Incomplete Markets and Calibrations for Coherence with Hedged Portfolios. Thought of the Day 154.0

 

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In complete market models such as the Black-Scholes model, probability does not really matter: the “objective” evolution of the asset is only there to define the set of “impossible” events and serves to specify the class of equivalent measures. Thus, two statistical models P1 ∼ P2 with equivalent measures lead to the same option prices in a complete market setting.

This is not true anymore in incomplete markets: probabilities matter and model specification has to be taken seriously since it will affect hedging decisions. This situation is more realistic but also more challenging and calls for an integrated approach between option pricing methods and statistical modeling. In incomplete markets, not only does probability matter but attitudes to risk also matter: utility based methods explicitly incorporate these into the hedging problem via utility functions. While these methods are focused on hedging with the underlying asset, common practice is to use liquid call/put options to hedge exotic options. In incomplete markets, options are not redundant assets; therefore, if options are available as hedging instruments they can and should be used to improve hedging performance.

While the lack of liquidity in the options market prevents in practice from using dynamic hedges involving options, options are commonly used for static hedging: call options are frequently used for dealing with volatility or convexity exposures and for hedging barrier options.

What are the implications of hedging with options for the choice of a pricing rule? Consider a contingent claim H and assume that we have as hedging instruments a set of benchmark options with prices Ci, i = 1 . . . n and terminal payoffs Hi, i = 1 . . . n. A static hedge of H is a portfolio composed from the options Hi, i = 1 . . . n and the numeraire, in order to match as closely as possible the terminal payoff of H:

H = V0 + ∑i=1n xiHi + ∫0T φdS + ε —– (1)

where ε is an hedging error representing the nonhedgeable risk. Typically Hi are payoffs of call or put options and are not possible to replicate using the underlying so adding them to the hedge portfolio increases the span of hedgeable claims and reduces residual risk.

Consider a pricing rule Q. Assume that EQ[ε] = 0 (otherwise EQ[ε] can be added to V0). Then the claim H is valued under Q as:

e-rTEQ[H] = V0 ∑i=1n xe-rTEQ[Hi] —– (2)

since the stochastic integral term, being a Q-martingale, has zero expectation. On the other hand, the cost of setting up the hedging portfolio is:

V0 + ∑i=1n xCi —– (3)

So the value of the claim given by the pricing rule Q corresponds to the cost of the hedging portfolio if the model prices of the benchmark options Hi correspond to their market prices Ci:

∀i = 1, …, n

e-rTEQ[Hi] = Ci∗ —– (4)

This condition is called calibration, where a pricing rule verifies the calibration of the option prices Ci, i = 1, . . . , n. This condition is necessary to guarantee the coherence between model prices and the cost of hedging with portfolios and if the model is not calibrated then the model price for a claim H may have no relation with the effective cost of hedging it using the available options Hi. If a pricing rule Q is specified in an ad hoc way, the calibration conditions will not be verified, and thus one way to ensure them is to incorporate them as constraints in the choice of the pricing measure Q.

Self-Financing and Dynamically Hedged Portfolio – Robert Merton’s Option Pricing. Thought of the Day 153.0

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As an alternative to the riskless hedging approach, Robert Merton derived the option pricing equation via the construction of a self-financing and dynamically hedged portfolio containing the risky asset, option and riskless asset (in the form of money market account). Let QS(t) and QV(t) denote the number of units of asset and option in the portfolio, respectively, and MS(t) and MV(t) denote the currency value of QS(t) units of asset and QV(t) units of option, respectively. The self-financing portfolio is set up with zero initial net investment cost and no additional funds are added or withdrawn afterwards. The additional units acquired for one security in the portfolio is completely financed by the sale of another security in the same portfolio. The portfolio is said to be dynamic since its composition is allowed to change over time. For notational convenience, dropping the subscript t for the asset price process St, the option value process Vt and the standard Brownian process Zt. The portfolio value at time t can be expressed as

Π(t) = MS(t) + MV(t) + M(t) = QS(t)S + QV(t)V + M(t) —– (1)

where M(t) is the currency value of the riskless asset invested in a riskless money market account. Suppose the asset price process is governed by the stochastic differential equation (1) in here, we apply the Ito lemma to obtain the differential of the option value V as:

dV = ∂V/∂t dt + ∂V/∂S dS + σ2/2 S22V/∂S2 dt = (∂V/∂t + μS ∂V/∂S σ2/2 S22V/∂S2)dt + σS ∂V/∂S dZ —– (2)

If we formally write the stochastic dynamics of V as

dV/V = μV dt + σV dZ —– (3)

then μV and σV are given by

μV = (∂V/∂t + ρS ∂V/∂S + σ2/2 S22V/∂S2)/V —– (4)

and

σV = (σS ∂V/∂S)/V —– (5)

The instantaneous currency return dΠ(t) of the above portfolio is attributed to the differential price changes of asset and option and interest accrued, and the differential changes in the amount of asset, option and money market account held. The differential of Π(t) is computed as:

dΠ(t) = [QS(t) dS + QV(t) dV + rM(t) dt] + [S dQS(t) + V dQV(t) + dM(t)] —– (6)

where rM(t)dt gives the interest amount earned from the money market account over dt and dM(t) represents the change in the money market account held due to net currency gained/lost from the sale of the underlying asset and option in the portfolio. And if the portfolio is self-financing, the sum of the last three terms in the above equation is zero. The instantaneous portfolio return dΠ(t) can then be expressed as:

dΠ(t) = QS(t) dS + QV(t) dV + rM(t) dt = MS(t) dS/S + MV(t) dV/V +  rM(t) dt —– (7)

Eliminating M(t) between (1) and (7) and expressing dS/S and dV/V in terms of their stochastic dynamics, we obtain

dΠ(t) = [(μ − r)MS(t) + (μV − r)MV(t)]dt + [σMS(t) + σV MV(t)]dZ —– (8)

How can we make the above self-financing portfolio instantaneously riskless so that its return is non-stochastic? This can be achieved by choosing an appropriate proportion of asset and option according to

σMS(t) + σV MV(t) = σS QS(t) + σS ∂V/∂S QV(t) = 0

that is, the number of units of asset and option in the self-financing portfolio must be in the ratio

QS(t)/QV(t) = -∂V/∂S —– (9)

at all times. The above ratio is time dependent, so continuous readjustment of the portfolio is necessary. We now have a dynamic replicating portfolio that is riskless and requires zero initial net investment, so the non-stochastic portfolio return dΠ(t) must be zero.

(8) becomes

0 = [(μ − r)MS(t) + (μV − r)MV(t)]dt

substituting the ratio factor in the above equation, we get

(μ − r)S ∂V/∂S = (μV − r)V —– (10)

Now substituting μfrom (4) into the above equation, we get the black-Scholes equation for V,

∂V/∂t + σ2/2 S22V/∂S2 + rS ∂V/∂S – rV = 0

Suppose we take QV(t) = −1 in the above dynamically hedged self-financing portfolio, that is, the portfolio always shorts one unit of the option. By the ratio factor, the number of units of risky asset held is always kept at the level of ∂V/∂S units, which is changing continuously over time. To maintain a self-financing hedged portfolio that constantly keeps shorting one unit of the option, we need to have both the underlying asset and the riskfree asset (money market account) in the portfolio. The net cash flow resulting in the buying/selling of the risky asset in the dynamic procedure of maintaining ∂V/∂S units of the risky asset is siphoned to the money market account.

Derivative Pricing Theory: Call, Put Options and “Black, Scholes'” Hedged Portfolio.Thought of the Day 152.0

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Fischer Black and Myron Scholes revolutionized the pricing theory of options by showing how to hedge continuously the exposure on the short position of an option. Consider the writer of a call option on a risky asset. S/he is exposed to the risk of unlimited liability if the asset price rises above the strike price. To protect the writer’s short position in the call option, s/he should consider purchasing a certain amount of the underlying asset so that the loss in the short position in the call option is offset by the long position in the asset. In this way, the writer is adopting the hedging procedure. A hedged position combines an option with its underlying asset so as to achieve the goal that either the asset compensates the option against loss or otherwise. By adjusting the proportion of the underlying asset and option continuously in a portfolio, Black and Scholes demonstrated that investors can create a riskless hedging portfolio where the risk exposure associated with the stochastic asset price is eliminated. In an efficient market with no riskless arbitrage opportunity, a riskless portfolio must earn an expected rate of return equal to the riskless interest rate.

Black and Scholes made the following assumptions on the financial market.

  1. Trading takes place continuously in time.
  2. The riskless interest rate r is known and constant over time.
  3. The asset pays no dividend.
  4. There are no transaction costs in buying or selling the asset or the option, and no taxes.
  5. The assets are perfectly divisible.
  6. There are no penalties to short selling and the full use of proceeds is permitted.
  7. There are no riskless arbitrage opportunities.

The stochastic process of the asset price St is assumed to follow the geometric Brownian motion

dSt/St = μ dt + σ dZt —– (1)

where μ is the expected rate of return, σ is the volatility and Zt is the standard Brownian process. Both μ and σ are assumed to be constant. Consider a portfolio that involves short selling of one unit of a call option and long holding of Δt units of the underlying asset. The portfolio value Π (St, t) at time t is given by

Π = −c + Δt St —– (2)

where c = c(St, t) denotes the call price. Note that Δt changes with time t, reflecting the dynamic nature of hedging. Since c is a stochastic function of St, we apply the Ito lemma to compute its differential as follows:

dc = ∂c/∂t dt + ∂c/∂St dSt + σ2/2 St2 ∂2c/∂St2 dt

such that

-dc + Δt dS= (-∂c/∂t – σ2/2 St2 ∂2c/∂St2)dt + (Δ– ∂c/∂St)dSt

= [-∂c/∂t – σ2/2 St2 ∂2c/∂St+ (Δ– ∂c/∂St)μSt]dt + (Δ– ∂c/∂St)σSdZt

The cumulative financial gain on the portfolio at time t is given by

G(Π (St, t )) = ∫0t -dc + ∫0t Δu dSu

= ∫0t [-∂c/∂u – σ2/2 Su22c/∂Su2 + (Δ– ∂c/∂Su)μSu]du + ∫0t (Δ– ∂c/∂Su)σSdZ—– (3)

The stochastic component of the portfolio gain stems from the last term, ∫0t (Δ– ∂c/∂Su)σSdZu. Suppose we adopt the dynamic hedging strategy by choosing Δu = ∂c/∂Su at all times u < t, then the financial gain becomes deterministic at all times. By virtue of no arbitrage, the financial gain should be the same as the gain from investing on the risk free asset with dynamic position whose value equals -c + Su∂c/∂Su. The deterministic gain from this dynamic position of riskless asset is given by

Mt = ∫0tr(-c + Su∂c/∂Su)du —– (4)

By equating these two deterministic gains, G(Π (St, t)) and Mt, we have

-∂c/∂u – σ2/2 Su22c/∂Su2 = r(-c + Su∂c/∂Su), 0 < u < t

which is satisfied for any asset price S if c(S, t) satisfies the equation

∂c/∂t + σ2/2 S22c/∂S+ rS∂c/∂S – rc = 0 —– (5)

This parabolic partial differential equation is called the Black–Scholes equation. Strangely, the parameter μ, which is the expected rate of return of the asset, does not appear in the equation.

To complete the formulation of the option pricing model, let’s prescribe the auxiliary condition. The terminal payoff at time T of the call with strike price X is translated into the following terminal condition:

c(S, T ) = max(S − X, 0) —– (6)

for the differential equation.

Since both the equation and the auxiliary condition do not contain ρ, one concludes that the call price does not depend on the actual expected rate of return of the asset price. The option pricing model involves five parameters: S, T, X, r and σ. Except for the volatility σ, all others are directly observable parameters. The independence of the pricing model on μ is related to the concept of risk neutrality. In a risk neutral world, investors do not demand extra returns above the riskless interest rate for bearing risks. This is in contrast to usual risk averse investors who would demand extra returns above r for risks borne in their investment portfolios. Apparently, the option is priced as if the rates of return on the underlying asset and the option are both equal to the riskless interest rate. This risk neutral valuation approach is viable if the risks from holding the underlying asset and option are hedgeable.

The governing equation for a put option can be derived similarly and the same Black–Scholes equation is obtained. Let V (S, t) denote the price of a derivative security with dependence on S and t, it can be shown that V is governed by

∂V/∂t + σ2/2 S22V/∂S+ rS∂V/∂S – rV = 0 —– (7)

The price of a particular derivative security is obtained by solving the Black–Scholes equation subject to an appropriate set of auxiliary conditions that model the corresponding contractual specifications in the derivative security.

The original derivation of the governing partial differential equation by Black and Scholes focuses on the financial notion of riskless hedging but misses the precise analysis of the dynamic change in the value of the hedged portfolio. The inconsistencies in their derivation stem from the assumption of keeping the number of units of the underlying asset in the hedged portfolio to be instantaneously constant. They take the differential change of portfolio value Π to be

dΠ =−dc + Δt dSt,

which misses the effect arising from the differential change in Δt. The ability to construct a perfectly hedged portfolio relies on the assumption of continuous trading and continuous asset price path. It has been commonly agreed that the assumed Geometric Brownian process of the asset price may not truly reflect the actual behavior of the asset price process. The asset price may exhibit jumps upon the arrival of a sudden news in the financial market. The interest rate is widely recognized to be fluctuating over time in an irregular manner rather than being constant. For an option on a risky asset, the interest rate appears only in the discount factor so that the assumption of constant/deterministic interest rate is quite acceptable for a short-lived option. The Black–Scholes pricing approach assumes continuous hedging at all times. In the real world of trading with transaction costs, this would lead to infinite transaction costs in the hedging procedure.

Algorithmic Trading. Thought of the Day 151.0

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One of the first algorithmic trading strategies consisted of using a volume-weighted average price, as the price at which orders would be executed. The VWAP introduced by Berkowitz et al. can be calculated as the dollar amount traded for every transaction (price times shares traded) divided by the total shares traded for a given period. If the price of a buy order is lower than the VWAP, the trade is executed; if the price is higher, then the trade is not executed. Participants wishing to lower the market impact of their trades stress the importance of market volume. Market volume impact can be measured through comparing the execution price of an order to a benchmark. The VWAP benchmark is the sum of every transaction price paid, weighted by its volume. VWAP strategies allow the order to dilute the impact of orders through the day. Most institutional trading occurs in filling orders that exceed the daily volume. When large numbers of shares must be traded, liquidity concerns can affect price goals. For this reason, some firms offer multiday VWAP strategies to respond to customers’ requests. In order to further reduce the market impact of large orders, customers can specify their own volume participation by limiting the volume of their orders to coincide with low expected volume days. Each order is sliced into several days’ orders and then sent to a VWAP engine for the corresponding days. VWAP strategies fall into three categories: sell order to a broker-dealer who guarantees VWAP; cross the order at a future date at VWAP; or trade the order with the goal of achieving a price of VWAP or better.

The second algorithmic trading strategy is the time-weighted average price (TWAP). TWAP allows traders to slice a trade over a certain period of time, thus an order can be cut into several equal parts and be traded throughout the time period specified by the order. TWAP is used for orders which are not dependent on volume. TWAP can overcome obstacles such as fulfilling orders in illiquid stocks with unpredictable volume. Conversely, high-volume traders can also use TWAP to execute their orders over a specific time by slicing the order into several parts so that the impact of the execution does not significantly distort the market.

Yet, another type of algorithmic trading strategy is the implementation shortfall or the arrival price. The implementation shortfall is defined as the difference in return between a theoretical portfolio and an implemented portfolio. When deciding to buy or sell stocks during portfolio construction, a portfolio manager looks at the prevailing prices (decision prices). However, several factors can cause execution prices to be different from decision prices. This results in returns that differ from the portfolio manager’s expectations. Implementation shortfall is measured as the difference between the dollar return of a paper portfolio (paper return) where all shares are assumed to transact at the prevailing market prices at the time of the investment decision and the actual dollar return of the portfolio (real portfolio return). The main advantage of the implementation shortfall-based algorithmic system is to manage transactions costs (most notably market impact and timing risk) over the specified trading horizon while adapting to changing market conditions and prices.

The participation algorithm or volume participation algorithm is used to trade up to the order quantity using a rate of execution that is in proportion to the actual volume trading in the market. It is ideal for trading large orders in liquid instruments where controlling market impact is a priority. The participation algorithm is similar to the VWAP except that a trader can set the volume to a constant percentage of total volume of a given order. This algorithm can represent a method of minimizing supply and demand imbalances (Kendall Kim – Electronic and Algorithmic Trading Technology).

Smart order routing (SOR) algorithms allow a single order to exist simultaneously in multiple markets. They are critical for algorithmic execution models. It is highly desirable for algorithmic systems to have the ability to connect different markets in a manner that permits trades to flow quickly and efficiently from market to market. Smart routing algorithms provide full integration of information among all the participants in the different markets where the trades are routed. SOR algorithms allow traders to place large blocks of shares in the order book without fear of sending out a signal to other market participants. The algorithm matches limit orders and executes them at the midpoint of the bid-ask price quoted in different exchanges.

Handbook of Trading Strategies for Navigating and Profiting From Currency, Bond, Stock Markets

BASEL III: The Deflationary Symbiotic Alliance Between Governments and Banking Sector. Thought of the Day 139.0

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The Bank for International Settlements (BIS) is steering the banks to deal with government debt, since the governments have been running large deficits to deal with the catastrophe of BASEL 2-inspired mortgaged-backed securities collapse. The deficits are ranged anywhere between 3 to 7 per cent of the GDP, and in cases even higher. These deficits were being used to create a floor under growth by stimulating the economy and bailing out financial institutions that got carried away by the wholesale funding of real estate. And this is precisely what BASEL 2 promulgated, i.e. encouraging financial institutions to hold mortgage-backed securities for investments.

In comes the BASEL 3 rules that implore than banks must be in compliance with these regulations. But, who gets to decide these regulations? Actually, banks do, since they then come on board for discussions with the governments, and such negotiations are catered to bail banks out with government deficits in order to oil the engine of economic growth. The logic here underlines the fact that governments can continue to find a godown of sorts for their deficits, while the banks can buy government debt without any capital commitment and make a good spread without the risk, thus serving the interests of the both parties involved mutually. Moreover, for the government, the process is political, as no government would find it acceptable to be objective in its viewership of letting a bubble deflate, because any process of deleveraging would cause the banks to offset their lending orgy, which is detrimental to the engineered economic growth. Importantly, without these deficits, the financial system could go down the deflationary spiral, which might turn out to be a difficult proposition to recover if there isn’t any complicity in rhyme and reason accorded to this particular dysfunctional and symbiotic relationship. So, whats the implication of all this? The more government debt banks hold, the less overall capital they need. And who says so? BASEL 3.

But, the mesh just seems to be building up here. In the same way that banks engineered counterfeit AAA-backed securities that were in fact an improbable financial hoax, how can countries that have government debt/GDP ratio to the tune of 90 – 120 per cent get a Standard&Poor’s ratings of a double-A? They have these ratings because they belong to a apical club that gives their members exclusive rights to a high rating even if they are irresponsible with their issuing of debts. Well, is that this simple? Yes and no. Yes, as is above, and no is merely clothing itself in a bit of an economic jargon, in that these are the countries where the government debt can be held without any capital against it. In other words, if a debt cannot be held, it cannot be issued, and that is the reason why countries are striving for issuing debts that have a zero weighting.

Let us take snippets across gradations of BASEL 1, 2 and 3. In BASEL 1, the unintended consequences were that banks were all buying equity in cross-owned companies. When the unwinding happened, equity just fell apart, since any beginning of a financial crisis is tailored to smash bank equities to begin with. Thats the first wound to rationality. In BASEL 2, banks were told to hold as much AAA-rated paper as they wanted with no capital against it. What happened if these ratings were downgraded? It would trigger a tsunami cutting through pension and insurance schemes to begin with forcing them to sell their papers and pile up huge losses meant to absorbed by capital, which doesn’t exist against these papers. So whatever gets sold is politically cushioned and buffered for by the governments, for the risks cannot be afforded to get any more denser as that explosion would sound the catastrophic death knell for the economy. BASEL 3 doesn’t really help, even if it mandated to hold a concentrated portfolio of government debt without any capital against it, for absorption of losses in case of a crisis hitting would have to exhumed through government bail-outs in scenarios where government debts are a century plus. So, are the banks in-stability, or given to more instability via BASEL 3?  The incentives to ever more hold government securities increase bank exposure to sovereign bonds, adding to existing exposure of government securities via repurchase transactions, investments and trading inventories. A ratings downgrade results in a fall in value of bonds triggering losses. Banks would then face calls for additional collateral, which would drain liquidity, and which would then require additional capital as way of compensation. where would this capital come in from, if not for the governments to source it? One way out would be recapitalization through government debt. On the other hand, the markets are required to hedge against the large holdings of government securities and so short stocks, currencies and insurance companies are all made to stare in the face of volatility that rips through them, of which the net resultant is falling liquidity. So, this vicious cycle would continue to cycle its way through any downgrades. And thats why the deflationary symbiotic alliance between the governments and banking sector isn’t anything more than high-fatigue tolerance….

Defaultable Bonds. Thought of the Day 133.0

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Defaultable bonds are bonds that have a positive possibility of default.  Most corporate bonds and some government bonds are defaultable.  When a bond defaults, its coupon and principal payments will be altered.  Most of the time, only a portion of the principal, and sometimes, also a portion of the coupon, will be paid. A defaultable (T, x) – bond with maturity T > 0 and credit rating x ∈ I ⊆ [0, 1], is a financial contract which pays to its holder 1 unit of currency at time T provided that the writer of the bond hasn’t bankrupted till time T. The set I stands for all possible credit ratings. The bankruptcy is modeled with the use of a so called loss process {L(t), t ≥ 0} which starts from zero, increases and takes values in the interval [0, 1]. The bond is worthless if the loss process exceeds its credit rating. Thus the payoff profile of the (T, x) – bond is of the form

1{LT ≤ x}

The price P(t, T, x) of the (T, x) – bond is a stochastic process defined by

P(t, T, x) = 1{LT ≤ x}e−∫tT f(t, u, x)du, t ∈ [0, T] —– (1)

where f (·, ·, x) stands for an x-forward rate. The value x = 1 corresponds to the risk-free bond and f(t, T, 1) determines the short rate process via f(t, t, 1), t ≥ 0.

The (T, x) – bond market is thus fully determined by the family of x-forward rates and the loss process L. This is an extension of the classical non-defaultable bond market which can be identified with the case when I is a singleton, that is, when I = {1}.

The model of (T, x) – bonds does not correspond to defaultable bonds which are directly traded on a real market. For instance, in this setting the bankruptcy of the (T, x2) – bond automatically implies the bankruptcy of the (T, x1) – bond if x1 < x2. In reality, a bond with a higher credit rating may, however, default earlier than that with a lower one. The (T, x) – bonds are basic instruments related to the pool of defaultable assets called Collateralized Debt Obligations (CDOs), which are actually widely traded on the market. In the CDO market model, the loss process L(t) describes the part of the pool which has defaulted up to time t > 0 and F(LT), where F as some function, specifies the CDO payoff at time T > 0. In particular, (T, x) – bonds may be identified with the digital-type CDO payoffs with the function F of the form

F(z) = Fx(z) := 1[0,x](z), x ∈ I, z ∈ [0,1]

Then the price of that payoff pt(Fx(LT)) at time t ≤ T equals P(t, T, x). Moreover, each regular CDO claim can be replicated, and thus also priced, with a portfolio consisting of a certain combination of (T, x) – bonds. Thus it follows that the model of (T, x) – bonds determines the structure of the CDO payoffs. The induced family of prices

P(t, T, x), T ≥ 0, x ∈ I

will be a CDO term structure. On real markets the price of a claim which pays more is always higher. This implies

P(t, T, x1) = pt(Fx1(LT)) ≤ pt(Fx2(LT)) = P(t, T, x2), t ∈ [0, T], x1 < x2, x1, x2 ∈ I —– (2)

which means that the prices of (T, x) – bonds are increasing in x. Similarly, if the claim is paid earlier, then it has a higher value and hence

P(t, T1, x) = pt(Fx(LT1)) ≥ pt(Fx(LT2)) = P(t, T2, x), t ∈ [0, T1], T1 < T2, x ∈ I —– (3)

which means that the (T, x) – bond prices are decreasing in T. The CDO term structure is monotone if both (2) and (3) are satisfied. Surprisingly, monotonicity of the (T, x) – bond prices is not always preserved in mathematical models even if they satisfy severe no-arbitrage conditions.

Tranche Declension.

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With the CDO (collateralized debt obligation) market picking up, it is important to build a stronger understanding of pricing and risk management models. The role of the Gaussian copula model, has well-known deficiencies and has been criticized, but it continues to be fundamental as a starter. Here, we draw attention to the applicability of Gaussian inequalities in analyzing tranche loss sensitivity to correlation parameters for the Gaussian copula model.

We work with an RN-valued Gaussian random variable X = (X1, … , XN), where each Xj is normalized to mean 0 and variance 1, and study the equity tranche loss

L[0,a] = ∑m=1Nlm1[xm≤cm] – {∑m=1Nlm1[xm≤cm] – a}

where l1 ,…, lN > 0, a > 0, and c1,…, cN ∈ R are parameters. We thus establish an identity between the sensitivity of E[L[0,a]] to the correlation rjk = E[XjXk] and the parameters cj and ck, from where subsequently we come to the inequality

∂E[L[0,a]]/∂rjk ≤ 0

Applying this inequality to a CDO containing N names whose default behavior is governed by the Gaussian variables Xj shows that an increase in name-to-name correlation decreases expected loss in an equity tranche. This is a generalization of the well-known result for Gaussian copulas with uniform correlation.

Consider a CDO consisting of N names, with τj denoting the (random) default time of the jth name. Let

Xj = φj-1(Fjj))

where Fj is the distribution function of τj (relative to the market pricing measure), assumed to be continuous and strictly increasing, and φj is the standard Gaussian distribution function. Then for any x ∈ R we have

P[Xj ≤ x] = P[τj ≤ Fj-1j(x))] = Fj(Fj-1j(x))) = φj(x)

which means that Xj has standard Gaussian distribution. The Gaussian copula model posits that the joint distribution of the Xj is Gaussian; thus,

X = (X1, …., Xn)

is an RN-valued Gaussian variable whose marginals are all standard Gaussian. The correlation

τj = E[XjXk]

reflects the default correlation between the names j and k. Now let

pj = E[τj ≤ T] = P[Xj ≤ cj]

be the probability that the jth name defaults within a time horizon T, which is held constant, and

cj = φj−1(Fj(T))

is the default threshold of the jth name.

In schematics, when we explore the essential phenomenon, the default of name j, which happens if the default time τis within the time horizon T, results in a loss of amount lj > 0 in the CDO portfolio. Thus, the total loss during the time period [0, T] is

L = ∑m=1Nlm1[xm≤cm]

This is where we are essentially working with a one-period CDO, and ignoring discounting from the random time of actual default. A tranche is simply a range of loss for the portfolio; it is specified by a closed interval [a, b] with 0 ≤ a ≤ b. If the loss x is less than a, then this tranche is unaffected, whereas if x ≥ b then the entire tranche value b − a is eaten up by loss; in between, if a ≤ x ≤ b, the loss to the tranche is x − a. Thus, the tranche loss function t[a, b] is given by

t[a, b](x) = 0 if x < a; = x – a, if x ∈ [a, b]; = b – a; if x > b

or compactly,

t[a, b](x) = (x – a)+ – (x – b)+

From this, it is clear that t[a, b](x) is continuous in (a, b, x), and we see that it is a non-decreasing function of x. Thus, the loss in an equity tranche [0, a] is given by

t[0,a](L) = L − (L − a)+

with a > 0.

Indecent Bazaars. Thought of the Day 113.0

centerperiphery

Peripheral markets may be defined as markets which generate only a small proportion of their financial inflows from local business and investors, but which attract the interest of ‘global’ investors. Emerging markets and markets for financial exotica such as financial derivatives are examples of such peripheral markets. Because emerging markets are largely dependent upon attracting international funds in order to generate increases in securities prices and capital gains which will attract further funds, they are particularly good examples of the principles of Ponzi finance at work in securities markets.

A common characteristic feature of peripheral markets is that they have no history of returns to financial investment on the scale on which finance is drawn to those markets in a time of capital market inflation. Such returns in the future have to be inferred on the basis of conjecture and fragmentary information. Investment decisions are therefore more dependent on sentiment, rather than reason. Any optimism is quickly justified by the rapid increase in asset prices in response to even a modest excess net inflow of money into such a market.

Emerging markets illustrate this very clearly. Such markets exist in developing and semi-industrialized countries with relatively undeveloped pensions and insurance institutions, principally because only a small proportion of households earn enough to be able to put aside long-term savings. The first fund manager comes upon such a market in the conviction that a change of government or government policy, or some temporary change in commodity prices, has opened a cornucopia of profitable opportunities and therefore warrants the dismissal of a history of economic, financial and political instability. If he or she is able with buying and enthusiasm to attract other speculators and fund managers to enter the market, they may drive up asset prices and make the largest capital gains. The second and third fund managers to buy into that market also make capital gains. The emulatory competition of trading on reputation while competing for returns makes international investment managers especially prone to this kind of ‘herd’ investment.

For a while such capital inflows into the market make everyone happy: international fund managers are able to show good returns from the funds in their care; finance theorists can reassure themselves that greater financial risks are compensated by higher returns; the government of the country in which the emerging market is located can sell its bonds and public sector enterprises to willing foreign investors and use the proceeds to balance its budget and repay its debts; the watchdogs of financial prudence in the International Monetary Fund can hail the revival of finance, the government’s commitment to private enterprise and apparent fiscal responsibility; state enterprises, hitherto stagnating because of under-investment by over-indebted governments, suddenly find themselves in the private sector commanding seemingly limitless opportunities for raising finance; the country’s currency after years of depreciation acquires a gilt-edged stability as dollars (the principal currency of international investment) flow in to be exchanged for local currency with which to buy local securities; the central bank accumulates dollars in exchange for the local currency that it issues to enable foreign investors to invest in the local markets and, with larger reserves, secures a new ease in managing its foreign liabilities; the indigenous middle and professional classes who buy financial and property (real estate) assets in time for the boom are enriched and for once cease their perennial grumbling at the sordid reality of life in a poor country. In this conjuncture the most banal shibboleths of enterprise and economic progress under capitalism appear like the very essence of worldly wisdom.

Only in such a situation of capital market inflation are the supposed benefits of foreign direct investment realized. Such investment by multinational companies is widely held to improve the ‘quality’ or productivity of local labour, management and technical know-how in less developed countries, whose technology and organization of labour lags behind that of the more industrialized countries. But only the most doltish and ignorant peasant would not have his or her productivity increased by being set to work with a machine of relatively recent vintage under the guidance of a manager familiar with that machine and the kind of work organization that it requires. It is more doubtful whether the initial increase in productivity can be realized without a corresponding increase in the export market (developing countries have relatively small home markets). It is even more doubtful if the productivity increase can be repeated without the replacement of the machinery by even newer machinery.

The favourable conjuncture in the capital markets of developing countries can be even more temporary. There are limits on the extent to which even private sector companies may take on financial liabilities and privatization is merely a system for transferring such liabilities from the government to the private sector without increasing the financial resources of the companies privatized. But to sustain capital gains in the emerging stock market, additional funds have to continue to flow in buying new liabilities of the government or the private sector, or buying out local investors. When new securities cease to attract international fund managers, the inflow stops. Sometimes this happens when the government privatization drive pauses, because the government runs out of attractive state enterprises or there are political and procedural difficulties in selling them. A fall in the proceeds from privatization may reveal the government’s underlying fiscal deficit, causing the pundits of international finance to sense the odour of financial unsoundness. More commonly rising imports and general price inflation, due to the economic boom set off by the inflow of foreign funds, arouse just such an odour in the noses of those pundits. Such financial soundness is a subjective view. Even if nothing is wrong in the country concerned, the prospective capital gain and yield in some other market need only rise above the expected inflation and yield of the country, to cause a capital outflow which will usually be justified in retrospect by an appeal to perceived, if not actual, financial disequilibrium.

Ponzi financial structures are characterized by ephemeral liquidity. At the time when money is coming into the markets they appear to be just the neo-classical ideal of market perfection, with lots of buyers and sellers scrambling for bargains and arbitrage profits. At the moment when disinvestment takes hold the true nature of peripheral markets and their ephemeral liquidity is revealed as trades which previously sped through in the frantic paper chase for profits are now frustrated. This too is particularly apparent in emerging markets. In order to sell, a buyer is necessary. If the majority of investors in a market also wish to sell, then sales cannot be executed for want of a buyer and the apparently perfect market liquidity dries up. The crash of the emerging stock market is followed by the fall in the exchange value of the local currency. Those international investors that succeeded in selling now have local currency which has to be converted into dollars if the proceeds of the sale are to be repatriated, or invested elsewhere. Exchange through the local banking system may now be frustrated if it has inadequate dollar reserves: a strong possibility if the central bank has been using dollars to service foreign debts. In spite of all the reassurance that this time it will be different because capital inflows are secured on financial instruments issued by the private sector, international investors are at this point as much at the mercy of the central bank and the government of an emerging market as international banks were at the height of the sovereign debt crisis. Moreover, the greater the success of the peripheral market in attracting funds, and hence the greater the boom in prices in that market, the greater is the desired outflow when it comes. With the fall in liquidity of financial markets in developing countries comes a fall in the liquidity of foreign direct investment, making it difficult to secure appropriate local financial support or repatriate profits.

Another factor which contributes to the fragility of peripheral markets is the opaqueness of financial accounting in them, in the sense that however precise and discriminating may be the financial accounting conventions, rules and reporting, they do not provide accurate indicators of the financial prospects of particular investments. In emerging markets this is commonly supposed to be because they lack the accounting regulations and expertise which supports the sophisticated integrated financial markets of the industrialized countries. In those industrialized countries, where accounting procedures are supposed to be much more transparent, peripheral markets such as venture capital and financial futures still suffer from accounting inadequacies because financial innovation introduces liabilities that have no history and which are not included in conventional accounts (notably the so-called ‘off-balance sheet’ liabilities). More important than these gaps in financial reporting is the volatility of profits from financial investment in such peripheral markets, and the absence of any stable relationship between profits from trading in their instruments and the previous history of those instruments or the financial performance of the company issuing them. Thus, even where financial records are comprehensive, accurate and revealed, they are a poor indicator of prospective returns from investments in the securities of peripheral markets.

With more than usually unreliable financial data, trading in those markets is much more based on reputation than on any systematic financial analysis: the second and third investor in such a market is attracted by the reputation of the first and subsequently the second investor. Because of the direct connection between financial inflows and values in securities markets, the more trading takes place on the basis of reputation the less of a guide to prospective returns is afforded by financial analysis. Peripheral markets are therefore much more prone to ‘ramping’ than other markets.

Why would such a crisis of withdrawal not occur, at least not on such a scale, in the more locally integrated capital markets of the advanced industrialised countries? First of all, integrated capital markets such as those of the UK, and the US are the domestic base for international investors. In periods of financial turbulence, they are more likely to have funds repatriated to them than to have funds taken out of them. Second, institutional investors tend to be more responsive to pressure to be ‘responsible investors’ in their home countries. In large measure this is because home securities make up the vast majority of investment fund portfolios. Ultimately, investment institutions will use their liquidity to protect the markets in which most of their portfolio is based. Finally, the locally integrated markets of the advanced industrialized countries have investing institutions with far greater wealth than the developing or semi-industrialized countries. Those markets are home for the pension funds which dominate the world markets. Among their wealth are deposits and other liquid assets which may be easily converted to support a stock market by buying securities. The poorer countries of the world have even poorer pension funds, which could not support their markets against an outflow due to portfolio switches by international investors.

Thus integrated markets are more ‘secure’ in that they are less prone to collapse than emerging or, more generally, peripheral markets. But precisely because of the large amount of trade already concentrated in the integrated markets, prices in them are much less likely to respond to investment fund inflows from abroad. Pension and insurance fund practice is to extrapolate those capital gains into the future for the purposes of determining the solvency of those funds. However, those gains were obtained because of a combination of inflation, the increased scope of funded pensions and the flight of funds from peripheral markets.

Haircuts and Collaterals.

In+addition,+new+collateral+requirements+are+approaching…

In a repo-style securities financing transaction, the repo buyer or lender is exposed to the borrower’s default risk for the whole duration with a market contingent exposure, framed on a short window for default settlement. A margin period of risk (MPR) is a time period starting from the last date when margin is met to the date when the defaulting counterparty is closed out with completion of collateral asset disposal. MPR could cover a number of events or processes, including collateral valuation, margin calculation, margin call, valuation dispute and resolution, default notification and default grace period, and finally time to sell collateral to recover the lent principal and accrued interest. If the sales proceeds are not sufficient, the deficiency could be made a claim to the borrower’s estate, unless the repo is non-recourse. The lender’s exposure in a repo during the MPR is simply principal plus accrued and unpaid interest. Since the accrued and unpaid interest is usually margined at cash, repo exposure in the MPR is flat.

A flat exposure could apply to OTC derivatives as well. For an OTC netting, the mark-to-market of the derivatives could fluctuate as its underlying prices move. The derivatives exposure is formally set on the early termination date which could be days behind the point of default. The surviving counterparty, however, could have delta hedged against market factors following the default so that the derivative exposure remains a more manageable gamma exposure. For developing a collateral haircut model, what is generally assumed is a constant exposure during the MPR.

The primary driver of haircuts is asset volatility. Market liquidity risk is another significant one, as liquidation of the collateral assets might negatively impact the market, if the collateral portfolio is illiquid, large, or concentrated in certain asset sectors or classes. Market prices could be depressed, bid/ask spreads could widen, and some assets might have to be sold at a steep discount. This is particularly pronounced with private securitization and lower grade corporates, which trade infrequently and often rely on valuation services rather than actual market quotations. A haircut model therefore needs to capture liquidity risk, in addition to asset volatility.

In an idealized setting, we therefore consider a counterparty (or borrower) C’s default time at t, when the margin is last met, an MPR of u during which there is no margin posting, and the collateral assets are sold at time t+u instantaneously on the market, with a possible liquidation discount g.

Let us denote the collateral market value as B(t), exposure to the defaulting counterparty C as E(t). At time t, one share of the asset is margined properly, i.e., E(t) = (1-h)B(t), where h is a constant haircut, 1 >h ≥0. The margin agreement is assumed to have a zero minimum transfer amount. The lender would have a residual exposure (E(t) – B(t+u)(1-g))+, where g is a constant, 1 > g ≥ 0. Exposure to C is assumed flat after t. We can write the loss function from holding the collateral as follows,

L(t + u) = Et(1 – Bt+u/Bt (1 – g)/(1 – h))+ = (1 – g)Bt(1 – Bt+u/Bt (h – g)/(1 – g))+ —– (1)

Conditional on default happening at time t, the above determines a one-period loss distribution driven by asset price return B(t+u)/B(t). For repos, this loss function is slightly different from the lender’s ultimate loss which would be lessened due to a claim and recovery process. In the regulatory context, haircut is viewed as a mitigation to counterparty exposure and made independent of counterparty, so recovery from the defaulting party is not considered.

Let y = (1 – Bt+u/Bt) be the price decline. If g=0, Pr(y>h) equals to Pr(L(u)>0). There is no loss, if the price decline is less or equal to h. A first rupee loss will occur only if y > h. h thus provides a cushion before a loss is incurred. Given a target rating class’s default probability p, the first loss haircut can be written as

hp = inf{h > 0:Pr(L(u) > 0) ≤ p} —– (2)

Let VaRq denote the VaR of holding the asset, an amount which the price decline won’t exceed, given a confidence interval of q, say 99%. In light of the adoption of the expected shortfall (ES) in BASEL IV’s new market risk capital standard, we get a chance to define haircut as ES under the q-quantile,

hES = ESq = E[y|y > VaRq]

VaRq = inf{y0 > 0 : Pr(y > y0) ≤ 1 − q} —– (3)

Without the liquidity discount, hp is the same as VaRq. If haircuts are set to VaRq or hES, the market risk capital for holding the asset for the given MPR, defined as a multiple of VaR or ES, is zero. This implies that we can define a haircut to meet a minimum economic capital (EC) requirement C0,

hEC = inf{h ∈ R+: EC[L|h] ≤ C0} —– (4)

where EC is measured either as VaR or ES subtracted by expected loss (EL). For rating criteria employing EL based target per rating class, we could introduce one more definition of haircuts based on EL target L0,

hEL = inf{h ∈ R+: E[L|h] ≤ L0} —– (5)

The expected loss target L0 can be set based on EL criteria of certain designated high credit rating, whether bank internal or external. With an external rating such as Moody’s, for example, a firm can set the haircut to a level such that the expected (cumulative) loss satisfies the expected loss tolerance L0 of some predetermined Moody’s rating target, e.g., ‘Aaa’ or ‘Aa1’. In (4) and (5), loss L’s holding period does not have to be an MPR. In fact, these two definitions apply to the general trading book credit risk capital approach where the standard horizon is one year with a 99.9% confidence interval for default risk.

Different from VaRq, definitions hp, hEL, and hEC are based on a loss distribution solely generated by collateral market risk exposure. As such, we no longer apply the usual wholesale credit risk terminology of probability of default (PD) and loss given default (LGD) to determine EL as product of PD and LGD. Here EL is directly computed from a loss distribution originated from market risk and the haircut intends to be wholesale counterparty independent. For real repo transactions where repo haircuts are known to be counterparty dependent, these definitions remain fit, when the loss distribution incorporates the counterparty credit quality.