Unified Model

What Quants Do

• Model prices and cash flows.
• Fit model parameters to market data.
• Compute expected values and derivatives.
• Specify hedges/trading strategies.
• Provide measures of how good a hedge is.

What Quants Don’t

• Provide realistic models.
• Specify when to hedge.
• Use parameters traders understand.
• Manage risk.

Holdings

Trades

Price

Prices depend on time, the instruments, the entities, and the amount the buyer wants to acquire, among other things.

• The trade (t;i,a'X,e;i',a',e') is available to e from e', where X = X(t;i,e;i',a',e').
• “Among other things” is denoted \Omega, the sample space.

Cash Flow

Stocks have dividends, bonds have coupons, futures have margin adjustments, commodities have holding costs, …

• Holding (i,a) at time t results in a cash flow aC(t;i;i') of i' at time t.
• Usually i' is a currency, but payment-in-kind is possible.
• Cash flows are zero except at discrete times.
• Cash flows do not depend on entities, only on the issuer of i.

Market (Model)

A market model specifies possible future prices and cash flows of instruments.

• Price: X\colon T\times I\times E\times I' \times A'\times E' (\times \Omega)\to\bm{R}.
• Cash Flow: C\colon T\times I\times I'(\times\Omega)\to\bm{R}.
• The trade (t,i,a'X(t;i,e;i',a',e'),e;i',a',e') is available to the buyer.
• Assuming price does not depend on amount and e, e' are understood…
• … X(t;i,e;i',a',e') is a vector-valued stochastic process X_t\colon\Omega\to\bm{R}^I indexed by instruments

Trading

A trading strategy specifies when, what, and how much to trade.

• Trading times τ_0 < τ_1 < \cdots…
• .. and amounts Γ_j\colon\Omega\to A^I.
• Position: Δ_t = \sum_{τ_j < t} Γ_j = \sum_{s < t} Γ_s.
• Value: V_t = (Δ_t + Γ_t)\cdot X_t is the mark-to-market.
• Amount: A_t = Δ_t\cdot C_t - Γ_t\cdot X_t shows up in the trade blotter.

Arbitrage

Arbitrage exists (for a model) if there exists a trading strategy that eventually closes out with A_{τ_0} > 0 and A_t\ge 0, t > τ_0.

• Make money up front and never lose money until strategy is closed.
• Closed out means the position is zero after some finite time.
• This definition does not depend on a measure.
• Traders consider RoI: A_0/|Γ_0|\cdot |X_0|.

FTAP

The Fundamental Theorem of Asset Pricing. No arbitrage if and only if there exist deflators D_t\colon\Omega\to(0,\infty) such that X_tD_t = E_t[X_u D_u + \sum_{t < s \le u} C_s D_s], where E_t is conditional expectation at time t.

• If C_t = 0, deflated prices are a martingale.
• As u\to\infty, value is discounted future cash flows.

Lemma

If X_tD_t = E_t[X_u D_u + \sum_{t < s \le u} C_s D_s], then V_tD_t = E_t[V_u D_u + \sum_{t < s \le u} A_s D_s].

• Price \leftrightarrow value, cash flow \leftrightarrow amount.
• Trading strategies create synthetic market instruments.

FTAP…

Easy direction: deflators imply no arbitrage.

If A_t \ge 0, t > 0, then V_0 D_0 = E_0\left[\sum_{t > 0} A_t D_t\right] \ge 0. Since V_0 = Γ_0\cdot X_0 = -A_0, A_0\le 0 so there is no arbitrage.

…FTAP

Hard direction: no need to prove!

Given any vector-valued martingale M_t and any deflator D_t X_t D_t = M_t - \sum_{s \le t} C_s D_s is an arbitrage free model.

Proof: More high school algebra.

Canonical Deflator

If the market has repos then D_t = e^{-\int_0^t f_s\,ds} where f_s is the repo/short rate at time s.

• The repo/short rate determines the value of all fixed income instruments.

Zero

A zero coupon bond D(u) has C_u = 1.

• X_t D_t = E_t[D_u] so X_t D_t = D_t(u)D_t = E_t[D_u], X_t = D_t(u) = E_t[D_u]/D_t = E_t[e^{-\int_t^u f(s)\,ds}].
• Risky zero D^{R,T}(u) has cash flow C_u = 1(T > u) or C_T = R1(T \le u).

FRA

A forward rate agreement F^δ(t,u) has C_t = -1, C_u = 1 + fδ, where δ is the dcf for the interval from t to u.

• 0 = X_0 = E[-D_t + (1 + fδ)D_u], so
• f = (D(t)/D(u) - 1)/δ.
• Par forward F^δ_s(t,u) at s defined by 0 = X_s = E_s[-D_t + (1 + F_s(t,u)δ(t,u))D_u],
• so F_s(t, u) = (D_s(t)/D_s(u) - 1)/δ(t,u)

Arrears

A FRA \bar{F}^δ(t,u) paying in arrears has one cash flow C_u = (f - F_t(t,u))δ.

• Model: M_t = (E_t[D_u])_u, u \ge t, all zeros indexed by maturity.
• Trades: τ_0 = t, Γ_t = +1_u, t_1 = u, Γ_u = -1_u.
• FRA + trades = arrears.

B-M/S

The Black-Merton/Scholes model is M_t = (r, s e^{σ B_t - σ^2t/2}), D_t = e^{-ρt}, so X_t = M_t/D_t = (e^{ρt}, se^{σ B_t + (ρ - σ^2)t/2}).

• Fixed dividends C_{t_j} = d.
• Proportional dividends C_{t_j} = pS_{t_j}.
• Equilibrium arguments, self-financing portfolios, Ito’s formula not required.

Hedge

Given a set of amounts A_j at t_j how do we find a trading strategy that produces these?

• V_0 = E[\sum_j A_j D_{t_j}] = Γ_0\cdot X_0 so D_{X_0}V_0= Γ_0.
• V_j = E_j[\sum_{k > j} A_k D_{t_k}] = (Δ_j + Γ_j)\cdot X_j so D_{X_j}V_j = Δ_j + Γ_j (Frechet derivative.)
• Note Δ_{j + 1} - Δ_j = Γ_j, so Δ is delta and Γ is gamma.
• This is never a perfect hedge.

Remarks

Trajectory of mathematical finance is increasingly accurate models covering all instruments.

Incorporate trading strategies when measuring risk.

Monte Carlo all the things! We have Moore’s Law on our side and can put the ML headcount to good use.

Provide real-time valuation and risk reporting across all asset classes.

Links

• Unified Model
• Unified Finance
• Excel add-in libary