# Value at Risk

Topic: Discuss and investigate VaR and its characteristics when applied to options. You must produce example calculations on: European and American style options

Long and short positions in these

Portfolio of at least three different options (more is better)

Introduction

All financial institutions bear some sort of risk while dealing with different financial instruments, whether it be corporate treasurers, fund managers or financial institutions, they are all exposed to a certain market risks while carrying out their daily trading activities. There is a possibility that the institution makes a blunder in forecasting the future value of its trade and this may lead to major losses that have to be incurred by the institution. The exact number of this loss that the institution might bear is unknown and this had bothered many regulators with the fear of major banks going bankrupt, so mitigate the risk of bankruptcy the Basle committee on banking supervision, in April 1995, announced that capital adequacy requirements for commercial banks would be based on VaR. Later in the same year there was proposal issued by the security and exchange commission that required publicly traded U.S corporations to disclose information about derivative activity, with VaR measure as one of three possible methods for making such disclosures.(Jorion,P, 1996) Value at Risk

Value at risk (VaR) is an attempt to provide a single number summarizing the total risk in the portfolio of financial assets. Var is the maximum loss that the institution will face on a given day. The basic variables involved in VaR measure is the time horizon and the confidence level e.g. an analyst calculating the VaR of a instrument(securities or derivatives) is a certain ‘X’ percent sure that his loss over the next ‘N’ days will not cross a certain value. In this ‘X’ percent would correspond to the confidence level that is assumed by the analyst, it could be any number from 1-100. If the analyst chooses a 90% confidence level then the VaR would correspond to the (100-X, i.e.90 in this case)th percentile of the distribution of the change in the value of the portfolio over the next N days. Var basically helps answer the question “how bad can things get?” for the financial institutions.(hull) Regulators usually require a bank’s capital for market risk to be at least three times the 10-day 99% VaR. Since, N day VaR = 1-day VaR x N , therefore the minimum capital level is 3 x 10 = 9.49 times the 1-day 99% VaR.(hull) An option is an instrument which gives the holder (owner) the right to buy or sell a particular asset (underlying asset) at a particular price at a certain date. A put option gives the holder the right to sell the underlying asset at a certain price at a certain date whereas a call option gives the holder the right to buy the underlying asset at a certain date for a certain date. The payoffs for the call option is ST –K (ST = price of underlying at maturity; K = exercise price) and for put is K – ST. There are mainly two types of options, European option and American option. A European option is a typical option where the underlying price is already decided and the maturity date is fixed and it can only be exercised on the date of maturity. An American option is similar to an European option with the only difference being that it can be exercised at any given point during the life (duration) of the option.(hull) While calculating the VaR for options the method adopted is far more complex than the ones used for equities as the VaR for options is based on the underlying asset of the option (stocks, bonds, futures, foreign currency, etc). While calculating the VaR of options we can either use the linear form or the non-linear form. The linear or delta normal form for calculating VaR considers a delta change in the portfolio of options. Delta is the rate of change of the value of the portfolio with respect to change in the value of...

References: Jorion,P (1996) “Measuring The Risk In Value At Risk” financial analyst journal

Hoadley,P (2011) “Option Pricing Models and the "Greeks”

Hull,j,c. (2010) “options, futures and derivatives”

An excellent exposition

of the Monte Carlo method is given by Hammersley and Handscomb (1964).

Shreider (1966). Fishman (1973) and Meyer (1956) provide additional useful

references.

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