. The second is early exercise, which is inherently backward-looking and therefore awkward for a forward simulation — hence the specialised approaches for American Monte Carlo. And its sensitivities, the Greeks, cannot be read off the way they can from a grid: they must be estimated by re-pricing with perturbed inputs, which introduces its own error. The practical picture for a learner is therefore a division of labour rather than a competition. Lattices and grids for single-name options with early exercise; Monte Carlo for path-dependent structures and for anything with several sources of uncertainty; and analytical formulas where their assumptions happen to hold, mainly as a cross-check. Two systems disagreeing in the third decimal place on a standard contract is not evidence that one is wrong; it is what different discretisations of the same mathematics look like. What matters at the level of this subject is the thing all three share: none of them forecasts direction. Every method estimates the cost of replicating the payoff by trading the underlying, which is why the expected return on the stock never appears as an input and why a change in the view about the company’s prospects changes the option price only through its volatility. • Lattices and grids handle early exercise directly; Monte Carlo does not, natively. • Monte Carlo is general and converges slowly, and its Greeks must be estimated. • Two systems differing in the last decimal are discretisations, not errors. • None of the methods takes a view on direction; all price the cost of replication. A practical reason to know the method: the Greeks a platform reports are computed by one of these techniques, and their precision is not uniform. Deep out-of-the-money contracts with a grid produce small sensitivities that are mostly numerical noise, which is why delta on a far strike can jump around between systems.

Sources

• The one-step binomial model and risk-neutral pricingHull, “Options, Futures and Other Derivatives”
• Black-Scholes as the limit of the replicating portfolio argumentBlack & Scholes (1973), “The Pricing of Options and Corporate Liabilities”
• Delta hedging and why the hedge, not the forecast, sets the priceNatenberg, “Option Volatility and Pricing”