Unified Model

A derivative security is a contract. A buyer pays a seller to make payments on specified future dates. The seller quotes a price to the buyer and is obligated make future payments if the contract is executed.

The Nobel Prize winning theory of Fischer Black, Myron Scholes, and Robert Merton III demonstrated how to dynamically replicate payoffs that were a function of some underlying instrument price. The value of the option is the cost of setting up the initial replicating hedge. They showed the value of the initial hedge can be computed by taking the risk-neutral expectation of discounted future payoffs and that the initial hedge in the underlying is the derivative of the value with respect to the underlying price, delta.

Suppose you are a trader and your boss asks for a quote on a European call option. The B-S/M option value is v = s N(d_1) - ke^{-\rho t}N(d_2), where s is the current stock price, k is the call option strike, \rho is the risk-free rate, t is the time in years to expiration$, N is the standard normal cumulative distribution function, {d_1 = (\log s/k + \rho t + \sigma^2t/2)/\sigma\sqrt{t}}, and {d_2 = d_1 - \sigma\sqrt{t}}, where \sigma is the volatility. To turn this equation into a number that you can show to your boss you need to know the current “price”, the strike, the “risk-free” rate, the “time in years” to expiration, and the “volatility.”

The quoted words are not well defined. The strike and expiration are defined in the call contract, but if the time to expiration is 3 months how do you convert that into a time in years? What is the price of a stock that rarely trades? What is the risk-free rate if you are a day trader using your credit card for financing? Finally, if the underlying has no listed options how do you devine the volatility? We know from market data that a single volatility cannot be used to price options at all strikes.

Suppose your boss agrees with the quote you provided and the buyer executes the trade. Now you have to satisfy the contract obligation by coming up with \max\{S_t - k, 0\} at expiration t. The B-S/M theory suggests you put on an initial delta hedge by purchasing \partial v/\partial s = N(d_1) shares of the stock. Note d_1 and d_2 depend on s so this is non-trivial to prove.

Time moves on and the stock price changes. When do you change your hedge, and by how much? The B-S/M theory answer is that you hedge continuously, but that is untenable. It is a mathematical artifact of using Ito processes to model prices and hedges. Every hedge involves only a finite number of trades.

It long past time to stop whistling past the graveyard and confront the realities of trading head on. The Unified Model puts your nose directly into the most important ongoing problem of mathematical finance: how do you value, hedge, and manage the risk of derivatives using only a finite number of trades? It does not provide a solution to this, only a simple and mathematically rigorous framework for studying the problem based on Stephen Ross’(Ross 1978) “A Simple Approach to the Valuation of Risky Streams”. According to Ross

If there are no arbitrage opportunities in the capital markets, then there exists a (not generally unique) valuation operator, L.

He used the Hahn-Banach theorem to show the existence of L and observed it is not unique if the market is not complete. We extend the theory to any portfolio of instruments and define L in terms of deflators: positive measures indexed by time that depend only on available information. His crucial insight was to show the absence of arbitrage was purely geometric. Positive measures having mass one make a showing, but their “risk-neutral probability” does not shed any light on real-world probability.

The Unified Model does not require the Hahn-Banach theorem, much less Ito processes or partial differential equations. Instruments have prices and cash flows associated with holding them. Every arbitrage-free model is determined by a martingale measure and a deflator. Multiple market instruments can be used for hedging, not just one underlying. The Unified Model generalizes the B-S/M delta hedge as a Frechet derivative of the hedge value with respect to the vector of instruments used in the hedge. The model allows for futures, options, and limit orders to be used as hedging instruments.

Matt Levin likes to say “Everything is securities fraud.” My mantra is “Everything is a derivative.” Just like a derivative, managing a portfolio consists of a finite sequence of trades. The various XVA calculations can be understood as a type of hedge. We are both wrong, but hopefully in a useful way that piques your interest to read furthur.