From 5d10bb8071b16820057bf7499cf66ac0673f460c Mon Sep 17 00:00:00 2001 From: "Keith A. Lewis" Date: Thu, 25 Apr 2024 11:45:57 -0400 Subject: [PATCH] sync --- docs/123.html | 4 +- docs/a.html | 2 +- docs/a_z.html | 4 +- docs/apl.html | 4 +- docs/bach.html | 4 +- docs/basic.html | 4 +- docs/bayes.html | 4 +- docs/bigo.html | 4 +- docs/bio.html | 2 +- docs/blurb.html | 2 +- docs/bm.html | 4 +- docs/bonds.html | 4 +- docs/bqn.html | 4 +- docs/bsm.html | 4 +- docs/capm.html | 4 +- docs/cat.html | 4 +- docs/cdf.html | 4 +- docs/ce.html | 4 +- docs/cm.html | 4 +- docs/copula.html | 4 +- docs/coro.html | 4 +- docs/curry.html | 4 +- docs/db.html | 4 +- docs/de.html | 4 +- docs/det.html | 4 +- docs/div.html | 4 +- docs/dual.html | 4 +- docs/ec.html | 4 +- docs/ecr.html | 4 +- docs/eebo.html | 4 +- docs/eli5.html | 2 +- docs/ep.html | 4 +- docs/ep_cover.html | 4 +- docs/ep_title.html | 4 +- docs/eps.html | 2 +- docs/eth.html | 4 +- docs/factors.html | 4 +- docs/falg.html | 4 +- docs/fc.html | 4 +- docs/fd.html | 4 +- docs/ff.html | 4 +- docs/fi.html | 4 +- docs/fin.html | 2 +- docs/fof.html | 4 +- docs/fsa.html | 2 +- docs/funny.html | 4 +- docs/ga.html | 4 +- docs/gla.html | 4 +- docs/gop.html | 4 +- docs/ho-lee.html | 4 +- docs/ice.html | 2 +- docs/int.html | 4 +- docs/ip.html | 4 +- docs/is.html | 4 +- docs/ito.html | 4 +- docs/kal.html | 4 +- docs/kh.html | 4 +- docs/la.html | 4 +- docs/lagd.html | 4 +- docs/lambda.html | 4 +- docs/lfc.html | 4 +- docs/lo.html | 4 +- docs/math.css | 2 +- docs/mcmc.html | 4 +- docs/measure.html | 4 +- docs/mf.html | 4 +- docs/mfm.html | 4 +- docs/ml.html | 4 +- docs/mo.html | 4 +- docs/monad.html | 4 +- docs/njr.html | 4 +- docs/ode.html | 4 +- docs/op.html | 4 +- docs/op2.html | 4 +- docs/opm.html | 4 +- docs/opm2.html | 4 +- docs/optics.html | 4 +- docs/pc.html | 2 +- docs/pcp.html | 2 +- docs/pdv.html | 4 +- docs/pes.html | 4 +- docs/pnl.html | 4 +- docs/pq.html | 2 +- docs/prob.html | 4 +- docs/profile.html | 2 +- docs/ps.html | 4 +- docs/pt.html | 4 +- docs/raii.html | 4 +- docs/rel.html | 4 +- docs/remf.html | 2 +- docs/res.html | 4 +- docs/rl.html | 4 +- docs/ro.html | 4 +- docs/rw.html | 4 +- docs/set.html | 4 +- docs/sf.html | 4 +- docs/shimko.html | 2 +- docs/simfin.html | 2 +- docs/simple.html | 4 +- docs/sofr.html | 4 +- docs/sofrsg.html | 4 +- docs/sp.html | 4 +- docs/stat.html | 4 +- docs/tilfp.html | 2 +- docs/tim.html | 4 +- docs/tlh.html | 4 +- docs/tmt.html | 4 +- docs/tnbt.html | 2 +- docs/tos.html | 4 +- docs/tpm.html | 4 +- docs/trading.html | 4 +- docs/ts.html | 4 +- docs/u.html | 4 +- docs/uf.html | 4 +- docs/um.html | 4 +- docs/um0.html | 4 +- docs/um1.html | 267 ++++++++++++++++++++++++-------------------- docs/um2.html | 4 +- docs/um_short.html | 4 +- docs/um_slides.html | 4 +- docs/umm.html | 4 +- docs/ums.html | 4 +- docs/vs.html | 4 +- docs/vs0.html | 4 +- docs/vswap.html | 4 +- docs/wc.html | 4 +- docs/wif.html | 4 +- docs/wmpl.html | 4 +- docs/wtd.html | 4 +- docs/ycm.html | 4 +- docs/yo.html | 4 +- docs/zcb.html | 4 +- 132 files changed, 393 insertions(+), 362 deletions(-) diff --git a/docs/123.html b/docs/123.html index ebf1edf..f2e1268 100644 --- a/docs/123.html +++ b/docs/123.html @@ -5,7 +5,7 @@ - + 1-2-3 Model + /* CSS for citations */ + div.csl-bib-body { } + div.csl-entry { + clear: both; + margin-bottom: 0em; + } + .hanging-indent div.csl-entry { + margin-left:2em; + text-indent:-2em; + } + div.csl-left-margin { + min-width:2em; + float:left; + } + div.csl-right-inline { + margin-left:2em; + padding-left:1em; + } + div.csl-indent { + margin-left: 2em; + } @@ -47,18 +67,20 @@

Unified Model

Keith A. Lewis

-

April 23, 2024

+

April 25, 2024

Abstract
Value, hedge, and manage the risk of instruments

This note assumes you are familiar with measure theory and stochastic -processes, but are not necesarily an expert. We provide a mathematically -rigorous 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. It does not provide a solution, only an initial framework -for for further research.

+processes, but are not necessarily an expert. We provide a +mathematically rigorous model that extends (Ross 1978) without involving the +Hahn-Banach theorem. The Unified Model can be used on any set of +instruments to value, hedge, and understand how poorly risk-neutral +pricing can be used for managing risk. It does not provide a solution, +only an initial framework for further research.

Unified Model

Let T be the set of trading times, @@ -70,14 +92,14 @@

Unified Model

If {\mathcal{A}} is a finite algebra of sets on \Omega then the atoms of {\mathcal{A}}, \bar{{\mathcal{A}}}, form a partition of -\Omega. A function \underline{{\mathcal{A}}}, form a partition +of \Omega. A function X\colon\Omega\to\boldsymbol{R} is {\mathcal{A}} measurable if and only if it is constant on atoms of {\mathcal{A}} so X\colon\bar{{\mathcal{A}}}\to\boldsymbol{R} -is a function.

+class="math inline">X\colon\underline{{\mathcal{A}}}\to\boldsymbol{R} +is a function.

If P is a probability measure on \Omega and X\colon\Omega\to\boldsymbol{R} is a random @@ -86,49 +108,30 @@

Unified Model

class="math inline">Y(P|A) = (XP)|{\mathcal{A}}.

Market

-
-
Price – market price assuming perfect liquidity
-
-X_t\colon\bar{{\mathcal{A}}_t}\to\boldsymbol{R}^I -
-
Cash flow – dividends, coupons, margin adjustments for -futures
-
-C_t\colon\bar{{\mathcal{A}}_t}\to\boldsymbol{R}^I -
-
+

Price – X_t\colon\underline{{\mathcal{A}}}_t\to\boldsymbol{R}^I +market prices assuming perfect liquidity.

+

Cash flow – C_t\colon\underline{{\mathcal{A}}}_t\to\boldsymbol{R}^I +dividends, coupons, margin adjustments for futures.

Trading

-
-
Trading Strategy – increasing stopping times
-
-\tau_0 < \cdots < \tau_n and -trades \Gamma_j\colon{\mathcal{A}}_{\tau_j}\to\boldsymbol{R}^I -
-
Position – accumulate trades not including last trade
-
-\Delta_t = \sum_{\tau_j < t}\Gamma_j = -\sum_{s < t} \Gamma_s -
-
+

Trading Strategy – \tau_0 < +\cdots < \tau_n stopping times and trades \Gamma_j\colon\underline{{\mathcal{A}}}_{\tau_j}\to\boldsymbol{R}^I

+

Position – \Delta_t = \sum_{\tau_j +< t}\Gamma_j = \sum_{s < t} \Gamma_s accumulate trades +not including last trades.

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 -
-
+

Value – V_t = (\Delta_t + +\Gamma_t)\cdot X_t mark-to-market existing positions and current +trades at current prices.

+

Account – A_t = \Delta_t\cdot C_t +- \Gamma_t\cdot X_t receive cash flows proportional to position +and pay for current trades.

Arbitrage

@@ -140,18 +143,21 @@

Arbitrage

money.

The Fundamental Theorem of Asset Pricing states there is no arbitrage if and only if there exist deflators, positive measures D_t\colon{\mathcal{A}}_t\to(0,\infty), {t\in T} on \Omega, with +class="math inline">D_t on {\mathcal{A}}_t, {t\in T}, with \tag{1} X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|{{\mathcal{A}}_t} - Note if cash flows are zero then deflated prices are a -martingale. If there are a finite number of cash flows then prices are -determined by deflated future cash flows.

+ A martingale measure satisfies M_t = M_u|{\mathcal{A}}_t for t \le u. Note if cash flows are zero then +deflated prices are a martingale measure. If X_u D_u goes to zero as u goes to infinity then prices are determined +by deflated future cash flows.

Lemma. If X_t D_t = M_t - -\sum_{s\le t} C_s D_s where M_t = -M_u|{{\mathcal{A}}_t}, t \le u, -then there is no arbitrage.

+\sum_{s\le t} C_s D_s where M_t +is a martingale measue then there is no arbitrage.

Lemma. For any arbitrage free model and any trading strategy \tag{2} V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s @@ -168,61 +174,61 @@

Arbitrage

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\overline{A}_j be paid at times \overline{\tau}_j. If there is a trading +strategy (\tau_j, \Gamma_j) with A_{\overline{\tau}_j} = \overline{A}_j for +all j and A_t += 0 otherwise (aka self-financing) then a “perfect hedge” +exists1.

Note V_t D_t= (\sum_{\tau_j > t} -\bar{A}_j D_{\bar{\tau_j}})|{\mathcal{A}}_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.

+\overline{A}_j D_{\overline{\tau_j}})|{\mathcal{A}}_t can be +computed from the derivative contract specification and the deflators +D_t. Since we also have V_t = (\Delta_t + \Gamma_t)\cdot X_t the +Frechet derivative D_{X_t}V_t of option +value with respect to X_t is \Delta_t + \Gamma_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 obligation.

+

Note the Unified Model does not require Ito’s formula, much less a +proof involving partial differential equations and change of measure. +One simply writes down a martingale and deflator then uses equation (2) +to value, hedge, and manage the risk of realistic trading strategies. +The notion of “continuous time” hedging is a mathematical myth.

Black-Scholes/Merton

The Black-Scholes/Merton model uses M_t = (r, s\exp(\sigma B_t - \sigma^2t/2)P, where B_t is Brownian motion and P is Weiner measure The deflator is D_t = \exp(-\rho t).

-

Note the Unified Model does not require Ito’s formula and a proof -involving partial differential equations. One just writes down a -martingale and deflator then uses equation (2) to value, hedge, and -manage the risk of trading strategies that can be performed. The notion -of “continuous time” hedging is a mathematical myth.

+class="math inline">B_t is Brownian motion, P is Wiener measure, and the deflator is +D_t = \exp(-\rho t)P.

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 [t_j, t_{j+1}] 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}|{\mathcal{A}}_j. If {\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}|{\mathcal{A}}_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.

+class="math inline">{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.

The continuous time analog is D_t = \exp(-\int_0^t f(s)\,ds) where f -is the continuously componded instantaneous forward rate.

+is the continuously compounded instantaneous forward rate.

Zero Coupon Bond

@@ -248,21 +254,23 @@

Forward Rate Agreement

A forward rate agreement with coupon f over the interval [u,v] having day count convention \delta has two cash flows: -1 at time t -and 1 + f\delta(u,v) at time u. The par forward coupon at time -t, F_t^\delta(u,v) is the coupon for which the -price is 0 at time t. By equation (1) -we have 0 = (-D_u + (1 + +class="math inline">\delta2 has two cash flows: +-1 at time t and 1 + +f\delta(u,v) at time u. The +par forward coupon at time t, +F_t^\delta(u,v) is the coupon for which +the price is 0 at time t. By equation +(1) we have 0 = (-D_u + (1 + F_t^\delta(u,v))D_v|{\mathcal{A}}_t so F_t^\delta(u,v) = \frac{1}{\delta(u,v)}\bigl(\frac{D_t(u)}{D_t(v)} - 1\bigr).

A swap is a collection of back-to-back forward rate -agreements. The swap par coupon makes the price 0 at time (t_j). The +swap par coupon makes the price 0 at time t so F_t^\delta(t_0,\dots,t_n) = \frac{D_t(t_0) - D_t(t_n)}{\sum_{j=1}^n\delta(t_{j-1},t_j)D_t(t_j)}. @@ -271,20 +279,20 @@

Forward Rate Agreement

Risky Bonds

Companies can default and may pay only a fraction of the notional -owed on bonds they issued. A simple model for this is to assume the time -of default is a random variable T and -recovery R is a constant fraction -between 0 and 1.2 The sample space for the default -time is [0,\infty) indicating the time -of default. The information available at time t is the partition consisting of singletons -\{s\}, s < +owed on bonds they issued. A simple model3 for +this is to assume the time of default T +and recovery R are random variables. +The sample space for the default time is [0,\infty) indicating the time of default. +The information available at time t is +the partition consisting of singletons \{s\}, s < t and the set [t, \infty). If default occurs prior to t then we know -exactly when it happend. If default has not occured by time t then we only know it can be any time after -that.

+exactly when it happened. If default has not occurred by time t then we only know it can occur any time +after that.

The cash flows for a risky bond D^{T,R}(u) are 1 at time u if T > @@ -292,13 +300,22 @@

Risky Bonds

class="math inline">T if T \le t. For the model to be arbitrage-free we must have - D_t^{R,T}(u) D_t = (D_u 1(T > u) + RD_t 1(T\le -u)|{\mathcal{A}}_t. - If the interest rate is 0 then D_t = -1 and D_0^{R,T} = P(T > u) + RP(T\le -u).

-

We need to extend the sample space to model the default time, \Omega' = \Omega\times [0,\infty).

+D_t^{R,T}(u) D_t = \bigl(1(T > u)D_u + 1(T\le u)R +D_T\bigr)|{\mathcal{A}}_t. + If the deflators are independent of the stopping time and +recovery is RD(T,u) at T\le u for some constant R then using D_T(u)D_T = D_u|{\mathcal{A}}_t for T \le u we have + D_t^{T,R}(u) D_t P(T > t) = \bigl(P(T > u) + R P(t < T \le +u)\bigr)D_u|{\mathcal{A}}_t. + The credit spread \lambda_t^{T,R}(u) is defined by {D_t^{T,R}(u) = D_t(u)\exp(-\lambda_t^{T,R}(u) (u - +t))}. Note if T = \infty or +R = 1 then the credit spread is +zero.

+
+
+Ross, Stephen A. 1978. “A Simple +Approach to the Valuation of +Risky Streams.” The Journal of +Business 51 (3): 453–75. https://www.jstor.org/stable/2352277. +
+
Risky Bonds

  1. Perfect hedges never exist.↩︎

    2. -

  2. This is a very simplified model.

    The day count fraction \delta(u, v) is approximately v - u in years.↩︎

    3. +

  3. This is a very simplified model.↩︎

diff --git a/docs/um2.html b/docs/um2.html index 8666590..ccf77ba 100644 --- a/docs/um2.html +++ b/docs/um2.html @@ -5,7 +5,7 @@ - + Unified Model for Derivative Instruments