Market Risk Reporting

For

Equity Options

Using

The Delta-Plus Method

July, 4 2001

Equities

All equity and equity funds positions are reported at their current market value. Market risk amounts for 8% of the principal and specific risk are set to 8% for each issuer (§2.2, un-diversified portfolio: 8%, liquid diversified: 4%, index contracts: 2%). Specific risk of equities is defined as the part of the price fluctuation that is not correlated with the market (represented by the equity index). Netting between long and short positions is possible whenever identical shares or equity index contracts are involved (§2.1.1). Equity options are reported through their delta equivalent (option add-ons due to Gamma and Vega effects come up separately).

Options Four valuation methods are proposed in the EBK paper (34, 1998):

‚Simplified Procedure‘ (for long positions only): where capital requirements are based on a comparison between the market value of the option and a conventional fraction of the underlying financial asset (smallest amount between both).

‚Delta-Plus Procedure‘: based on a numerical approximation of the potential change in value of the portfolio using the Delta, Gamma and Vega of the portfolio.

‚Scenario Analysis‘: using the ‘Matrix Sensitivity’ method (scenarios combining changes in the underlying and in market volatility) to obtain the worst case for potential losses.

‚Internal Models’: providing a full simulation of the whole portfolio of the bank. The second method (§5.3.2) used by Bank Leumi in Zurich.

Black & Scholes Option Model: The option valuation model derived in 1973 by Fisher Black and Myron Scholes for common

stocks is used (N: being the cumulative normal distribution function, : the volatility of the underlying, t: the time to expiration of the option, and Today and Strike: the prices of the underlying). The discount factor between today and the expiration date is assumed to be 1.

)()(

)1)(()1)(()(

2

1)/ln(

)()(

12

21

12

2

1

21

dNTodaydNStrikePut

dNStrikedNTodayStrikeTodayCallPut

tdd

t

tStrikeToday

d

dNStrikedNTodayCall

Reporting Procedures Market risk is basically a change in value of the option (Delta-Plus-Procedure § 5.3.2).

Changes due to the time-decay () and interest rate sensitivity () are neglected here.

VolVegaNotionalSGammaNotionalSDeltaNotionalV 2

)(2

1)(

where:

V = Possible change in value of the portfolio

Notional = Principal amount of the underlying (equity: of contracts x denomination)

S = Change in the price of the underlying

Vol = Change in the implied volatility

Hence, the risk generated by an option is approximately the sum of a Delta contribution (linear risk), together with a Gamma effect (non-linear part) and, a Vega effect due to the stochastic nature of the instrument.

Delta Equivalent (§ 5.3.2 a): First, the Delta equivalent is calculated (the value like if it would be a cash position):

DeltaNotionalequivalentDelta _

Notional = Principal amount of the underlying (equity: of contracts x denomination)

Leading to the risk associated with the Delta:

SequivalentDeltaeffectDelta __

S = Interest rates: risk-weighting factor according to table 1 in §1.3.1, typically 3.25% (5y gap) Equity/Index: 8% of the underlying price (market risk) FX and PM: 10% of the underlying price Commodities: 20% of the underlying price.

Gamma effect (§ 5.3.2 b):

2)(

2

1_ SGammaNotionaleffectGamma

Notional = Principal amount of the underlying (equity: of contracts x denomination)

S = Interest rates: risk-weighting factor according to table 1 in §1.3.1, typically 3.25% (5y gap) Equity/Index: 8% of the underlying price (market risk) FX and PM: 10% of the underlying price Commodities: 20% of the underlying price

Gamma effects from instruments belonging to the same market can be netted (e.g. instruments from the same currency and the same time gap or from the same currency pair or the same stock index, §5.3.2 b). Only negative Gamma positions are taken into account.

Vega effect (§ 5.3.2 c):

VolVegaNotionaleffectVega _ Vol = 0.25 implied volatility

Vega effects from instruments related to the same market can be netted (§5.3.2 c).

Naming Conventions for Options

A Call option is the right for the holder to buy a specific asset at a price agreed at time of purchase. In the case of an interest rate option, it is the right to pay a fixed interest rate.

10 Call

Delta

A Cap is an over the counter option (treated as a call), which sets a ceiling on the upward movement of a floating interest rate index. A Floor is an option (treated as a put), which grants the right to receive a minimum interest rates, should the floating index fall below the strike level. The Delta of an option, also called the hedge ratio, is a dimensionless number indicating the change in the option’s cash value resulting from a change in the value of the underlying. The Delta equivalent position generated by an option is the cash position in the underlying financial asset, which has the same sensitivity to market, changes as the option position. The Gamma is the sensitivity of an option’s delta to changes in the value of the underlying. The Gamma effect is the approximated change in value due to the convexity of the portfolio, assuming a small move in the price of the underlying instrument. The implied volatility of a financial asset is the standard deviation of a financial asset induced by option prices quoted on the marketplace. The maturity date of the underlying is the date when the investor receives the principal associated with the security, along with the last coupon payment. The option expiration date is the last date that an option holder can exercise his option; hence the time to expiration is the outstanding lifetime of the option. The present value (PV) is the value today of a series of cash flows discounted against the market yield curve. The net present value (NPV) is the present day value of a series of cash flows less the initial investment. A Put option is the right for the holder to sell a specific asset at a price agreed at time of purchase. In the case of an interest rate option, it is the right to receive a fixed interest rate.

01 Put

Delta

The Put-Call parity is the relationship between the market price on a put and a call that have the same exercise price, expiration date, and underlying asset.

1putcall

DeltaDelta

The strike price (or exercise price) is the price at which the holder of the option may exercise his right to purchase or sell the underlying instrument. A Swaption is an option to enter an interest rate swap transaction at a given future date at a given fixed rate. A payer Swaption is the right to pay a fixed rate (treated as a call option), while a receiver Swaption is the right to receive a fixed interest rate (treated as a put option). The Vega is the sensitivity of an option to changes in the volatility of the underlying. The Vega effect is the approximated change in value of the portfolio, assuming a 25% change in the implied volatility of the underlying.