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

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 Q_S(t) and Q_V(t) denote the number of units of asset and option in the portfolio, respectively, and M_S(t) and M_V(t) denote the currency value of Q_S(t) units of asset and Q_V(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 The self-financing portfolio is set up with zero initial net investment cost and no additional funds are added or withdrawn afterwards.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 S_t, the option value process V_t and the standard Brownian process Z_t. The portfolio value at time t can be expressed as

Π(t) = M_S(t) + M_V(t) + M(t) = Q_S(t)S + Q_V(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 S² ∂²V/∂S² dt = (∂V/∂t + μS ∂V/∂S σ²/2 S² ∂²V/∂S²)dt + σS ∂V/∂S dZ —– (2)

If we formally write the stochastic dynamics of V as

dV/V = μ_V dt + σ_V dZ —– (3)

then μ_V and σ_V are given by

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

and

σ_V = (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) = [Q_S(t) dS + Q_V(t) dV + rM(t) dt] + [S dQ_S(t) + V dQ_V(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) = Q_S(t) dS + Q_V(t) dV + rM(t) dt = M_S(t) dS/S + M_V(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)M_S(t) + (μ_V − r)M_V(t)]dt + [σM_S(t) + σ_V M_V(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

σM_S(t) + σ_V M_V(t) = σS Q_S(t) + σS ∂V/∂S Q_V(t) = 0

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

Q_S(t)/Q_V(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)M_S(t) + (μ_V − r)M_V(t)]dt

substituting the ratio factor in the above equation, we get

(μ − r)S ∂V/∂S = (μ_V − r)V —– (10)

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

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

Suppose we take Q_V(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.

Advertisement

Bullish or Bearish. Note Quote.

The term spread refers to the difference in premiums between the purchase and sale of options. An option spread is the simultaneous purchase of one or more options contracts and sale of the equivalent number of options contracts, in a different series of the class of options. A spread could involve the same underlying: 

•  Buying and selling calls, or 
•  Buying and selling puts.

Combining puts and calls into groups of two or more makes it feasible to design derivatives with interesting payoff profiles. The profit and loss outcomes depend on the options used (puts or calls); positions taken (long or short); whether their strike prices are identical or different; and the similarity or difference of their exercise dates. Among directional positions are bullish vertical call spreads, bullish vertical put spreads, bearish vertical spreads, and bearish vertical put spreads. 

If the long position has a higher premium than the short position, this is known as a debit spread, and the investor will be required to deposit the difference in premiums. If the long position has a lower premium than the short position, this is a credit spread, and the investor will be allowed to withdraw the difference in premiums. The spread will be even if the premiums on each side results are the same. 

A potential loss in an option spread is determined by two factors: 

• Strike price 
• Expiration date 

If the strike price of the long call is greater than the strike price of the short call, or if the strike price of the long put is less than the strike price of the short put, a margin is required because adverse market moves can cause the short option to suffer a loss before the long option can show a profit.

A margin is also required if the long option expires before the short option. The reason is that once the long option expires, the trader holds an unhedged short position. A good way of looking at margin requirements is that they foretell potential loss. Here are, in a nutshell, the main option spreadings.

A calendar, horizontal, or time spread is the simultaneous purchase and sale of options of the same class with the same exercise prices but with different expiration dates. A vertical, or price or money, spread is the simultaneous purchase and sale of options of the same class with the same expiration date but with different exercise prices. A bull, or call, spread is a type of vertical spread that involves the purchase of the call option with the lower exercise price while selling the call option with the higher exercise price. The result is a debit transaction because the lower exercise price will have the higher premium.

• The maximum risk is the net debit: the long option premium minus the short option premium. 
• The maximum profit potential is the difference in the strike prices minus the net debit. 
• The breakeven is equal to the lower strike price plus the net debit. 

A trader will typically buy a vertical bull call spread when he is mildly bullish. Essentially, he gives up unlimited profit potential in return for reducing his risk. In a vertical bull call spread, the trader is expecting the spread premium to widen because the lower strike price call comes into the money first. 

Vertical spreads are the more common of the direction strategies, and they may be bullish or bearish to reflect the holder’s view of market’s anticipated direction. Bullish vertical put spreads are a combination of a long put with a low strike, and a short put with a higher strike. Because the short position is struck closer to-the-money, this generates a premium credit. 

Bearish vertical call spreads are the inverse of bullish vertical call spreads. They are created by combining a short call with a low strike and a long call with a higher strike. Bearish vertical put spreads are the inverse of bullish vertical put spreads, generated by combining a short put with a low strike and a long put with a higher strike. This is a bearish position taken when a trader or investor expects the market to fall. 

The bull or sell put spread is a type of vertical spread involving the purchase of a put option with the lower exercise price and sale of a put option with the higher exercise price. Theoretically, this is the same action that a bull call spreader would take. The difference between a call spread and a put spread is that the net result will be a credit transaction because the higher exercise price will have the higher premium. 

• The maximum risk is the difference in the strike prices minus the net credit. 
• The maximum profit potential equals the net credit. 
• The breakeven equals the higher strike price minus the net credit. 

The bear or sell call spread involves selling the call option with the lower exercise price and buying the call option with the higher exercise price. The net result is a credit transaction because the lower exercise price will have the higher premium.

A bear put spread (or buy spread) involves selling some of the put option with the lower exercise price and buying the put option with the higher exercise price. This is the same action that a bear call spreader would take. The difference between a call spread and a put spread, however, is that the net result will be a debit transaction because the higher exercise price will have the higher premium. 

• The maximum risk is equal to the net debit. 
• The maximum profit potential is the difference in the strike
  prices minus the net debit. 
• The breakeven equals the higher strike price minus the net debit.

An investor or trader would buy a vertical bear put spread because he or she is mildly bearish, giving up an unlimited profit potential in return for a reduction in risk. In a vertical bear put spread, the trader is expecting the spread premium to widen because the higher strike price put comes into the money first. 

In conclusion, investors and traders who are bullish on the market will either buy a bull call spread or sell a bull put spread. But those who are bearish on the market will either buy a bear put spread or sell a bear call spread. When the investor pays more for the long option than she receives in premium for the short option, then the spread is a debit transaction. In contrast, when she receives more than she pays, the spread is a credit transaction. Credit spreads typically require a margin deposit. 

Bank Recapitalization. Some Scattered Thoughts on Efficacies.

If we are still thinking of Demonetization and GST as speed breakers to economy, which entirely isn’t false, the what could one say of Bank Recapitalization? Is this a master stroke of sorts to salvaging sensibility before the present ruling dispensation of BJP is red-faced before 2019 GE? Or, is Bank Recapitalization is all about safeguarding the dismal dip in the growth and especially so when the world economy is on an ascent, despite warnings of a Minsky Moment? What are the challenges to Bank Recapitalization and how would these face up to the challenges of the NPAs and PSB consolidation? These are pressing questions that simply cannot be answered by a political will getting catalyzed, but requires a deeper economic drift and traction.

So, if Bank Recapitalization to the tune of Rs. 2.1 lakh crore infusion into the public sector banks were to come through, and which it would, the budgetary allocations are a mere chunk, while raising money from the market too isn’t that major a factor. The roost is to be ruled by recapitalization bonds, or recap bonds, in short. What then are the challenges of this methodology?

Technically, in the current context, there is really not much of a risk in issuing recapitalization bonds. The outside risk of recapitalization bonds is that this move may tighten liquidity in the system if all the surplus liquidity in the banking system goes into its capital. However, since recapitalization bonds are callable in nature, this risk should not be too great. Also, the debt markets are now sufficiently deep and broad and can support the funding needs of the India corporates and hence that is unlikely to be a major issue. The only concern is that rating agencies globally will look at recapitalization as a form of off-balance sheet financing, which does not give them too much comfort. Many rating agencies look at such bonds as a means of raising debt that is not visible in the fiscal deficit. This lack of visibility is what might be the hurdles race for the government. But, then is there a way out?

Alternatively, what if the government were not to recapitalize? Then, it can look to postponing its adherence to Basel III from 2019. But that will be seen by global markets as an admission by the Government of India that it does not have the liquidity to capitalize its banks. That may not go down well with foreign investors. Under these circumstances, infusing capital into the banks through the issue of recapitalization bonds may be the best option available!

What are the main economic ramifications as a result of these? The government’s plan at recapitalization would have little impact on its target to shrink the shortfall to 3.2 percent of the GDP because the IMF rules classify such debt as “below the line” financing. Only interest expenses would be added to the fiscal deficit, and this is estimated at about Rs. 90 billion or 0.4 percent of the total budgeted spending. Technically, however, India’s accounting rules require the bonds to be included in the budget deficit, so the government would reclassify them later as off-balance sheet items. The government is yet to disclose the details on the structure and pricing of the bonds, as well as how it would raise the rest of the cash. These will determine if there is a liquidity squeeze. If the measures do revive credit growth, inflation may accelerate as well, limiting scope to lower the policy rate. When it comes to the question of who would buy these bonds, the answer is probably banks themselves, who are flush with deposits following the note ban. Banks can then cleverly invest these funds in the recap bonds which will then be ultimately routed back as equity in the system. This would ensure that the bond market would not be impacted by such a large issuance for the private sector issuers.

Now, these are serious questions questioning some of the advocacy groups have to come to terms with. For one thing, in my opinion, mergers and acquisitions to consolidate PSBs are to be put back on the back foot, for recapitalization has at least punctuated to for the time being. Second is credit growth, or more precisely credit demand, which would be induced with an energy following this exercise. Third, and most importantly, the lending might gain velocity, but only after April 2018, since banks would require a correctional facility on their balance sheets. This lending would somehow be channeled towards infrastructure giants like Sagarmala and Bharatmala with a key difference being that the Government might prioritize Engineering, Procurement and Construction (EPC) over Hybrid Annuity Model like the PPP for the obvious risks associated with the latter subsequently feeding into the NPAs and/or stressed assets. 

Option Spread. Drunken Risibility.

The term spread refers to the difference in premiums between the purchase and sale of options. An option spread is the simultaneous purchase of one or more options contracts and sale of the equivalent number of options contracts, in a different series of the class of options. A spread could involve the same underlying:

•  Buying and selling calls, or
•  Buying and selling puts.

Combining puts and calls into groups of two or more makes it feasible to design derivatives with interesting payoff profiles. The profit and loss outcomes depend on the options used (puts or calls); positions taken (long or short); whether their strike prices are identical or different; and the similarity or difference of their exercise dates. Among directional positions are bullish vertical call spreads, bullish vertical put spreads, bearish vertical spreads, and bearish vertical put spreads.

If the long position has a higher premium than the short position, this is known as a debit spread, and the investor will be required to deposit the difference in premiums. If the long position has a lower premium than the short position, this is a credit spread, and the investor will be allowed to withdraw the difference in premiums. The spread will be even if the premiums on each side results are the same.

A potential loss in an option spread is determined by two factors:

• Strike price
• Expiration date

If the strike price of the long call is greater than the strike price of the short call, or if the strike price of the long put is less than the strike price of the short put, a margin is required because adverse market moves can cause the short option to suffer a loss before the long option can show a profit.

A margin is also required if the long option expires before the short option. The reason is that once the long option expires, the trader holds an unhedged short position. A good way of looking at margin requirements is that they foretell potential loss. Here are, in a nutshell, the main option spreadings.

A calendar, horizontal, or time spread is the simultaneous purchase and sale of options of the same class with the same exercise prices but with different expiration dates. A vertical, or price or money, spread is the simultaneous purchase and sale of options of the same class with the same expiration date but with different exercise prices. A bull, or call, spread is a type of vertical spread that involves the purchase of the call option with the lower exercise price while selling the call option with the higher exercise price. The result is a debit transaction because the lower exercise price will have the higher premium.

• The maximum risk is the net debit: the long option premium minus the short option premium.
• The maximum profit potential is the difference in the strike prices minus the net debit.
• The breakeven is equal to the lower strike price plus the net debit.

A trader will typically buy a vertical bull call spread when he is mildly bullish. Essentially, he gives up unlimited profit potential in return for reducing his risk. In a vertical bull call spread, the trader is expecting the spread premium to widen because the lower strike price call comes into the money first.

Vertical spreads are the more common of the direction strategies, and they may be bullish or bearish to reflect the holder’s view of market’s anticipated direction. Bullish vertical put spreads are a combination of a long put with a low strike, and a short put with a higher strike. Because the short position is struck closer to-the-money, this generates a premium credit.

Bearish vertical call spreads are the inverse of bullish vertical call spreads. They are created by combining a short call with a low strike and a long call with a higher strike. Bearish vertical put spreads are the inverse of bullish vertical put spreads, generated by combining a short put with a low strike and a long put with a higher strike. This is a bearish position taken when a trader or investor expects the market to fall.

The bull or sell put spread is a type of vertical spread involving the purchase of a put option with the lower exercise price and sale of a put option with the higher exercise price. Theoretically, this is the same action that a bull call spreader would take. The difference between a call spread and a put spread is that the net result will be a credit transaction because the higher exercise price will have the higher premium.

• The maximum risk is the difference in the strike prices minus the net credit.
• The maximum profit potential equals the net credit.
• The breakeven equals the higher strike price minus the net credit.

The bear or sell call spread involves selling the call option with the lower exercise price and buying the call option with the higher exercise price. The net result is a credit transaction because the lower exercise price will have the higher premium.

A bear put spread (or buy spread) involves selling some of the put option with the lower exercise price and buying the put option with the higher exercise price. This is the same action that a bear call spreader would take. The difference between a call spread and a put spread, however, is that the net result will be a debit transaction because the higher exercise price will have the higher premium.

• The maximum risk is equal to the net debit.
• The maximum profit potential is the difference in the strike
  prices minus the net debit.
• The breakeven equals the higher strike price minus the net debit.

An investor or trader would buy a vertical bear put spread because he or she is mildly bearish, giving up an unlimited profit potential in return for a reduction in risk. In a vertical bear put spread, the trader is expecting the spread premium to widen because the higher strike price put comes into the money first.

So, investors and traders who are bullish on the market will either buy a bull call spread or sell a bull put spread. But those who are bearish on the market will either buy a bear put spread or sell a bear call spread. When the investor pays more for the long option than she receives in premium for the short option, then the spread is a debit transaction. In contrast, when she receives more than she pays, the spread is a credit transaction. Credit spreads typically require a margin deposit.

Optimal Hedging…..

Risk management is important in the practices of financial institutions and other corporations. Derivatives are popular instruments to hedge exposures due to currency, interest rate and other market risks. An important step of risk management is to use these derivatives in an optimal way. The most popular derivatives are forwards, options and swaps. They are basic blocks for all sorts of other more complicated derivatives, and should be used prudently. Several parameters need to be determined in the processes of risk management, and it is necessary to investigate the influence of these parameters on the aims of the hedging policies and the possibility of achieving these goals.

The problem of determining the optimal strike price and optimal hedging ratio is considered, where a put option is used to hedge market risk under a constraint of budget. The chosen option is supposed to finish in-the-money at maturity in the, such that the predicted loss of the hedged portfolio is different from the realized loss. The aim of hedging is to minimize the potential loss of investment under a specified level of confidence. In other words, the optimal hedging strategy is to minimize the Value-at-Risk (VaR) under a specified level of risk.

A stock is supposed to be bought at time zero with price S₀, and to be sold at time T with uncertain price S_T. In order to hedge the market risk of the stock, the company decides to choose one of the available put options written on the same stock with maturity at time τ, where τ is prior and close to T, and the n available put options are specified by their strike prices K_i (i = 1, 2,··· , n). As the prices of different put options are also different, the company needs to determine an optimal hedge ratio h (0 ≤ h ≤ 1) with respect to the chosen strike price. The cost of hedging should be less than or equal to the predetermined hedging budget C. In other words, the company needs to determine the optimal strike price and hedging ratio under the constraint of hedging budget.

Suppose the market price of the stock is S₀ at time zero, the hedge ratio is h, the price of the put option is P₀, and the riskless interest rate is r. At time T, the time value of the hedging portfolio is

S₀e^rT + hP₀e^rT —– (1)

and the market price of the portfolio is

S_T + h(K − S_τ)⁺ e^r(T−τ) —– (2)

therefore the loss of the portfolio is

L = (S₀e^rT + hP₀e^rT) − (S_T +h(K−S_τ)⁺ e^r(T−τ)) —– (3)

where x⁺ = max(x, 0), which is the payoff function of put option at maturity.

For a given threshold v, the probability that the amount of loss exceeds v is denoted as

α = Prob{L ≥ v} —– (4)

in other words, v is the Value-at-Risk (VaR) at α percentage level. There are several alternative measures of risk, such as CVaR (Conditional Value-at-Risk), ESF (Expected Shortfall), CTE (Conditional Tail Expectation), and other coherent risk measures. The criterion of optimality is to minimize the VaR of the hedging strategy.

The mathematical model of stock price is chosen to be a geometric Brownian motion, i.e.

dS_t/S_t = μdt + σdB_t —– (5)

where S_t is the stock price at time t (0 < t ≤ T), μ and σ are the drift and the volatility of stock price, and B_t is a standard Brownian motion. The solution of the stochastic differential equation is

S_t = S₀ e^{σB_t + (μ−1/2σ2)t} —– (6)

where B₀ = 0, and S_t is lognormally distributed.

Proposition:

For a given threshold of loss v, the probability that the loss exceeds v is

Prob {L ≥ v} = E [I_{{X ≤ c1}} F_Y (g(X) − X)] + E [I_{{X ≥ c1}} F_Y (c₂ − X)] —– (7)

where E[X] is the expectation of random variable X. I_{{X < c}} is the index function of X such that I_{{X < c}} = 1 when {X < c} is true, otherwise I_{{X < c}} = 0. F_Y (y) is the cumulative distribution function of random variable Y , and

c₁ = 1/σ [ln(K/S₀) − (μ−1/2σ²)τ] ,

g(X) = 1/σ [(ln (S₀ + hP₀)e^rT − h (K − f(X)) e^r(T−τ) −v)/S₀ − (μ − 1/2σ²) T],

f(X) = S₀ e^σX + (μ−1/2σ²)τ,

c₂ = 1/σ [(ln (S₀ + hP₀) e^rT − v)/S₀ − (μ− 1/2σ²) T

X and Y are both normally distributed, where X ∼ N(0,√τ), Y ∼ N(0,√(T−τ).

For a specified hedging strategy, Q(v) = Prob {L ≥ v} is a decreasing function of v. The VaR under α level can be obtained from equation

Q(v) = α —– (8)

The expectations in Proposition can be calculated with Monte Carlo simulation methods, and the optimal hedging strategy which has the smallest VaR can be obtained from equation (8) by numerical searching methods….

Financial Forward Rate “Strings” (Didactic 1)

Imagine that Julie wants to invest $1 for two years. She can devise two possible strategies. The first one is to put the money in a one-year bond at an interest rate r₁. At the end of the year, she must take her money and find another one-year bond, with interest rate r_1/2 which is the interest rate in one year on a loan maturing in two years. The final payoff of this strategy is simply (1 + r₁)(1 + r_1/2). The problem is that Julie cannot know for sure what will be the one-period interest rate r_1/2 of next year. Thus, she can only estimate a return by guessing the expectation of r_1/2.

Instead of making two separate investments of one year each, Julie could invest her money today in a bond that pays off in two years with interest rate r₂. The final payoff is then (1 + r₂)². This second strategy is riskless as she knows for sure her return. Now, this strategy can be reinterpreted along the line of the first strategy as follows. It consists in investing for one year at the rate r₁ and for the second year at a forward rate f₂. The forward rate is like the r_1/2 rate, with the essential difference that it is guaranteed : by buying the two-year bond, Julie can “lock in” an interest rate f₂ for the second year.

This simple example illustrates that the set of all possible bonds traded on the market is equivalent to the so-called forward rate curve. The forward rate f(t,x) is thus the interest rate that can be contracted at time t for instantaneously riskless borrowing 1 or lending at time t + x. It is thus a function or curve of the time-to-maturity x², where x plays the role of a “length” variable, that deforms with time t. Its knowledge is completely equivalent to the set of bond prices P(t,x) at time t that expire at time t + x. The shape of the forward rate curve f(t,x) incessantly fluctuates as a function of time t. These fluctuations are due to a combination of factors, including future expectation of the short-term interest rates, liquidity preferences, market segmentation and trading. It is obvious that the forward rate f (t, x+δx) for δx small can not be very different from f (t,x). It is thus tempting to see f(t,x) as a “string” characterized by a kind of tension which prevents too large local deformations that would not be financially acceptable. This superficial analogy is in the follow up of the repetitious intersections between finance and physics, starting with Bachelier who solved the diffusion equation of Brownian motion as a model of stock market price fluctuations five years before Einstein, continuing with the discovery of the relevance of Lévy laws for cotton price fluctuations by Mandelbrot that can be compared with the present interest of such power laws for the description of physical and natural phenomena. The present investigation delves into how to formalize mathematically this analogy between the forward rate curve and a string. We formulate the term structure of interest rates as the solution of a stochastic partial differential equation (SPDE), following the physical analogy of a continuous curve (string) whose shape moves stochastically through time.

The equation of motion of macroscopic physical strings is derived from conservation laws. The fundamental equations of motion of microscopic strings formulated to describe the fundamental particles derive from global symmetry principles and dualities between long-range and short-range descriptions. Are there similar principles that can guide the determination of the equations of motion of the more down-to-earth financial forward rate “strings”?

Suppose that in the middle ages, before Copernicus and Galileo, the Earth really was stationary at the centre of the universe, and only began moving later on. Imagine that during the nineteenth century, when everyone believed classical physics to be true, that it really was true, and quantum phenomena were non-existent. These are not philosophical musings, but an attempt to portray how physics might look if it actually behaved like the financial markets. Indeed, the financial world is such that any insight is almost immediately used to trade for a profit. As the insight spreads among traders, the “universe” changes accordingly. As G. Soros has pointed out, market players are “actors observing their own deeds”. As E. Derman, head of quantitative strategies at Goldman Sachs, puts it, in physics you are playing against God, who does not change his mind very often. In finance, you are playing against Gods creatures, whose feelings are ephemeral, at best unstable, and the news on which they are based keep streaming in. Value clearly derives from human beings, while mass, charge and electromagnetism apparently do not. This has led to suggestions that a fruitful framework to study finance and economy is to use evolutionary models inspired from biology and genetics.

This does not however guide us much for the determination of “fundamental” equa- tions, if any. Here, we propose to use the condition of absence of arbitrage opportunity and show that this leads to strong constraints on the structure of the governing equations. The basic idea is that, if there are arbitrage opportunities (free lunches), they cannot live long or must be quite subtle, otherwise traders would act on them and arbitrage them away. The no-arbitrage condition is an idealization of a self-consistent dynamical state of the market resulting from the incessant actions of the traders (ar- bitragers). It is not the out-of-fashion equilibrium approximation sometimes described but rather embodies a very subtle cooperative organization of the market.

We consider this condition as the fundamental backbone for the theory. The idea to impose this requirement is not new and is in fact the prerequisite of most models developed in the academic finance community. Modigliani and Miller [here and here] have indeed emphasized the critical role played by arbitrage in determining the value of securities. It is sometimes suggested that transaction costs and other market imperfections make irrelevant the no-arbitrage condition. Let us address briefly this question.

Transaction costs in option replication and other hedging activities have been extensively investigated since they (or other market “imperfections”) clearly disturb the risk-neutral argument and set option theory back a few decades. Transaction costs induce, for obvious reasons, dynamic incompleteness, thus preventing valuation as we know it since Black and Scholes. However, the most efficient dynamic hedgers (market makers) incur essentially no transaction costs when owning options. These specialized market makers compete with each other to provide liquidity in option instruments, and maintain inventories in them. They rationally limit their dynamic replication to their residual exposure, not their global exposure. In addition, the fact that they do not hold options until maturity greatly reduces their costs of dynamic hedging. They have an incentive in the acceleration of financial intermediation. Furthermore, as options are rarely replicated until maturity, the expected transaction costs of the short options depend mostly on the dynamics of the order flow in the option markets – not on the direct costs of transacting. For the efficient operators (and those operators only), markets are more dynamically complete than anticipated. This is not true for a second category of traders, those who merely purchase or sell financial instruments that are subjected to dynamic hedging. They, accordingly, neither are equipped for dynamic hedging, nor have the need for it, thanks to the existence of specialized and more efficient market makers. The examination of their transaction costs in the event of their decision to dynamically replicate their options is of no true theoretical contribution. A second important point is that the existence of transaction costs should not be invoked as an excuse for disregarding the no-arbitrage condition, but, rather should be constructively invoked to study its impacts on the models…..

Forward, Futures Contracts and Options: Top Down or bottom Up Modeling?

The simulation of financial markets can be modeled, from a theoretical viewpoint, according to two separate approaches: a bottom up approach and (or) a top down approach. For instance, the modeling of financial markets starting from diffusion equations and adding a noise term to the evolution of a function of a stochastic variable is a top down approach. This type of description is, effectively, a statistical one.

A bottom up approach, instead, is the modeling of artificial markets using complex data structures (agent based simulations) using general updating rules to describe the collective state of the market. The number of procedures implemented in the simulations can be quite large, although the computational cost of the simulation becomes forbidding as the size of each agent increases. Readers familiar with Sugarscape Models and the computational strategies based on Growing of Artificial Societies have probably an idea of the enormous potentialities of the field. All Sugarscape models include the agents (inhabitants), the environment (a two-dimensional grid) and the rules governing the interaction of the agents with each other and the environment. The original model presented by J. Epstein & R. Axtell (considered as the first large scale agent model) is based on a 51 x 51 cell grid, where every cell can contain different amounts of sugar (or spice). In every step agents look around, find the closest cell filled with sugar, move and metabolize. They can leave pollution, die, reproduce, inherit sources, transfer information, trade or borrow sugar, generate immunity or transmit diseases – depending on the specific scenario and variables defined at the set-up of the model. Sugar in simulation could be seen as a metaphor for resources in an artificial world through which the examiner can study the effects of social dynamics such as evolution, marital status and inheritance on populations. Exact simulation of the original rules provided by J. Epstein & R. Axtell in their book can be problematic and it is not always possible to recreate the same results as those presented in Growing Artificial Societies. However, one would expect that the bottom up description should become comparable to the top down description for a very large number of simulated agents.

The bottom up approach should also provide a better description of extreme events, such as crashes, collectively conditioned behaviour and market incompleteness, this approach being of purely algorithmic nature. A top down approach is, therefore, a model of reduced complexity and follows a statistical description of the dynamics of complex systems.

Forward, Futures Contracts and Options: Let the price at time t of a security be S(t). A specific good can be traded at time t at the price S(t) between a buyer and a seller. The seller (short position) agrees to sell the goods to the buyer (long position) at some time T in the future at a price F(t,T) (the contract price). Notice that contract prices have a 2-time dependence (actual time t and maturity time T). Their difference τ = T − t is usually called time to maturity. Equivalently, the actual price of the contract is determined by the prevailing actual prices and interest rates and by the time to maturity. Entering into a forward contract requires no money, and the value of the contract for long position holders and strong position holders at maturity T will be

(−1)^p (S(T)−F(t,T)) (1)

where p = 0 for long positions and p = 1 for short positions. Futures Contracts are similar, except that after the contract is entered, any changes in the market value of the contract are settled by the parties. Hence, the cashflows occur all the way to expiry unlike in the case of the forward where only one cashflow occurs. They are also highly regulated and involve a third party (a clearing house). Forward, futures contracts and options go under the name of derivative products, since their contract price F(t, T) depend on the value of the underlying security S(T). Options are derivatives that can be written on any security and have a more complicated payoff function than the futures or forwards. For example, a call option gives the buyer (long position) the right (but not the obligation) to buy or sell the security at some predetermined strike-price at maturity. A payoff function is the precise form of the price. Path dependent options are derivative products whose value depends on the actual path followed by the underlying security up to maturity. In the case of path-dependent options, since the payoff may not be directly linked to an explicit right, they must be settled by cash. This is sometimes true for futures and plain options as well as this is more efficient.