sync

um0.md

@@ -22,7 +22,7 @@ What volatiltiy should be used for valuing an option?
 After putting on a hedge, when should that be adjusted?
 How well will a hedging strategy perform?
 
-This note does not solve these problems. It proposes a rigorous
+This note does not solve these problems. It only proposes a rigorous
 mathematical framework to allow for quantitative discussions.
 We use stochastic processes to model market instrument prices and cash
 flows but have no need for Brownian motion, much less Ito processes.

um1.md

@@ -1,10 +1,10 @@
 ---
-title: Unified Model for Derivative Instruments
+title: Unified Model
 author: Keith A. Lewis
 institute: KALX, LLC
 fleqn: true
 classoption: fleqn
-abstract: Value, hedge, and manage the risk of any instruments
+abstract: Value, hedge, and manage the risk of instruments
 ...
 
 \newcommand{\RR}{\boldsymbol{R}}
@@ -13,40 +13,105 @@ abstract: Value, hedge, and manage the risk of any instruments
 \newcommand{\Var}{\operatorname{Var}}
 \newcommand{\Cov}{\operatorname{Cov}}
 
-Let $T$ be trading times, $I$ the set of all market instruments, $\Omega$ the sample space of possible outcomes,
-and $(A_t)_{t\in T}$ be algebras of sets on $\Omega$ indicating the information available at each trading time.
+A mathematical model for any set of instruments that can be used to value,
+hedge, and understand how poorly risk-neutral pricing can be used for
+managing risk.
+
+Stephen Ross...
+
+## Unified Model
+
+Let $T$ be the set of trading times, $I$ the set of all market
+instruments, $\Omega$ the sample space of possible outcomes, and
+$(\AA_t)_{t\in T}$ be algebras of sets on $\Omega$ indicating the
+information available at each trading time.
+
+If $\AA$ is a finite algebra of sets on $\Omega$ then
+the _atoms_ of $\AA$, $\bar{\AA}$, form a partition of $\Omega$.
+A function $X\colon\Omega\to\RR$ is $\AA$ measurable if and
+only if it is constant on atoms of $\AA$
+so $X\colon\bar{\AA}\to\RR$ is a function.
+
+### Market
 
 _Price_ - market price assuming perfect liquidity
-: $X_t\colon\AA_t\to\RR^I$
+: $X_t\colon\bar{\AA_t}\to\RR^I$
 
 _Cash flow_ - dividends, coupons, margin adjustments for futures
-: $C_t\colon\AA_t\to\RR^I$
+: $C_t\colon\bar{\AA_t}\to\RR^I$
+
+### Trading
 
 _Trading Strategy_ - increasing stopping times
 : $\tau_0 < \cdots < \tau_n$ and trades $\Gamma_j\colon\AA_{\tau_j}\to\RR^I$
 
 _Position_ - accumulate trades not including last trade
 : $\Delta_t = \sum_{\tau_j < t}\Gamma_j = \sum_{s < t} \Gamma_s$
 
+### Valuation
+
 _Value_ - mark-to-market including last trade
 : $V_t = (\Delta_t + \Gamma_t)\cdot X_t$
 
 _Account_ - trading account blotter
 : $A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t$
 
+### Arbitrage
+
 _Arbitrage_ exists if there is a trading strategy
 with $A_{\tau_0} > 0$, $A_t \ge 0$, $t > \tau_0$, and $\sum_{j} \Gamma_j = 0$.
 
 The Fundamental Theorem of Asset Pricing states there is no arbitrage if and only
-if there exist positive measures $D_t\colon\AA_t\to(0,\infty)$, ${t\in T}$ on $\Omega$ with
+if there exist _deflators_, positive measures $D_t\colon\AA_t\to(0,\infty)$, ${t\in T}$ on $\Omega$, with
 $$
-	X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|_{\AA_t}
+\tag{1} X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|{\AA_t}
 $$
+where $|$ indicates restriction of measure to a subalgebra of sets.
+
+__Lemma__. If $X_t D_t = M_t - \sum_{s\le t} C_s D_s$ where $M_t = M_u|{\AA_t}$, $t \le u$,
+then there is no arbitrage.
 
 __Lemma__. For any arbitrage free model and any trading strategy
 $$
-	V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{\AA_t}
+\tag{2}	V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|{\AA_t}
 $$
 
-__Lemma__. If $X_t D_t = M_t - \sum_{s\le t} C_s D_s$ where $M_t = M_u|_{\AA_t}$, $t \le u$,
-then there is no arbitrage.
+Note how the value $V_t$ corresponds to price $X_t$ and account $A_t$
+corresponds to $C_t$ in equations (2) and (1) respectively.
+Trading strategies create synthetic market instruments.
+This is the skeleton key to pricing derivative securities.
+
+## Application
+
+Suppose a derivative security specifies amounts $\bar{A}_j$ be paid at times $\bar{\tau}_j$.
+If there is a trading strategy $(\tau_j, \Gamma_j)$
+with $A_{\bar{\tau}_j} = \bar{A}_j$ for all $j$ and $A_t = 0$ otherwise (aka self-financing) then
+a "perfect hedge" exists[^1].
+The value of the derivative is the cost of setting up the initial hedge.
+
+Note $V_t D_t= (\sum_{\tau_j > t} \bar{A}_j D_{\bar{\tau_j}})|\AA_t$
+can be computed from the deriviative contract specification and the deflators $D_t$.
+Since $V_t = (\Delta_t + \Gamma_t)\cdot X_t$
+we have $\Delta_t + \Gamma_t$, is the Frechet derivative $D_{X_t}V_t$
+of option value with respect to $X_t$.
+
+If time $T = \{t_j\}$ is discrete we can compute a possible hedge at each time,
+$\Gamma_j = D_{X_j}V_j - \Delta_j$, since $\Delta_j$ is known at $t_{j-1}$.
+In general this hedge will not exactly replicate the derivative contract obligations.
+
+[^1]: Perfect hedges never exist.
+
+### Deflator
+
+If repurchase agreements are available then a _canonical deflator_ exists.
+A repurchase agreement over the interval $[t_j, t_{j+}]$ is specified
+by a rate $f_j$ known at time $t_j$. The price at $t_j$ is $1$ and it
+has a cash flow of $\exp(f_j(t_{j+1} - t_j))$ at time $t_{j+1}$.
+By equation (1) we have $D_j = \exp(f_j\Delta t_j)D_{j+1}|\AA_j$.
+If $D_{j+1}$ is known at time $t_j$ then $D_{j+1}/D_j = \exp(-f_j\Delta t_j)$ and
+$D_j = \exp(-\sum_{i < j}f_i\Delta t_i)$ is the canonical deflator.
+
+### Binomial Model
+
+
+