April 25, 2024
Instruments have prices and cash flows. Trading instruments result in cash flows to the trading account; you buy and sell at prevailing prices and receive cash flows associated with each instrument proportional to your current position. The mark-to-market of a position and cash flows associated with trading correspond to the price and cash flows of a synthetic instrument.
A derivative is a contract specifying cash flows. If a trading strategy replicates the cash flows then the value of the derivative is the cost of the initial hedging position. Risk management involves measuring how well a trading strategy replicates cash flows specified by the contract.
A money market account is assumed to be available for financing trading strategies. The price of the money market at time t is denoted R_t and has no cash flows. Let D_t = 1/R_t be the deflator.
Given a deflator, every model is specified by a vector-valued martingale M_t indexed by instruments. Prices X_t and cash flows C_t must satisfy X_t D_t = M_t - \sum_{s\le t} C_s D_s in any arbitrage-free model. Note if there are no cash flows then this implies the deflated prices, X_t D_t, are a martingale. In general, C_t = 0 except at a discrete set of times.
For example, the Black-Scholes/Merton model has money market R_t = e^{\rho t} and martingale S_t = s e^{\sigma B_t - \sigma^2t/2} where B_t is standard Brownian motion.
Let I be a set of instruments and T be a (totally ordered) set of trading times. Assume the standard setup \langle\Omega,P,(\mathcal{A}_t)_{t\in T}\rangle where \Omega is the sample space of all possible outcomes, P is a probability measure on \Omega, and \mathcal{A}_t is an algebra of sets on \Omega indicating the information available at time t.
We use the notation X\colon\Omega\to\mathcal{A} to indicate X\colon\Omega\to\mathbf{R} is \mathcal{A}-measurable where \mathcal{A} is an algebra of sets. If the algebra is finite then X is \mathcal{A}-measurable if and only if it is constant on atoms of \mathcal{A} so X is a function from \Omega to the atoms of \mathcal{A}.
Let X_t^i\colon\Omega\to\mathcal{A}_t be the price and C_t^i\colon\Omega\to\mathcal{A}_t be the cash flow of instrument i\in I at time t\in T. We also write this as X_t,C_t\colon\Omega\to\mathcal{A}_t where X_t and C_t are vectors indexed by the set of instruments when that is understood.
A consequence of the formula X_t D_t = M_t + \sum_{s\le t} C_s D_s is X_t D_t = E_t[X_u D_u + \sum_{t\le s < u}C_s D_s] where E_t is conditional expectation at time t and u > t. Note if X_t D_t goes to zero as t goes to infinity then X_0 = E[\sum_{t \ge 0} C_t D_t]. In this case price is the expected value of deflated future cash flows.
The formula follows from \begin{aligned} E_t[X_u D_u + \sum_{t\le s < u}C_s D_s] &= E_t[(M_u - \sum_{s\le u}C_s D_s) + \sum_{t\le s < u}C_s D_s] \\ &= E_t[M_u - \sum_{s\le t}C_t D_s] \\ &= M_t - \sum_{s\le t}C_s D_s \\ &= X_t D_t \\ \end{aligned}
Market participants trade instruments. We assume an initial position M_0 in the money market instrument. A trading strategy is a finite sequence of strictly increasing (stopping) times \tau_j and vector-valued random variables \Gamma_j\colon\Omega\to\mathcal{A}_{\tau_j} of amounts amounts purchased in each instrument at these times. Trades accrue to a position \Delta_t = \sum_{s < t} \Gamma_s, where \Gamma_s = \Gamma_j if s = \tau_j and is zero otherwise. The trades show up in the trade account as cash flows A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t. You receive all cash flows in proportion to your existing position and pay for the trades just executed. Let V_t = (\Delta_t + \Gamma_t)\cdot X_t be the value, or marked-to-market, of the strategy at time t. It is the amount in terms of the money market instrument of unwinding the current position and trades just executed at prevailing prices.
Lemma. Using the definitions above V_t D_t = E_t[V_u D_u + \sum_{t\le s < u}A_s D_s]. Note how V_t and A_t play the role of price and cash flow respectively. Trading strategies create synthetic market instruments.
Proof. If u > t is sufficiently small then Δ_t + Γ_t = Δ_u is \mathcal{A}_t-measurable. Since X_t D_t = E_t[(X_u + C_u) D_u] we have \begin{aligned} V_t D_t &= (Δ_t + Γ_t)\cdot X_t D_t\\ &= Δ_u\cdot X_t D_t\\ &= Δ_u\cdot E_t[(X_u + C_u) D_u]\\ &= E_t[Δ_u\cdot(X_u + C_u) D_u]\\ &= E_t[(Δ_u\cdot X_u + Δ_u\cdot C_u) D_u]\\ &= E_t[(Δ_u\cdot X_u + Γ_u\cdot X_u + A_u) D_u] \\ &= E_t[(V_u + A_u)D_u],\\ \end{aligned} where we use Δ_u\cdot C_u = Γ_u\cdot X_u + A_u. The displayed formula above follows by induction. \blacksquare
A model is specified by a deflator D_t and a martingale M_t. The reciprocal of the deflator R_t = 1/D_t is the money market instrument and has no cash flows. Prices X_t and cash flows C_t under the model satisfy X_t D_t = M_t - \sum_{s\le t} C_s D_s. A consequence is X_t D_t = E_t[X_u D_u + \sum_{t\le s < u}C_s D_s]. A trading strategy (\tau_j, \Gamma_j) has value V_t = (\Delta_t + \Gamma_t)\cdot X_t and account cash flows A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t where \Delta_t = \sum_{\tau_j < t} \Gamma_j = \sum_{s < t} \Gamma_s. A consequence is V_t D_t = E_t[V_u D_u + \sum_{t\le s < u}A_s D_s].
Arbitrage exists (given D_t and M_t) if there is a finite trading strategy (\tau_j, \Gamma_j) with A_{\tau_0} > 0, A_t \ge 0 for t > \tau_0, and \sum \Gamma_j = 0; you make money on the initial trade and never lose money after that. The position must eventually be closed out, \sum_j \Gamma_j = 0.
For a trading strategy that closes out V_{τ_0} D_{τ_0} = E_{τ_0}[\sum_{t > τ_{0}}{A_{t}D_{t}] \ge 0}. Since V_{τ_0} = Γ_{τ_0} \cdot X_{τ_0}, A_{τ_0} = - Γ_{τ_0} \cdot X_{τ_0} and D_{τ_0} > 0 we have A_{τ_0} \le 0. This shows no arbitrage can exist for models where prices and cash flows satisfy X_t D_t = M_t - \sum_{s\le t}C_s D_s.
Given cash flows (A_t)_{t\in T} does there exist a trading strategy (\tau_j, \Gamma_j) producing such A_t? Assume T = \{t_j\} with t_j increasing. Using V_0 = (\Delta_0 + \Gamma_0)\cdot X_0 and V_0 = E[\sum_{t\ge 0}A_t D_t] we have \Delta_0 + \Gamma_0 = \mathcal{D}_{X_0} E[\sum_{t\ge 0}A_t D_t], where \mathcal{D}_X denotes the Fréchet derivative with respect to X. Similarly, \Delta_j + \Gamma_j = \mathcal{D}_{X_j} E_{t_j}[\sum_{t\ge t_j}A_t D_t]. Since \Delta_0 = 0 this gives necessary conditions for determining \Gamma_j. The value of a derivative is the cost of setting up the initial hedge. \Delta is the derivative of value with respect to underlying and \Gamma the amount by which the delta hedge must be changed.
These are not sufficient conditions. By definition we always have V_j = (\Delta_j + \Gamma_j)\cdot X_j. The position will accrue to (\Delta_j + \Gamma_j)\cdot X_{j+1} at time t_{j+1} but there is no guarantee this will equal V_{j+1}. The difference is the profit and loss for the hedge over the period from t_j to t_{j + 1}. Since V_t is a function of X_t and D_t we can write a Taylor expansion for \Delta V_t = V_{t + \Delta t} - V_t in terms of powers of \Delta X_t and \Delta D_t. Risk managers use the higher order derivatives to “explain” profit and loss. Traders have to deal with the fact that the imperfect hedge throws off their replication. Quants have not yet devised a coherent theory to help traders deal with the various heuristics they have invented to account for this.
We now apply the model to various instruments.
A futures contract is specified by an underlying and an expiration. If S_t is the price of the underlying at time t and T is the expiration of the futures then the futures quote \Phi_T at T is S_T. Prior to that the futures quote \Phi_t, t < T, is determined by the exchange issuing the futures contract.
When trading on an exchange an initial margin must be provided. The exchange specifies margin adjustment times t_j that cause cash flows in margin accounts C_{t_j} = \Phi_{t_j} - \Phi_{t_{j-1}}. The price of a futures is always zero so 0 = E_{t_{j-1}}[C_{t_j}D_{t_j}]. Assuming D_{t_j} is \mathcal{A}_{t_{j-1}}-measurable then \Phi_{t_{j-1}} = E_{t_{j-1}}[\Phi_{t_j}]. This provides justification for assuming futures quotes are a martingale, \Phi_t = E_t[S_T], t\le T.
One might think the problem of modeling instrument prices can be solved by specifying the price at some future time T and let S_t D_t = E_t[S_T] - \sum_{t < s \le T} C_s D_s. This only moves the question of price dynamics to computing the conditional expectations E_t. In order to do that we need to specify the filtration $\mathcal{A}_t) however this is a highly underdetermined problem, albiet worthwhile to consider.
A zero coupon bond D(t) has cash flow 1 at t. Its value at time zero is is V_0 = E[1 D_t]. Its value at time s \le t, D_s(t), satifies D_s(t) D_s = E_s[1 D_t] so D_s(t) = E_s[D_t]/D_s. We write D(t) for D_0(t) to usefully confuse it with the name of the instrument.
A forward rate agreement has two cash flows: C_t = -1 at the effective date t and C_u = 1 + f\delta(t,u) at the termination date u, where f is the rate and \delta(t,u) is the day count fraction. The day count fraction uses a day count basis to convert an interval of time into a number approximately equal to u - t in years. For example the Actual/360 day count basis is the difference in days from t to u divided by 360.
The par forward rate is the rate that makes the initial value zero 0 = E[-1 D_t + (1 + f\delta)D_u]. The par forward rate at time s\le t, F_s^\delta(t,u), satisfies 0 = E_s[-1 D_t + (1 + F_s^\delta(t,u)) D_u] so F_s^\delta(t,u) = (E_s[D_t]/E_s[D_u] - 1)/\delta(t,u) = (D_s(t)/D_s(u) - 1)/\delta(t,u).
Fixed leg… portfolio of zero coupon bonds.
Some stocks pay dividends that are determined by the issuer. A cash dividend is a fixed amount d_u paid at time u. The stock price satisfies S_t D_t = M_t - \sum_{t_i\le t} d_i D_{t_i} so E[S_t] = Let f_j = E[S_{t_j} D_{t_j}].
A proportional dividend pays \delta_u S_u at time u where S_u is the stock price at time u. Short term dividends are usually announced several months in advance. Longer term dividends are unknown but it is reasonable to assume the dividends will be larger if the stock increases in price and smaller if it decreases.
A hybrid model specifies both proportional and discrete dividends; the stock pays \delta_u S_u + d_u at time u. A schedule can be specified to weight short term cash dividends higher than proportional dividends and weight long term proportional dividends higher than cash dividends.
The stock price satisfies S_j D_j = M_j - \sum_{i\le j} (\delta_i S_i + d_i)D_i E[S_j D_j] = M_0 - \sum_{i\le j} \delta_i E[S_i D_i] + d_i E[D_i] f_t = E[S_t D_t], f_j = f_{t_j}, D(j) = D_{t_j} f_j = M_0 - \sum_{i\le j} \delta_i f_i + d_i D(i) f_j - f_{j-1} = - \delta_j f_j - d_j D(j) (1 + \delta_j)f_j = f_{j-1} - d_j D(j) f_j = (f_{j-1} - d_j D(j))/(1 + \delta_j) f_j = f_{j-1}/(1 + \delta_j) - d_j D(j)/(1 + \delta_j) f_j = \frac{f_{j-1}}{(1 + \delta_j)} - \frac{d_j D(j)}{(1 + \delta_j)} f_j = \frac{ \frac{f_{j-2}}{(1 + \delta_{j-1})} - \frac{d_{j-1} D({j-1})}{(1 + \delta_{j-1})} }{(1 + \delta_j)} - \frac{d_j D(j)}{(1 + \delta_j)} f_j = \frac{f_{j-2}}{(1 + \delta_{j-1})(1 + \delta_j)} - \frac{d_{j-1} D({j-1})}{(1 + \delta_{j-1})(1 + \delta_j)} - \frac{d_j D(j)}{(1 + \delta_j)} f_j = \frac{f_0}{\Pi_{i=0}^j (1 + \delta_i)} - \sum_{i=0}^j \frac{d_i D(i)}{\Pi_{k=i}^j (1 + \delta_k)}
Repurchase agreements determine the deflator.
Stocks have dividends, bonds have coupons, futures have daily margin adjustments. An European option has a single cash flow at expiration. Currencies and commodities do not have cash flows.
This model does not incorporate bid-ask spreads. The spread also depends on the amount being transacted and the credit quality of the two parties involved in the transaction. Cash flows are determined by the issuer of the instrument and are usually zero. Transactions often involve cash flows with third parties such as broker commissions or borrowing costs. These considerations will be ignored, for now.
Conditional expectation E[X|\mathcal{A}] is the Radon-Nykodym derivative of (XP)|_\mathcal{A} with respect to P.
Margin on futures and interest paid on those.