# stochastic review

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Stochastic Models in Finance and InsuranceScript by Ilya Molchanov and Michael Schmutz michael.schmutz@stat.unibe.ch Recommended books:Primary J.C. Hull, Options, Futures and other Derivatives, Prentice-Hall, 2009. M. Baxter, A. Rennie, Financial Calculus, Cambridge University Press, 2001. J. Cvitani, F. Zapatero, Introduction to the Economics and Mathematics of Financial Markets, MIT c Press, 2004. W. Hausmann, K. Diener, J. Ksler, Derivate, Arbitrage und Portfolio-Selektion, Vieweg, 2002. a A. Irle, Finanzmathematik. Die Bewertung von Derivaten, Teubner, 2003. Secondary R. Dobbins, S. Witt, J. Fielding, Portfolio Theory and Investment Management, Blackwell, 1994. E. Straub, Non-Life Insurance Mathematics, Springer, 1988. H.U. Gerber, Life Insurance Mathematics, Springer, 1990. S.N. Neftci, An Introduction to the Mathematics of Financial Derivatives, Academic Press, 1996. P. Wilmott, Derivatives. The Theory and Practice of Financial Engineering, Wiley, 1998. J.Y. Campbell, A.W. Lo, A.C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, 1997. H. Bhlmann, Mathematical Methods in Risk Theory, Springer, 1970. u Further reading More economical/actuarial ... R. Korn, E. Korn, Option pricing and Portfolio Optimization, Amer. Math. Society, 2001. R.W. Kolb, Understanding Futures Markets, Blackwell, 1997. R.W. Kolb, Practical Readings in Financial Derivatives, Blackwell, 1998. D. Winstone, Financial Derivatives, Chapman & Hall, 1995. E.J. Elton, M.J. Gruber, Modern Portfolio Theory and Investment Analysis, Wiley. C.D. Daykin, T. Pentikinen, M. Pesonen, Practical Risk Theory for Actuaries, Chapman & Hall, 1994. a

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More mathematical ... R. Korn, Optimal Portfolios, World Scientic, 1997. P. Wilmott, J. Dewynne, S. Howison, Option Pricing, Oxford Financial Press, 1993 (mostly deterministic approach). P. Wilmott, S. Howison, J. Dewynne, The Mathematics of Financial Derivatives, Cambridge University Press, 1995. S.P. Pliska, Introduction to Mathematical Finance, Blackwell, 1997 (mostly discrete). S.E. Shreve, Stochastic Calculus for Finance, Springer, 2004. T. Mikosch, Elementary Stochastic Calculus with Finance in View, World Scientic, 1998. N.H. Bingham, R. Kiesel, Risk-Neutral Valuation. Pricing and Hedging of Financial Derivatives, Springer, 1998. Y.K. Kwok, Mathematical Models of Financial Derivatives, Springer, 1998. D. Lamberton, B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall, 1996. M. Musiela, M. Rutkowski, Martingale Methods in Financial Modelling, Springer, 1997. T. Rolski et al, Stochastic Processes for Insurance and Finance, Wiley, 1999.

Plan Chapter Chapter Chapter Chapter

1. 2. 3. 4.

Basic concepts of nancial derivatives Stochastic models for stock prices Portfolios Risk and insurance.

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1. Basic concepts of asset returns, futures, options and other financial instruments

A nancial derivative is an instrument whose value depends on, or is derived from, the value of another asset. The underlying assets include stocks, currencies, interest rates, commodities, debt instruments, electricity, insurance payos, the weather, etc.

1. Asset returns1.1. Interest rates Consider an amount a invested for n years at an interest rate r per annum. If the rate is compounded once per annum, the terminal value of the investment is a(1 + r)n . If it is compounded m times per annum, the terminal rate of the investment is a(1 + r/m)mn . The limit as m tends to innity corresponds to continuous compounding aern . Let rc be the rate of continuous compounding and rm be the rate with compounding m times per annum. Then aerc n = a(1 + rm /m)mn , whence rc = m log(1 + rm /m) , rm = m(erc /m 1) .

For instance, with a single annual compounding m = 1 and r = r1 rc = log(1 + r) , 1.2. Forward rates The forward interest rate is the interest rate for a future period of time implied by the rates prevailing in the market today. r = erc 1 .

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Year 1 2 3 4 5

Zero rate 10.0 10.5 10.8 11.0 11.1

Forward rate 11.0 11.4 11.6 11.5

Example: if $100 invested for one year and then for another year, then 100e0.10 e0.11 = 123.37 = 100e10.52 .

2. Forwards and futures2.1. Forward contacts A forward contract is a contract that obligates the holder to buy or sell an asset for a predetermined delivery price at a predetermined future time. The main features of a Forward contract are: initiated now, performed later; involves exchange of assets; price set at time of contracting. 2.2. Some important words and concepts

buyer = long position; buying = going long seller = short position; selling = going short

portfolio = combination of several assets/securities, etc.Short selling involves selling an asset that is not owned with the intention of buying it later. Example: An investor contacts a broker to short 500 IBM shares. The broker borrows the shares from another client and sells them depositing the proceeds to the investors account. At some stage the investor instructs the broker to close out the position, the broker uses the funds from the investor to purchase 500 IBM shares and replaces them. If at any time, the broker runs out of shares, the investor is short-squeezed and must close the position immediately, even if not ready to do so. 2.3. Futures A futures contract is a standardised contract that obligates the holder to buy or sell an asset at a predetermined delivery price during a specied future period. The contract is settled daily.

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Futures: Futures Futures Futures Futures Futures Futures

started at Chicago Board of Trade, opened 1848 trade on organised exchanges. contracts have standardised contract terms. exchanges have associated clearinghouses to guarantee fullment of futures contract obligations. trading requires margin payment and daily settlement. positions can be closed easily. markets are regulated by identiable agencies, while forward markets are self-regulating.

Standardised contract terms cover the following issues: quantity quality expiration months delivery terms delivery dates (normally any day in a month) minimum price uctuation (tick is the smallest change in the price of a futures contract permitted by the exchange) daily price limit (restricts price movements in a single day) trading days and hours Clearinghouse guarantees fullment of the contract, acting as the seller to the buyer and as the buyer to the seller. Thus, the buyer and seller do not have to check credit worthiness. Margin provides a nancial safeguard to ensure that traders will perform on their contract obligations. Initial margin (deposit requested from trader before trading any futures, usually 5% of the commoditys value). Maintenance margin. If the trader sustains a loss, it is taken from his margin. When the value of the funds on deposit reaches the maintenance margin (usually 75% of the initial margin), the trader is required to replenish the margin (this demand is known as margin call). Closing a futures position Delivery or cash settlement (usually not more than 1% of all contracts end this way, for currencies this may be about 2%). Oset or reversing trade (the trader enters the reverse contract, so his net position is zero which is recognised by the clearinghouse; the reverse contract should match exactly the original contract entered). Types of futures contracts: agricultural and metallurgical contracts interest-earning assets (bonds, treasury bills, etc.) foreign currencies stock indices (they do not admit a possibility of actual delivery) Combination of several related futures is called a spread intramarket spread, also called calendar spread or time spread intermarket spread (dierent but closely related commodities) Abusive trading practices and manipulations Example: (The Hunt Silver Manipulation) Prices (per ounce in US dollars): 6 (1979), over 50 (Jan 1980), 12 (Mar 1980), 5-6 (1996). Amassed gigantic futures contracts and demanded delivery as they came due; at the same time, they bought big quantities of physical silver and held it o the market. The exchange imposed 5

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liquidation-only trading, meaning trade to only close existing futures positions. The exchange increased the margins on silver, then Hunts defaulted on their margin obligations. They tried to issue bonds backed by their physical silver holdings, which the market interpreted as act of desperation and the price crashed. Sued by their co-conspirators, became bankrupt by 1990. 2.4. Traders A speculator is a trader who enters the futures market in pursuit of prot, accepting the risk. A hedger is a trader who trades futures to reduce some preexisting risk exposure. They are often producers or major users of a given commodity (e.g., a farmer may hedge by selling his anticipated harvest even before the farmer plants). They often trade through a brokerage rm. Arbitrageurs enter several contracts in dierent markets to exploit price uctuations. If a good had two prices, a trader can get an arbitrage prot a sure prot with no investment. But prices may dier because of transportation costs, etc.

Arbitrage = trading without investment that guarantees no loss with probability one and potential income with a non-zero probabilityTrading orders: market order (buy or sell at the best price currently available); limit order (maximum and minimum prices specied); short sale; stop order (activated when the price of a stock reaches a predetermined limit).

3. Hedging using futuresA company that is due to sell an asset takes a short futures position (short hedge). If the price goes down, the company loses on the sale, but makes a gain on t

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