Stochastic Review

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<p>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 &amp; 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 &amp; Hall, 1994. a</p> <p>FS 2011</p> <p>Stoch Modelle FV</p> <p>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 &amp; 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.</p> <p>Plan Chapter Chapter Chapter Chapter</p> <p>1. 2. 3. 4.</p> <p>Basic concepts of nancial derivatives Stochastic models for stock prices Portfolios Risk and insurance.</p> <p>2</p> <p>1. Basic concepts of asset returns, futures, options and other financial instruments</p> <p>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.</p> <p>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) .</p> <p>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 .</p> <p>3</p> <p>FS 2011</p> <p>Stoch Modelle FV</p> <p>Year 1 2 3 4 5</p> <p>Zero rate 10.0 10.5 10.8 11.0 11.1</p> <p>Forward rate 11.0 11.4 11.6 11.5</p> <p>Example: if $100 invested for one year and then for another year, then 100e0.10 e0.11 = 123.37 = 100e10.52 .</p> <p>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</p> <p>buyer = long position; buying = going long seller = short position; selling = going short</p> <p>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.</p> <p>4</p> <p>FS 2011</p> <p>Stoch Modelle FV</p> <p>Futures: Futures Futures Futures Futures Futures Futures</p> <p>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.</p> <p>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</p> <p>FS 2011</p> <p>Stoch Modelle FV</p> <p>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.</p> <p>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).</p> <p>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 the short futures position. If the price goes up, the company loses on futures and gains on sales. A company that is due to buy an asset takes a long hedge. Note: a futures hedge does not necessarily improve the nal outcome, it even makes it worse roughly 50% of the time. But it reduces the risk by making the outcome more certain. Example. Italian company expects to receive 50 million USD at the end of July. USD contracts have deliveries Mar, Jun, Sep, Dec. The company shorts four 12.5 million Sep USD contracts. In the end of Jul the company receives USD and closes out futures position. If the Sep futures price in Mar was 0.78 and the spot and futures price in Jul are 0.72 and 0.7250, then the gain on futures is 0.78-0.725=0.055 cents per USD. The eective price per USD will be 0.72+0.055=0.775. Rolling the hedge forward is achieved by entering successive futures contracts and closing out previous contracts. Optimal hedge ratio Hedge ratio h is the ratio of the size of the position taken in futures contracts to the size of the exposure. Let S be the change in spot price, S, during the life of the hedge; let F be the change in futures price, F , S , F are standard deviations of S and F and is the coecient of correlation.</p> <p>6</p> <p>FS 2011</p> <p>Stoch Modelle FV</p> <p>For a short hedge the eective price obtained is S2 + h(F1 F2 ), so that change of total value S hF (and hF S for a long hedge). The variance of the change of value (in either case) is2 2 S + h2 F 2hS F ,</p> <p>which is minimised if h=</p> <p>S . F</p> <p>The optimal h = 1 if the futures prices mirror the spot price perfectly.</p> <p>4. Futures pricesNewspaper quotes First line: nearby contract, then distant or deferred contracts; opening, high, low prices; settlement price (at the close of trading); change in the settlement price from the previous day; open interest (the number of futures contracts for which delivery is currently obligated). Example. A buys and B sells one contract, the open interest is one; C buys and D sells 3 contracts, the open interest is 4; then A sells and D buys 1 contract, the open interest is 3. Basis is current cash price minus futures price. In a normal market the prices for more distant futures are higher, while in the inverted market the distant future prices are lower.</p> <p>Figure 1.1: Converging cash and futures prices (normal market) Assume that the market is perfect (no transaction costs and no restrictions on free contracting). The cost-of-carry or carrying charge is the total costs to carry a good forward in time: storage, insurance, transportation and nancing costs. We will mostly consider the nancing costs. 4.1. Examples Assume that interest rate is 10%</p> <p>7</p> <p>FS 2011</p> <p>Stoch Modelle FV</p> <p>Cash-and-Carry strategy Spot price of gold 1800; Future price of gold (delivery in 1 year) 2000 Transaction: t=0 Borrow 1800 for one year at 10% Buy 1 ounce of gold in the spot market for 1800 Short a futures contract for delivery in 1 year for 2000 t=1 Remove the gold from storage Deliver the ounce of gold, repay the loan, including interest Total cash flow: .... Reversed Cash-and-Carry Strategy Spot price of gold 1800; Future price of gold (delivery in 1 year) 1950 Transaction: t=0 Sell ounce of gold short Lend 1800 for 1 year at 10% Enter a long position in a future for buying 1 ounce of gold in one year t=1 Collect proceeds from the loan Accept delivery of the futures contract Total cash flow: .... 4.2. Pricing rule The futures price must equal the spot price plus the cost of carrying the spot commodity forward to the delivery date of the futures contract F0,T = S0 (1 + C) or F0,T = S0 ecT ,</p> <p>where C (resp. c) measures the storage cost plus the interest that is paid to nance the asset less the income earned on the asset. If the above equations hold, the market is a full carry market. Example: Silver Sep 41.20, Silver Dec 41.95, Bankers Acceptance Rate 90 days 7.5%. Then (41.95/41.20)4 = 1.0748 corresponds roughly to the interest rate. If an asset provides income I during the life of a contract, then F0,T = (S0 I)erT . It is important to remember that I is the discounted price (i.e. price at time zero) of all future cash ows. Reason: buy one unit of the asset and enter a short forward contract to sell it for F0,T . This costs S0 and should match cash inow I + F0,T erT . 4.3. Pricing under transaction costs The transaction costs and other market imperfections lead to the appearance of two prices: the ask price at which the goods are sold and the bid price at which they are purchased. The ask price is always greater than or equal to the bid price. For nancing purposes, this corresponds to the two interest rates: the lending rate and the borrowing rate. If T denotes transaction costs, then S0 (1 + CL )(1 T ) F0,T S0 (1 + CB )(1 + T ) , 8</p> <p>FS 2011</p> <p>Stoch Modelle FV</p> <p>where CL is the lending rate and CB is the borrowing rate. There are normally restrictions on short selling. If only a part f of the funds may be used, then S0 (1 + f CL )(1 T ) F0,T S0 (1 + CB )(1 + T ) .</p> <p>5. BondsInterest rates: Treasury rates - risk-free. LIBOR rates (London Interbank Oer Rate) - not necessarily risk-free (usually...</p>