Option valuation toolkit

Four option problems recur in valuation work.

Employee options are a claim on equity. They have to be priced and subtracted before you divide by shares. The equity of a deeply indebted firm behaves like a call option on the firm's assets, which is why a stock keeps trading after a discounted cash flow model says the company is worth nothing. A patent, a licence or an undeveloped reserve can carry value that a static DCF misses. And some options get exercised early, which a European formula cannot see.

The arithmetic here is exact. The inputs usually are not. When the underlying asset does not trade, no replicating portfolio can be built and no arbitrage disciplines the answer. Treat the output as an estimate with a wide band around it, and never quote it to the dollar.

The script

resources/options.py — pure standard library, no installation needed. Every subcommand reads JSON on stdin (or --in FILE) and prints JSON.

python3 resources/options.py <subcommand> --example      # show the input shape
python3 resources/options.py <subcommand> --in payload.json
python3 resources/options.py selftest                    # verify the engine

selftest runs 18 checks and should report 18 passed. It holds put-call parity to nine decimals. It confirms that a 500-step binomial tree converges to the Black-Scholes value. It checks that an American put is never worth less than its European twin. It also verifies that the dilution loop converges and that implied volatility inverts the pricer. Run it after touching anything.

Before you price a real option, run the three tests

Real options are everywhere. Most of them are worth nothing. The premium is admitted only when three tests pass in order, and the burden of proof sits with whoever wants the premium.

Test 2 is the one that kills most claims. Exclusivity is a spectrum, and the value you may claim scales with it. Ranked from weakest barrier to strongest: first-mover advantage, technological edge, brand name, telecom licences, pharmaceutical patents. Multiply the full option value by an exclusivity factor between 0 and 1 rather than pretending the barrier is absolute. A patent stops copies; it does not stop a rival developing a different drug for the same disease.

Opportunities are not options. An opportunity is something you might do. An option is a right that others do not have, with a defined underlying, a defined strike and a defined expiry. "Real options" belongs in the same family as "synergy" and "strategic considerations" — language reached for when a price has already been decided and needs justifying. If test 1 or test 2 fails, add nothing to the DCF value. If test 3 fails alone, use a decision tree instead and treat any option number as an order of magnitude.

Whatever you do, guard against double counting. If you value a patent separately as an option, remove the growth that same patent would produce from the DCF of the existing business.

Mapping a real option onto the script

There is no separate real-option subcommand. You map the business facts onto black-scholes or binomial inputs yourself.

The dividend_yield field is the single most consequential input in real-option work and the one most often left at zero. It represents value leaking out of the underlying while you wait. For a right that expires in n years, 1/n is the working default: each year of delay is one year of cash flows you will never collect. Real-option lives run five to seventeen years, so omitting this yield overstates the answer badly.

Match the riskfree rate to the option's life. A seventeen-year option takes a seventeen-year government bond rate, not a short rate.

Worked example: a pharmaceutical patent

Biogen's patent on Avonex. The underlying is the drug: the present value of cash flows from launching now is $3,422m, and development would cost $2,875m in present-value dollars. The patent runs 17 years. Biotech industry variance of log value is 0.224, so volatility is sqrt(0.224) = 0.4733. The 17-year bond rate is 6.7%. Cost of delay is 1/17 = 5.89%.

echo '{"spot": 3422, "strike": 2875, "time_to_expiry": 17, "volatility": 0.47328637,
       "riskfree_rate": 0.067, "dividend_yield": 0.0589, "option_type": "call"}' \
  | python3 resources/options.py black-scholes

The result is $905m, against an intrinsic value of $547m. The published figure is $907m; the difference comes from reading N(d) off a table rounded to four decimals. Waiting is worth roughly $358m more than developing today.

Optimal exercise falls out of repeating the calculation. Hold while the option value exceeds spot − strike; exercise once it drops below. Re-run with time_to_expiry set to 16, 15, 14 and so on, and compare each result with the constant $547m. For Avonex the lines cross at about eight years of remaining patent life. Past that point the cost of delay dominates and holding destroys value.

When a binomial tree beats Black-Scholes

Black-Scholes is the continuous-time limit of the binomial model. Both give the same answer when their assumptions hold, which the selftest confirms to within a fifth of a cent at 500 steps. Use the tree when one of those assumptions breaks.

Early exercise is the rule for real options rather than the exception, because a firm takes a project the moment waiting stops paying. A European formula cannot represent that choice and will understate the option.

echo '{"spot": 100, "strike": 100, "time_to_expiry": 1, "volatility": 0.3,
       "riskfree_rate": 0.05, "steps": 200, "option_type": "put", "american": true}' \
  | python3 resources/options.py binomial

That returns 9.863. The same put priced European is 9.354, so the right to exercise early is worth about half a point, roughly 5% of the option. For a call on an asset with no payout the premium is zero, since exercising early only surrenders time value.

Choose steps by watching the answer settle. The value oscillates at low step counts and converges from there: 9.969 at 25 steps, 9.856 at 100, 9.863 at 200, 9.867 at 500. A few hundred steps is enough for most work, and the cost grows with the square of the count. If the script refuses to run because the risk-neutral probability falls outside [0, 1], the volatility is too low relative to the rates for the step size — raise steps.

The tree is also the bridge to decision-tree analysis. A binomial tree and a capital budgeting decision tree have the same shape. They differ in the discount rate, not the logic. Where the pricing test fails but the option is real, build the decision tree instead. Do not build both and add them together.

Employee options

Outstanding employee options reduce value per share. The reason is not dilution as such. Options are exercised only when they are in the money, so the firm issues shares below the prevailing price, and existing shareholders pay the difference. Options also matter before exercise, because today's value must already carry the probability and cost of exercise later.

Three methods are in circulation. Two are biased in a predictable direction.

The treasury-stock shortcut is what accounting uses for diluted earnings per share, which is why it keeps appearing in valuations. Look at the XYZ column: a firm grants 10 million at-the-money options on a $10 stock and the treasury method says value per share is unchanged at $10.00. That cannot be right. Options at the money have real time value, and someone paid for them. The method also mishandles out-of-the-money options in both directions. Include them and the strike proceeds get added while the value does not, which perversely raises value per share. Exclude them and you have declared time value to be zero.

So value the options and subtract them as a separate claim. That is what employee-options does.

Why the calculation has to iterate

Exercise creates new shares. New shares lower the value of every share. A lower share value lowers the option's own payoff. So the underlying that belongs in the pricer is not the share value today but a diluted one:

adjusted spot = (equity value + total option value) / (shares + options)

The total option value appears on both sides. There is no closed-form solution, so the script starts from an undiluted guess and iterates to a fixed point. Convergence takes under a dozen passes and the output reports iterations and converged.

echo '{"equity_value": 1000, "shares_outstanding": 100, "options_outstanding": 10,
       "average_strike_price": 10, "average_time_to_expiry": 10,
       "volatility": 0.40, "riskfree_rate": 0.04}' \
  | python3 resources/options.py employee-options

XYZ has equity of $1,000m and 100 million shares, so $10.00 per share before the grant. Ten million at-the-money options with a $10 strike and ten years to run, at 40% volatility, converge to an adjusted spot of $9.584 and a value of $5.423 per option. The total claim is $54.2m, leaving $945.8m of equity in common stock and $9.46 per share. The grant cost shareholders 5.4%, not the 10% a share-count adjustment would suggest and not the zero the treasury method reports.

Choosing the inputs

Which equity value. The equity_value field sets the underlying. Passing share count times the current market price reproduces the published models and keeps the option value anchored to something observable. Passing your own DCF equity value keeps the valuation internally consistent, at the cost of making the option value move with your assumptions. The two answers diverge exactly when you disagree with the market, which is when it matters. Pick one, say which, and recompute the options whenever the share value changes.

Maturity. Use expected life, not contractual life. Employees exercise early and often, and Black-Scholes is a European model. Shortening the maturity is the standard proxy for that behaviour. A ten-year grant exercised on average in six years takes six.

Volatility. Take the stock's own history, then adjust downward if the firm is maturing. A large option pool relative to shares makes the answer sensitive to this input.

Vesting and tax. The script does not apply either. Multiply total_option_value by the probability of vesting when a material share of the pool is unvested. Multiply by (1 − marginal tax rate) when exercise creates a deduction for the firm — and only then.

Repricing. If the plan allows the strike to be reset after a price fall, the option is worth more than the model says, and the extra comes straight out of shareholders' pockets. Note it even if you cannot price it.

What not to do afterwards

Subtract the option value and divide by actual shares outstanding. Doing both — netting out the option value and using a diluted count — charges shareholders the same cost twice.

Options already granted are a claim on existing equity, so they belong in the bridge. Options expected to be granted in future are compensation, so they belong in forecast operating expenses. Never put the same grant in both places, and do not add back stock-based compensation as a non-cash charge.

Equity as a call option on the firm

Shareholders own a residual claim protected by limited liability. If firm value at the debt's maturity exceeds what is owed, they keep the difference. If it falls short, they hand over the keys and lose no more than their stake. That payoff is max(firm value − debt owed, 0), which is a call option struck at the face value of debt and expiring when the debt matures.

This is why the stock of an insolvent company still trades. A DCF says the firm is worth $800m against $1,000m of debt, so the equity is worth nothing. The option model disagrees. Firm value is volatile and the debt is not due yet. There is still a path where the assets recover past $1,000m, and shareholders capture all of the upside on that path while bearing none of the extra downside.

echo '{"firm_value": 800, "face_value_of_debt": 1000, "debt_maturity": 5,
       "firm_value_volatility": 0.40, "riskfree_rate": 0.05}' \
  | python3 resources/options.py equity-as-option

Equity is worth $283m even though the firm owes $200m more than it is worth. The implied value of the debt is the remainder, $517m, and the yield lenders are demanding on that is 14.1%. The probability of default, read off 1 − N(d2), is 66%. All of these come from one calculation, which is a useful discipline: an equity value you like and a default probability you refuse to believe cannot both be kept.

When to reach for this. Two conditions together. Earnings are negative because of debt rather than because the business is young, and market-value debt to capital is above 50%. Below that threshold the ordinary DCF is the better answer. Run the DCF first regardless — it is where firm_value comes from.

Risk shifting falls out of the model. Raise firm_value_volatility from 0.40 to 0.80 and equity rises from $283m to $507m while the implied debt value drops from $517m to $293m. Firm value has not changed. Volatility alone moved $224m from lenders to shareholders. The model explains three things at once. Managers of a distressed firm take gambles their lenders hate. Lenders write covenants to stop them. And shareholders sometimes fight a rescue that reduces risk, because it hands their option value back to the lenders.

Inputs for this subcommand:

Firm value volatility is the weakest input in the calculation. Vary it and report the range rather than a single figure.

Treat the answer as one branch. The option approach and the probability-weighted distress approach are alternative ways to price the same survival risk. Use one. Do not also raise the discount rate for failure risk, and do not shrink the cash flows for it.

Implied volatility

implied-vol inverts the pricer by bisection. Feed it an observed option price and it returns the volatility that reproduces it, along with the recomputed price so you can see the fit.

echo '{"spot": 100, "strike": 100, "time_to_expiry": 1, "riskfree_rate": 0.05,
       "option_price": 14.23, "option_type": "call"}' \
  | python3 resources/options.py implied-vol

Two uses. Traded warrants or listed options on the company give a market view of volatility that beats a historical estimate for pricing employee options. And running an implied volatility on your own equity-as-option inputs shows what the market believes about firm value volatility, which is often more informative than arguing about your own proxy.

The search covers volatilities from near zero up to 500%. A price outside that range makes the script stop and report the achievable span, which almost always means the spot, strike or maturity is wrong.

Handoffs to the other engines

The equity bridge is circular in the same way the dilution loop is: the DCF needs an option value and the option value needs an equity value. One pass is enough in practice. Take bridge.equity_value from a DCF run with employee_options_value set to zero, price the options, then re-run the DCF with the answer in place.

Run valuation-consistency-checks over the finished artifacts before accepting anything.

Sanity checks before quoting a number

Six inputs drive every option value, and their directions are facts rather than estimates. Check each against the table before believing an output.

Volatility raises both calls and puts, because the downside is truncated either way. A real option value that falls when you raise volatility means a sign error or a mistaken input.

Two more checks. The time_value field should be positive for any live option; a European option can show negative time value only if the underlying pays out heavily, and it usually signals a wrong yield. And a deep out-of-the-money real option can still be worth a lot, which is exactly why the exclusivity test matters so much.

Common failures