Option Pricing

Option Pricing

Option Pricing: Black-Scholes-Merton, Binomial Trees, Employee Options, Strategies and Monte Carlo

Scope and conventions

Every analysis uses one currency, time in years, and rates per year with continuous compounding: an amount grows as e^rT, and a yield y compounded annually is entered as ln(1 + y). The stock pays a continuous dividend yield q (zero for none); for an option on a currency q is the foreign interest rate, and for an option on a futures price q = r. Volatility σ is the annual standard deviation of the log return. The values are those of the models named below from the inputs given; they are not forecasts and not investment advice.

Black-Scholes-Merton

With the stock price lognormal under the risk-neutral measure, the European call and put are c = S e^−qT N(d₁) − K e^−rT N(d₂) and p = K e^−rT N(−d₂) − S e^−qT N(−d₁), with d₁ = [ln(S/K) + (r − q + σ²/2)T]/(σ√T), d₂ = d₁ − σ√T and N the standard normal distribution function (Black and Scholes 1973; the dividend yield is Merton's 1973 extension). N(d₂) is the risk-neutral probability that the call is exercised. The tool evaluates N by a convergent series near zero and a continued fraction in the tails, so deep in- and out-of-the-money values keep their relative accuracy.

Sensitivities, per unit of each variable: delta e^−qTN(d₁) for the call and −e^−qTN(−d₁) for the put; gamma e^−qTφ(d₁)/(Sσ√T); vega S e^−qTφ(d₁)√T; theta −S e^−qTφ(d₁)σ/(2√T) − rK e^−rTN(d₂) + qS e^−qTN(d₁) for the call and −S e^−qTφ(d₁)σ/(2√T) + rK e^−rTN(−d₂) − qS e^−qTN(−d₁) for the put; rho KT e^−rTN(d₂) and −KT e^−rTN(−d₂); vanna −e^−qTφ(d₁)d₂/σ; volga vega · d₁d₂/σ (φ is the normal density). The first table prints them in trading units (vega and rho per percentage point, theta per calendar day); the second sets each closed form beside a central finite difference of the price, which is a check a reader can repeat.

Put-call parity, c − p = S e^−qT − K e^−rT, follows from the payoffs alone and holds for any model; the bounds max(S e^−qT − K e^−rT, 0) ≤ c ≤ S e^−qT and the mirror for the put are the limits at zero and infinite volatility. An observed price inside the bounds corresponds to exactly one volatility because the value rises with σ; the implied volatility is found by Newton's method on σ from the Manaster and Koehler starting point, each step kept inside a bracket that is halved whenever the Newton step would leave it, so the solve cannot diverge.

Cash dividends of known amount are handled by the escrowed-dividend model: the present value of the dividends paid before expiry is taken off the stock price and the remainder is given the lognormal process. Enter a dividend either as a cash amount or in the yield, not both. The break-even price at expiry is K + c for a call bought at c and K − p for a put.

A public table to check against

Internal Revenue Service Revenue Procedure 2003-68 prints a safe-harbor table of Black-Scholes values as a percentage of the stock price for three volatility classes, fourteen ratios of stock price to strike and eleven terms. The table does not state its volatilities, interest rate or dividend yield; fitting them shows that it was computed with volatilities of 30, 50 and 70 percent, an interest rate of 5 percent and a dividend yield of 1 percent, and with those values the formula above reproduces all 462 printed cells exactly after rounding to the printed tenth of a percent. The default case of the tool is one of those cells: at the money, one year, 30 percent volatility, 13.6 percent of the stock price.

Binomial trees

A tree replaces the lognormal process by n steps of length Δt = T/n in which the price moves to S u or S d; the value at a node is the discounted risk-neutral expectation of the two following nodes, e^−rΔt[p V_u + (1 − p)V_d], and an American option takes the larger of that and its exercise value. Cox, Ross and Rubinstein (1979) use u = e^σ√Δt, d = 1/u and p = (e^(r−q)Δt − d)/(u − d). Jarrow and Rudd (1983) use equal probabilities with the drift in the moves, u, d = exp[(r − q − σ²/2)Δt ± σ√Δt]. Leisen and Reimer (1996) choose the probabilities from the Peizer-Pratt inversion of the binomial distribution, p = h(d₂) and p′ = h(d₁) with u = e^(r−q)Δtp′/p and d = (e^(r−q)Δt − p u)/(1 − p), which centres the tree on the strike at an odd number of steps and makes the European value converge with the square of the step count instead of oscillating. The convergence table shows the three side by side.

The early-exercise boundary is the stock price at which exercise and holding are worth the same; the tool locates it at each step between the two nodes that straddle it and averages consecutive steps, whose nodes interleave. A put is exercised below its boundary, which rises to the strike at expiry; a call is exercised early only when the stock pays dividends. The quadratic approximation of Barone-Adesi and Whaley (1987) is printed beside the tree: the European value plus an early-exercise premium A (S/S*)^q whose critical price S* solves a value-matching equation. It is an approximation (within about 2 percent of a finite-difference solution in the tool's checks) and is weakest for long terms.

Employee stock options

Options granted to employees cannot be sold, vest over time and are usually exercised before they expire, so a closed-form value uses an expected term in place of the contractual one. SEC Staff Accounting Bulletin No. 107 accepts for plain vanilla options the simplified method, expected term = (vesting term + original contractual term)/2, with the vesting term of a graded grant averaged over its tranches: ten years and four annual tranches give (2.5 + 10)/2 = 6.25 years, the bulletin's own example. Bulletin No. 110 continues to accept the method where a company does not have sufficient historical exercise data. The tool values the grant at that single term and, for comparison, tranche by tranche and at the full contractual term.

Volatility may be entered, computed from a pasted history of closing prices as the sample standard deviation of the log returns ln(P_i/P_i−1) multiplied by the square root of the number of intervals in a year, or taken as the mean of the volatilities of comparable public companies. Bulletin 107 discusses the length of history (generally at least the expected term) and the interval of the observations; a volatility from n returns has a standard error of about σ/√(2n).

The lattice follows the form of Hull and White (2004) on a Cox-Ross-Rubinstein tree over the contractual term. Before vesting the option cannot be exercised. After vesting the holder exercises as soon as the stock price reaches a multiple M of the strike, and in every step a fraction eΔt of the holders leaves the company, exercising a vested in-the-money option and losing an out-of-the-money one. With no multiple, no exits and no dividends the lattice returns the European value at the contractual term. Forfeiture before vesting is kept out of the value per option and applied to the number of options expected to vest, tranche by tranche, as granted × (1 − f)^years.

Exercise of employee options issues new shares. Where the stock price does not yet reflect the grant, the warrant form of Galai and Schneller (1978) values each of M options on N shares as N/(N + M) times a call on the equity per old share including the options, S + (M/N)W, an equation in W solved numerically; the plain factor N/(N + M) is printed beside it. The cost schedule spreads the cost of the options expected to vest over the years of service: straight-line over the whole award (and not less than the cost of the tranches already vested), or graded, each tranche over its own vesting period, which recognises more in the early years. The schedule is a planning estimate, not an accounting determination.

Strategies

A strategy is a list of legs: calls, puts and stock with signed quantities. Its profit at expiry is the sum of quantity × payoff less the net cost of opening it, a piecewise-linear function whose corners are the strikes; the tool evaluates it at zero and at every strike, takes the slope above the highest strike to decide whether profit or loss is unlimited, and finds the break-even prices as its zeros. A premium left blank is the Black-Scholes-Merton value at the common volatility. The sensitivities of the strategy are the quantity-weighted sums of those of the legs (stock has a delta of one and no others). The chart draws the profit at expiry and the profit now if the position were closed at model values. Early exercise of American-style legs, assignment, margin, commissions and taxes are outside the calculation.

Monte Carlo

The risk-neutral price at a later date is S exp[(r − q − σ²/2)Δt + σ√Δt z] with z standard normal; the value of an option is the discounted mean payoff over many simulated paths (Boyle 1977). Each draw z is used again as −z (antithetic variates) and the pair's payoffs are averaged, which lowers the variance when the payoff is monotonic in z; the standard error is that of the pair averages and falls with the square root of their number. For the European option the estimate is compared with the closed form. For the Asian option the payoff is on the arithmetic mean of the prices at m equally spaced dates, for which no closed form exists; the option on the geometric mean has one, because the log of a geometric mean of lognormal prices is normal with mean ln S + (r − q − σ²/2)T(m + 1)/(2m) and variance σ²T(m + 1)(2m + 1)/(6m²) (Kemna and Vorst 1990 for the continuous case), and it serves as a control variate: the arithmetic estimate less the geometric estimate from the same paths plus the geometric closed form.

Limits

Constant volatility and interest rate; no volatility smile, stochastic volatility or jumps; cash dividends in the Black-Scholes-Merton analysis only; trees of up to 5,000 steps; the employee-option lattice does not model blackout periods, reload features or performance conditions; barrier, lookback and other exotic options are not offered. Tax and accounting treatment of any option is a matter for the reporting entity and its advisers.