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

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 S_t is assumed to follow the geometric Brownian motion

dS_t/S_t = μ dt + σ dZ_t —– (1)

where μ is the expected rate of return, σ is the volatility and Z_t 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 Δ_t units of the underlying asset. The portfolio value Π (S_t, t) at time t is given by

Π = −c + Δ_t S_t —– (2)

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

dc = ∂c/∂t dt + ∂c/∂S_t dS_t + σ²/2 S_t² ∂²c/∂S_t² dt

such that

-dc + Δ_t dS_t = (-∂c/∂t – σ²/2 S_t² ∂²c/∂S_t²)dt + (Δ_t – ∂c/∂S_t)dS_t

= [-∂c/∂t – σ²/2 S_t² ∂²c/∂S_t² + (Δ_t – ∂c/∂S_t)μS_t]dt + (Δ_t – ∂c/∂S_t)σS_t dZ_t

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

G(Π (S_t, t )) = ∫₀^t -dc + ∫₀^t Δ_u dS_u

= ∫₀^t [-∂c/∂u – σ²/2 S_u² ∂²c/∂S_u² + (Δ_u – ∂c/∂S_u)μS_u]du + ∫₀^t (Δ_u – ∂c/∂S_u)σS_u dZ_u —– (3)

The stochastic component of the portfolio gain stems from the last term, ∫₀^t (Δ_u – ∂c/∂S_u)σS_u dZ_u. Suppose we adopt the dynamic hedging strategy by choosing Δ_u = ∂c/∂S_u 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 + S_u∂c/∂S_u. The deterministic gain from this dynamic position of riskless asset is given by

M_t = ∫₀^tr(-c + S_u∂c/∂S_u)du —– (4)

By equating these two deterministic gains, G(Π (S_t, t)) and M_t, we have

-∂c/∂u – σ²/2 S_u² ∂²c/∂S_u² = r(-c + S_u∂c/∂S_u), 0 < u < t

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

∂c/∂t + σ²/2 S² ∂²c/∂S² + 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 S² ∂²V/∂S² + 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 + Δ_t dS_t,

which misses the effect arising from the differential change in Δ_t. 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.

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Collateral Debt Obligations. Thought of the Day 111.0

A CDO is a general term that describes securities backed by a pool of fixed-income assets. These assets can be bank loans (CLOs), bonds (CBOs), residential mortgages (residential- mortgage–backed securities, or RMBSs), and many others. A CDO is a subset of asset- backed securities (ABS), which is a general term for a security backed by assets such as mortgages, credit card receivables, auto loans, or other debt.

To create a CDO, a bank or other entity transfers the underlying assets (“the collateral”) to a special-purpose vehicle (SPV) that is a separate legal entity from the issuer. The SPV then issues securities backed with cash flows generated by assets in the collateral pool. This general process is called securitization. The securities are separated into tranches, which differ primarily in the priority of their rights to the cash flows coming from the asset pool. The senior tranche has first priority, the mezzanine second, and the equity third. Allocation of cash flows to specific securities is called a “waterfall”. A waterfall is specified in the CDO’s indenture and governs both principal and interest payments.

1: If coverage tests are not met, and to the extent not corrected with principal proceeds, the remaining interest proceeds will be used to redeem the most senior notes to bring the structure back into compliance with the coverage tests. Interest on the mezzanine securities may be deferred and compounded if cash flow is not available to pay current interest due.

One may observe that the creation of a CDO is a complex and costly process. Professionals such as bankers, lawyers, rating agencies, accountants, trustees, fund managers, and insurers all charge considerable fees to create and manage a CDO. In other words, the cash coming from the collateral is greater than the sum of the cash paid to all security holders. Professional fees to create and manage the CDO make up the difference.

CDOs are designed to offer asset exposure precisely tailored to the risk that investors desire, and they provide liquidity because they trade daily on the secondary market. This liquidity enables, for example, a finance minister from the Chinese government to gain exposure to the U.S. mortgage market and to buy or sell that exposure at will. However, because CDOs are more complex securities than corporate bonds, they are designed to pay slightly higher interest rates than correspondingly rated corporate bonds.

CDOs enable a bank that specializes in making loans to homeowners to make more loans than its capital would otherwise allow, because the bank can sell its loans to a third party. The bank can therefore originate more loans and take in more origination fees. As a result, consumers have more access to capital, banks can make more loans, and investors a world away can not only access the consumer loan market but also invest with precisely the level of risk they desire.

1: To the extent not paid by interest proceeds.

2: To the extent senior note coverage tests are met and to the extent not already paid by interest proceeds. If coverage tests are not met, the remaining principal proceeds will be used to redeem the most senior notes to bring the structure back into compliance with the coverage tests. Interest on the mezzanine securities may be deferred and compounded if cash flow is not available to pay current interest due.

The Structured Credit Handbook provides an explanation of investors’ nearly insatiable appetite for CDOs:

Demand for [fixed income] assets is heavily bifurcated, with the demand concentrated at the two ends of the safety spectrum . . . Prior to the securitization boom, the universe of fixed-income instruments issued tended to cluster around the BBB rating, offering neither complete safety nor sizzling returns. For example, the number of AA and AAA-rated companies is quite small, as is debt issuance of companies rated B or lower. Structured credit technology has evolved essentially in order to match investors’ demands with the available profile of fixed-income assets. By issuing CDOs from portfolios of bonds or loans rated A, BBB, or BB, financial intermediaries can create a larger pool of AAA-rated securities and a small unrated or low-rated bucket where almost all the risk is concentrated.

CDOs have been around for more than twenty years, but their popularity skyrocketed during the late 1990s. CDO issuance nearly doubled in 2005 and then again in 2006, when it topped $500 billion for the first time. “Structured finance” groups at large investment banks (the division responsible for issuing and managing CDOs) became one of the fastest-growing areas on Wall Street. These divisions, along with the investment banking trading desks that made markets in CDOs, contributed to highly successful results for the banking sector during the 2003–2007 boom. Many CDOs became quite liquid because of their size, investor breadth, and rating agency coverage.

Rating agencies helped bring liquidity to the CDO market. They analyzed each tranche of a CDO and assigned ratings accordingly. Equity tranches were often unrated. The rating agencies had limited manpower and needed to gauge the risk on literally thousands of new CDO securities. The agencies also specialized in using historical models to predict risk. Although CDOs had been around for a long time, they did not exist in a significant number until recently. Historical models therefore couldn’t possibly capture the full picture. Still, the underlying collateral could be assessed with a strong degree of confidence. After all, banks have been making home loans for hundreds of years. The rating agencies simply had to allocate risk to the appropriate tranche and understand how the loans in the collateral base were correlated with each other – an easy task in theory perhaps, but not in practice.

The most difficult part of valuing a CDO tranche is determining correlation. If loans are uncorrelated, defaults will occur evenly over time and asset diversification can solve most problems. With low correlation, an AAA-rated senior tranche should be safe and the interest rate attached to this tranche should be close to the rate for AAA-rated corporate bonds. High correlation, however, creates nondiversifiable risk, in which case the senior tranche has a reasonable likelihood of becoming impaired. Correlation does not affect the price of the CDO in total because the expected value of each individual loan remains the same. Correlation does, however, affect the relative price of each tranche: Any increase in the yield of a senior tranche (to compensate for additional correlation) will be offset by a decrease in the yield of the junior tranches.

Synthetic Structured Financial Instruments. Note Quote.

An option is common form of a derivative. It’s a contract, or a provision of a contract, that gives one party (the option holder) the right, but not the obligation to perform a specified transaction with another party (the option issuer or option writer) according to specified terms. Options can be embedded into many kinds of contracts. For example, a corporation might issue a bond with an option that will allow the company to buy the bonds back in ten years at a set price. Standalone options trade on exchanges or Over The Counter (OTC). They are linked to a variety of underlying assets. Most exchange-traded options have stocks as their underlying asset but OTC-traded options have a huge variety of underlying assets (bonds, currencies, commodities, swaps, or baskets of assets). There are two main types of options: calls and puts:

• Call options provide the holder the right (but not the obligation) to purchase an underlying asset at a specified price (the strike price), for a certain period of time. If the stock fails to meet the strike price before the expiration date, the option expires and becomes worthless. Investors buy calls when they think the share price of the underlying security will rise or sell a call if they think it will fall. Selling an option is also referred to as ”writing” an option.
• Put options give the holder the right to sell an underlying asset at a specified price (the strike price). The seller (or writer) of the put option is obligated to buy the stock at the strike price. Put options can be exercised at any time before the option expires. Investors buy puts if they think the share price of the underlying stock will fall, or sell one if they think it will rise. Put buyers – those who hold a “long” – put are either speculative buyers looking for leverage or “insurance” buyers who want to protect their long positions in a stock for the period of time covered by the option. Put sellers hold a “short” expecting the market to move upward (or at least stay stable) A worst-case scenario for a put seller is a downward market turn. The maximum profit is limited to the put premium received and is achieved when the price of the underlyer is at or above the option’s strike price at expiration. The maximum loss is unlimited for an uncovered put writer.

Coupon is the annual interest rate paid on a bond, expressed as percentage of the face value.

Coupon rate or nominal yield = annual payments ÷ face value of the bond

Current yield = annual payments ÷ market value of the bond

The reason for these terms to be briefed here through their definitions from investopedia lies in the fact that these happen to be pillars of synthetic financial instruments, to which we now take a detour.

According to the International Financial Reporting Standards (IFRS), a synthetic instrument is a financial product designed, acquired, and held to emulate the characteristics of another instrument. For example, such is the case of a floating-rate long-term debt combined with an interest rate swap. This involves

• Receiving floating payments
• Making fixed payments, thereby synthesizing a fixed-rate long-term debt

Another example of a synthetic is the output of an option strategy followed by dealers who are selling synthetic futures for a commodity that they hold by using a combination of put and call options. By simultaneously buying a put option in a given commodity, say, gold, and selling the corresponding call option, a trader can construct a position analogous to a short sale in the commodity’s futures market.

Because the synthetic short sale seeks to take advantage of price disparities between call and put options, it tends to be more profitable when call premiums are greater than comparable put premiums. For example, the holder of a synthetic short future will profit if gold prices decrease and incur losses if gold prices increase.

By analogy, a long position in a given commodity’s call option combined with a short sale of the same commodity’s futures creates price protection that is similar to that gained through purchasing put options. A synthetic put seeks to capitalize on disparities between call and put premiums.

Basically, synthetic products are covered options and certificates characterized by identical or similar profit and loss structures when compared with traditional financial instruments, such as equities or bonds. Basket certificates in equities are based on a specific number of selected stocks.

A covered option involves the purchase of an underlying asset, such as equity, bond, currency, or other commodity, and the writing of a call option on that same asset. The writer is paid a premium, which limits his or her loss in the event of a fall in the market value of the underlying. However, his or her potential return from any increase in the asset’s market value is conditioned by gains limited by the option’s strike price.

The concept underpinning synthetic covered options is that of duplicating traditional covered options, which can be achieved by both purchase of the underlying asset and writing of the call option. The purchase price of such a product is that of the underlying, less the premium received for the sale of the call option.

Moreover, synthetic covered options do not contain a hedge against losses in market value of the underlying. A hedge might be emulated by writing a call option or by calculating the return from the sale of a call option into the product price. The option premium, however, tends to limit possible losses in the market value of the underlying.

Alternatively, a synthetic financial instrument is done through a certificate that accords a right, based on either a number of underlyings or on having a value derived from several indicators. This presents a sense of diversification over a range of risk factors. The main types are

• Index certificates
• Region certificates
• Basket certificates

By being based on an official index, index certificates reflect a given market’s behavior. Region certificates are derived from a number of indexes or companies from a given region, usually involving developing countries. Basket certificates are derived from a selection of companies active in a certain industry sector.

An investment in index, region, or basket certificates fundamentally involves the same level of potential loss as a direct investment in the corresponding assets themselves. Their relative advantage is diversification within a given specified range; but risk is not eliminated. Moreover, certificates also carry credit risk associated to the issuer.

Also available in the market are compound financial instruments, a frequently encountered form being that of a debt product with an embedded conversion option. An example of a compound financial instrument is a bond that is convertible into ordinary shares of the issuer. As an accounting standard, the IFRS requires the issuer of such a financial instrument to present separately on the balance sheet the

• Equity component
• Liability component

On initial recognition, the fair value of the liability component is the present value of the contractually determined stream of future cash flows, discounted at the rate of interest applied at that time by the market to substantially similar cash flows. These should be characterized by practically the same terms, albeit without a conversion option. The fair value of the option comprises its

• Time value
• Intrinsic value (if any)

The IFRS requires that on conversion of a convertible instrument at maturity, the reporting company derecognizes the liability component and recognizes it as equity. Embedded derivatives are an interesting issue inasmuch as some contracts that themselves are not financial instruments may have financial instruments embedded in them. This is the case of a contract to purchase a commodity at a fixed price for delivery at a future date.

Contracts of this type have embedded in them a derivative that is indexed to the price of the commodity, which is essentially a derivative feature within a contract that is not a financial derivative. International Accounting Standard 39 (IAS 39) of the IFRS requires that under certain conditions an embedded derivative is separated from its host contract and treated as a derivative instrument. For instance, the IFRS specifies that each of the individual derivative instruments that together constitute a synthetic financial product represents a contractual right or obligation with its own terms and conditions. Under this perspective,

• Each is exposed to risks that may differ from the risks to which other financial products are exposed.
• Each may be transferred or settled separately.

Therefore, when one financial product in a synthetic instrument is an asset and another is a liability, these two do not offset each other. Consequently, they should be presented on an entity’s balance sheet on a net basis, unless they meet specific criteria outlined by the aforementioned accounting standards.

Like synthetics, structured financial products are derivatives. Many are custom-designed bonds, some of which (over the years) have presented a number of problems to their buyers and holders. This is particularly true for those investors who are not so versatile in modern complex instruments and their further-out impact.

Typically, instead of receiving a fixed coupon or principal, a person or company holding a structured note will receive an amount adjusted according to a fairly sophisticated formula. Structured instruments lack transparency; the market, however, seems to like them, the proof being that the amount of money invested in structured notes continues to increase. One of many examples of structured products is the principal exchange-rate-linked security (PERLS). These derivative instruments target changes in currency rates. They are disguised to look like bonds, by structuring them as if they were debt instruments, making it feasible for investors who are not permitted to play in currencies to place bets on the direction of exchange rates.

For instance, instead of just repaying principal, a PERLS may multiply such principal by the change in the value of the dollar against the euro; or twice the change in the value of the dollar against the Swiss franc or the British pound. The fact that this repayment is linked to the foreign exchange rate of different currencies sees to it that the investor might be receiving a lot more than an interest rate on the principal alone – but also a lot less, all the way to capital attrition. (Even capital protection notes involve capital attrition since, in certain cases, no interest is paid over their, say, five-year life cycle.)

Structured note trading is a concept that has been subject to several interpretations, depending on the time frame within which the product has been brought to the market. Many traders tend to distinguish between three different generations of structured notes. The elder, or first generation, usually consists of structured instruments based on just one index, including

• Bull market vehicles, such as inverse floaters and cap floaters
• Bear market instruments, which are characteristically more leveraged, an example being the superfloaters

Bear market products became popular in 1993 and 1994. A typical superfloater might pay twice the London Interbank Offered Rate (LIBOR) minus 7 percent for two years. At currently prevailing rates, this means that the superfloater has a small coupon at the beginning that improves only if the LIBOR rises. Theoretically, a coupon that is below current market levels until the LIBOR goes higher is much harder to sell than a big coupon that gets bigger every time rates drop. Still, bear plays find customers.

Second-generation structured notes are different types of exotic options; or, more precisely, they are yet more exotic than superfloaters, which are exotic enough in themselves. There exist serious risks embedded in these instruments, as such risks have never been fully appreciated. Second-generation examples are

• Range notes, with embedded binary or digital options
• Quanto notes, which allow investors to take a bet on, say, sterling London Interbank Offered Rates, but get paid in dollar.

There are different versions of such instruments, like you-choose range notes for a bear market. Every quarter the investor has to choose the “range,” a job that requires considerable market knowledge and skill. For instance, if the range width is set to 100 basis points, the investor has to determine at the start of the period the high and low limits within that range, which is far from being a straight job.

Surprisingly enough, there are investors who like this because sometimes they are given an option to change their mind; and they also figure their risk period is really only one quarter. In this, they are badly mistaken. In reality even for banks you-choose notes are much more difficult to hedge than regular range notes because, as very few people appreciate, the hedges are both

• Dynamic
• Imperfect

There are as well third-generation notes offering investors exposure to commodity or equity prices in a cross-category sense. Such notes usually appeal to a different class than fixed-income investors. For instance, third-generation notes are sometimes purchased by fund managers who are in the fixed-income market but want to diversify their exposure. In spite of the fact that the increasing sophistication and lack of transparency of structured financial instruments sees to it that they are too often misunderstood, and they are highly risky, a horde of equity-linked and commodity-linked notes are being structured and sold to investors. Examples are LIBOR floaters designed so that the coupon is “LIBOR plus”:

The pros say that flexibly structured options can be useful to sophisticated investors seeking to manage particular portfolio and trading risks. However, as a result of exposure being assumed, and also because of the likelihood that there is no secondary market, transactions in flexibly structured options are not suitable for investors who are not

• In a position to understand the behavior of their intrinsic value
• Financially able to bear the risks embedded in them when worst comes to worst

It is the price of novelty, customization, and flexibility offered by synthetic and structured financial instruments that can be expressed in one four-letter word: risk. Risk taking is welcome when we know how to manage our exposure, but it can be a disaster when we don’t – hence, the wisdom of learning ahead of investing the challenges posed by derivatives and how to be in charge of risk control.

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….