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Stochastic Models in Finance and Insurance

Script by Ilya Molchanovwww.imsv.unibe.ch/∼ilya

Michael [email protected]

Recommended books:

Primary

• J.C. Hull, Options, Futures and other Derivatives, Prentice-Hall, 1999.

• M. Baxter, A. Rennie, Financial Calculus, Cambridge University Press, 1997.

• J. Cvitanic, F. Zapatero, Introduction to the Economics and Mathematics of Financial Markets, MITPress, 2004.

• W. Hausmann, K. Diener, J. Kasler, Derivate, Arbitrage und Portfolio-Selektion, Vieweg, 2002.

• 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 UniversityPress, 1997.

• H. Buhlmann, Mathematical Methods in Risk Theory, Springer, 1970.

Further readingMore 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. Pentikainen, M. Pesonen, Practical Risk Theory for Actuaries, Chapman & Hall, 1994.

HS 2009 Stoch Modelle FV

More mathematical ...

• R. Korn, Optimal Portfolios, World Scientific, 1997.

• P. Wilmott, J. Dewynne, S. Howison, Option Pricing, Oxford Financial Press, 1993 (mostly deterministicapproach).

• P. Wilmott, S. Howison, J. Dewynne, The Mathematics of Financial Derivatives, Cambridge UniversityPress, 1995.

• S.P. Pliska, Introduction to Mathematical Finance, Blackwell, 1997 (mostly discrete).

• T. Mikosch, Elementary Stochastic Calculus with Finance in View, World Scientific, 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.

PlanChapter 1. Basic concepts of financial derivativesChapter 2. Stochastic models for stock pricesChapter 3. PortfoliosChapter 4. Risk and insurance.

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1. Basic concepts of asset returns, futures, options

and other financial instruments

1. Asset returns

1.1. Interest rates

Consider an amount a invested for n years at an interest rate r per annum. If the rate is compounded once perannum, the terminal value of the investment is a(1+ r)n. If it is compounded m times per annum, the terminalrate of the investment is

a(1 + r/m)mn .

The limit as m tends to infinity 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

aercn = a(1 + rm/m)mn ,

whencerc = 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) , r = erc − 1 .

1.2. Forward rates

Year Zero rate Forward rate

1 10.0

2 10.5 11.0

3 10.8 11.4

4 11.0 11.6

5 11.1 11.5

Example: if $100 invested for one year and then for another year, then 100e0.10e0.11 = 123.37 = 100e0.105×2.

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2. Forwards and futures

2.1. Forward contacts

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 longseller = 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 anotherclient and sells them depositing the proceeds to the investor’s account. At some stage the investor instructs thebroker to close out the position, the broker uses the funds from the investor to purchase 500 IBM shares andreplaces them. If at any time, the broker runs out of shares, the investor is short-squeezed and must close theposition immediately, even if not ready to do so.

2.3. Futures

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

Example: Exporter sells to US and paid in US$; he wants to avoid the risk of currency fluctuation, so buysa Swiss Francs futures contract. The contract may be sold by another trader in the opposite position or by aspeculator. The terms are usually less favourable than the current ones, but risk is avoided.

2.4. Traders

A speculator is a trader who enters the futures market in pursuit of profit, accepting the risk.

A hedger is a trader who trades futures to reduce some preexisting risk exposure. They are often producersor major users of a given commodity (e.g., a farmer may hedge by selling his anticipated harvest even beforethe farmer plants). They often trade through a brokerage firm.

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Arbitrageurs enter several contracts in different markets to exploit price fluctuations. If a good had twoprices, a trader can get an arbitrage profit — a sure profit with no investment. But prices may differ becauseof transportation costs, etc.

Arbitrage = sure profit without investment

Trading orders: market order (buy or sell at the best price currently available); limit order (maximum andminimum prices specified); short sale; stop order (activated when the price of a stock reaches a predeterminedlimit).

2.5. Contracts and margins

Standardised contract terms cover the following issues:• quantity• quality• expiration months• delivery terms• delivery dates (normally any day in a month)• minimum price fluctuation (tick is the smallest change in the price of a futures contract permitted by theexchange)• daily price limit (restricts price movements in a single day)• trading days and hours

Clearinghouse guarantees fulfilment of the contract, acting as the seller to the buyer and as the buyer to theseller. Thus, the buyer and seller do not have to check credit worthiness.

Margin provides a financial 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 commodity’svalue).

• Maintenance margin. If the trader sustains a loss, it is taken from his margin. When the value of the fundson deposit reaches the maintenance margin (usually 75% of the initial margin), the trader is required toreplenish 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 thismay be about 2%).

• Offset or reversing trade (the trader enters the reverse contract, so his net position is zero which isrecognised 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)

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Combination of several related futures is called a spread• intramarket spread, also called calendar spread or time spread• intermarket spread (different but closely related commodities)

Abusive trading practices and manipulationsExample: (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 thesame time, they bought big quantities of physical silver and held it off the market. The exchange imposedliquidation-only trading, meaning trade to only close existing futures positions. The exchange increased themargins on silver, then Hunts defaulted on their margin obligations. They tried to issue bonds backed by theirphysical silver holdings, which the market interpreted as act of desperation and the price crashed. Sued by theirco-conspirators, became bankrupt by 1990.

3. Hedging using futures

A company that is due to sell an asset takes a short futures position (short hedge). If the price goes down, thecompany loses on the sale, but makes a gain on the short futures position. If the price goes up, the companyloses 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 final outcome, it even makes it worse roughly 50% ofthe time. But it reduces the risk by making the outcome more certain.

Example. US company expects to receive 50 million Yen at the end of July. Yen contract have deliveries Mar,Jun, Sep, Dec. The company shorts four 12.5 million Sep Yen contracts. In the end of Jul the company receivesYen and closes out futures position. If the Sep futures price in Mar was 0.78 and the spot and futures price inJul are 0.72 and 0.7250, then the gain on futures is 0.78-0.725=0.055 cents per yen. The effective price per Yenwill be 0.72+0.055=0.775.

Rolling the hedge forward is achieved by entering successive futures contracts and closing out previouscontracts.

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 , σS ,σF are standard deviations of ∆S and ∆F and ρ is the coefficient of correlation.

For a short hedge the effective price obtained is S2 + h(F1 − F2), so that change of total value

∆S − h∆F

(and h∆F −∆S for a long hedge). The variance of the change of value (in either case) is

σ2s + h2σ2

F − 2hρσSσF ,

which is minimised ifh = ρ

σS

σF.

The optimal h = 1 if the futures prices mirror the spot price perfectly.

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4. Futures prices

Newspaper quotesFirst 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 openinterest 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 arehigher, while in the inverted market the distant future prices are lower.

Prices

Time

Expiration

Futures

Cash

Figure 1.1: Converging cash and futures prices

Assume that the market is perfect (no transaction costs and no restrictions on free contracting). The cost-of-carryor carrying charge is the total costs to carry a good forward in time: storage, insurance, transportation andfinancing costs. We will mostly consider the financing costs.

4.1. Examples

Assume that Interest rate is 10%

Cash-and-Carry strategySpot price of gold 400; Future price of gold (delivery in 1 year) 450Transaction: t=0 Borrow 400 for one year at 10%

Buy 1 ounce of gold in the spot market for 400Sell a futures contract for 450 for delivery in 1 year

t=1 Remove the gold from storageDeliver the ounce of gold, repay the loan, including interest

Total cash flow: ....

Reversed Cash-and-Carry StrategySpot price of gold 420; Future price of gold (delivery in 1 year) 450Transaction: t=0 Sell 1 ounce of gold short

Lend 420 for 1 year at 10% and buy 1 ounce of gold for delivery in one yeart=1 Collect proceeds from the loan

Accept delivery of the futures contractTotal cash flow: ....

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4.2. Pricing rule

The futures price must equal the spot price plus the cost of carrying the spot commodity forward to thedelivery date of the futures contract

F0,t = S0(1 + C) or F0,t = S0ect ,

where C (resp. c) measures the storage cost plus the interest that is paid to finance the asset less the incomeearned on the asset.

If the above equations hold, the market is a full carry market.

Example: Gold Sep 410.20, Gold Dec 417.90, Banker’s Acceptance Rate 90 days 7.8%. Then (417.90/410.20)4 =1.0772 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 flows.

Reason: buy one unit of the asset and enter a short forward contract to sell it for F0,t. This costs S0 and shouldmatch cash inflow I + F0,te

−rt.

4.3. Pricing under transaction costs

The transaction costs and other market imperfections lead to the appearance of two prices: the ask price atwhich the goods are sold and the bid price at which they are purchased. The ask price is always greater thanor equal to the bid price. For financing purposes, this corresponds to the two interest rates: the lending rateand the borrowing rate.

If T denotes transaction costs, then

S0(1 + CL)(1− T ) ≤ F0.t ≤ S0(1 + CB)(1 + T ) ,

where CL is the lending rate and CB is the borrowing rate. There are normally restrictions on short selling. Ifonly a part f of the funds may be used, then

S0(1 + fCL)(1− T ) ≤ F0.t ≤ S0(1 + CB)(1 + T ) .

5. Bonds

5.1. Pure discount bonds

Interest rates:

• Treasury rates - risk-free.

• LIBOR rates (London Interbank Offer Rate) - not necessarily risk-free (usually quoted on loans expressedin dollars; used by numerous financial institutions).

• Repo rate (repurchase agreement) - very little risk (one party sells a security to another party and agreesto buy it back at a specified time/price).

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A pure discount bond promises to pay a certain amount (face value) at a specified time in the future and issold for less that this promised future payment. Its price:

P =C

(1 + r)tor P = e−rtC ,

where C is the cash to be paid at the maturity of the bond at time t, r is the annualised (resp. continuouslycompounded) yield to maturity.

5.2. Coupon bonds

Coupon bonds make regular scheduled payments between the date of issue and the maturity date.

• Treasury bills (T-bills) (up to one year maturity). Treasury bonds and Treasury notes (longer maturity).

• Eurodollar CD (Certificate of Deposit) - issued by banks outside the US to attract dollar denominatedfunds (US$ deposited in foreign banks).

• Repurchase agreement or repos, when one party sells a security to another part and agrees to buy it backat a specified time/price.

Example: Treasury zero rates are rates for investments that start today and last some time without intermediatepayments and with continuous compounding. For example, 0.5 years - 5.0%, 1.0 - 5.8%, 1.5 - 6.4%, 2.0 - 6.8%.Consider two-year treasury bond of $100 with coupons at rate 6% p.a. with the coupons being paid semiannually.The theoretical price of this bond is

3e−0.05×0.5 + 3e−0.058×1.0 + 3e−0.064×1.5 + 103e−0.068×2.0 = 98.39 .

To find the yield on the bond we have to solve

3e−y×0.5 + 3e−y×1.0 + 3e−y×1.5 + 103e−y×2.0 = 98.39

that gives y = 6.76%.

To find its par yield c solve

c

2e−0.05×0.5 +

c

2e−0.058×1.0 +

c

2e−0.064×1.5 + (100 +

c

2)e−0.068×2.0 = 100

that gives c = 6.87 per annum with semiannual compounding.

5.3. Currency futures

The futures price is given byF0,t = S0e

(r−rf )t ,

where rf is the risk-free interest rate in the foreign country. Reason: the holder may invest the foreign currencyin a foreign-denominated bond, which brings extra income.

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6. Major Market Indices

Index =∑

Pi

Divisor,

where Pi is price of stock i. The divisor keeps the value from changing due to stock splits or stock dividendsor stock substitution. Absolute changes in value are only important (change from 1 to 2 has the same effect asthe change from 100 to 101).

Important market indices:• Dow Jones Industrial Average (DJIA);• S&P 100 and S&P 500 Indices;• FTSE 100 (25 pounds per index point, a tick is 12.50);• The Nikkei index (225 largest Japanese firms) - equally valued and so more volatile than the US indices;• and naturally SMI.

The price for stock index futures must equal the price of the shares underlying the stock index plus the cost ofcarrying the stock to the future expiration, minus the future value of the dividends the stocks will pay beforeexpiration

F0,t = S0(1 + C)−∑

Di(1 + ri) .

6.1. Pricing and hedging with indices

The index can be considered as an asset providing a continuous dividend yield at rate q. Then

F0,t = S0e(r−q)t .

Example. Hedging the value of a portfolioCash market Futures market5 AprilHolds a balanced portfolio of UK Sells twenty June FTSE 100 contractsshares valued at 1 million pounds at a price of 2000 each (25x2000x20), sobut fears a fall in its value. committing himself to the notional saleThe current FTSE is 2000. of 1 million of stock at equity prices

implied by the futures price on 5 Apr10 MayThe FTSE 100 index has fallen Closes out the futures position buingto 1900. The value of portfolio twenty June FTSE 100 contracts at a pricedeclines to 950,000. of 1900. There is a gain of 200 ticks on

each contract.Loss of 50,000 Gain of 50,000

The other action can be undertaken if the hedger anticipates receipt of cash and fears a rise in the index. Thenone buys futures index contracts.

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7. Options

Call option gives the owner the right to buy a particular good at a certain price.Put option gives the owner the right to sell a particular good at a certain price.

The seller of an option is also known as the writer of an option. The act of selling is called writing an option.Each option contract stipulates a price that will be paid if the option is exercised, known as exercise price orstrike price. Every option is valid for only a limited time (until maturity). Every option involves a paymentfrom the buyer to seller (price or premium of an option).

American options can be executed at any time prior to maturity.European options can be executed only at maturity.

In the USA, options are traded at the Chicago Board Options Exchange, Philadelphia Exchange, the AmericanStock Exchange and the Pacific Stock Exchange. Stock options on over 500 different stocks (in lots of 100 shares)mostly American; foreign currency options (depending on currency, e.g. 31,250 pounds per option) Americanand European; index options (S&P 100 American and S&P 500 European at 100 times the index); options onfutures (the underlying asset is a futures contract - when the holders of a call/put option exercise they acquirea long/short position in the futures contract plus a cash amount equal to the excess of the futures/strike priceover the strike/futures price).

Newspaper quotes on stock options:the first column - the company and the closing stock price (per share, so it is to be multiplied by 100). Thestrike price and maturity appear in the second and third columns. If traded before, then the volume and, finally,the prices for call and put options are included.

Margin accounts are always required for writers. When the option is purchased the option price must be paidin full (and not from margin accounts). The Options Clearing Corporation (OCC) performs the same role asclearinghouses for futures. When the option is exercised, the broker notifies the OCC; it randomly selects amember with an outstanding short position in the same option. The member using an established procedure,elects a particular client who has written an option. The client is said to be assigned.

Profit/Loss

Price0

17

240

Profit/Loss

Price0

9

240

Figure 1.2: Profit/loss profile of a call/put option at expiry. The strike price is 240. For writers of an optionthe graphs are opposite.

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7.1. Hedging with options

Positions in options trading:• A long position is a call position.• A long position is a put position.• A short position is a call position.• A short position is a put position.

Profit/Loss

Price0

(a) Long position in a Stock andshort position in a Call

Profit/Loss

Price0

(b) Short position in a Stock andlong position in a Call

Profit/Loss

Price0

(c) Long position in a Stock andLong position in a Put

Profit/Loss

Price0

(d) Short position in a Stock andshort position in a Put

Figure 1.3: Strategies involving a single option and a stock

7.2. Spreads

A spread trading strategy involves taking a position in two or more options of the same type.

A bull spread can be created by buying a call option on a stock with a certain strike price and selling a calloption on the same stock with a higher strike price. Both options have the same expiration date. (It can bealso created by buying a put with a low strike price and selling a put with a higher strike price).

Reason: Limits the trader’s upside as well as downside risk. The trader gives up some upside potential byselling a call option with higher strike price X2. If St ≥ X2, then the total payoff is X2 −X1; if X1 < St < X2,then the payoff is St − X1; if St ≤ X1, the payoff is zero. The gain is obtained as payoff minus the price ofoption bought plus the price of option sold.

Bull spread with calls involves an initial investment, as the price of the option sold is less than the price of theoption purchased.

A trader who enters into a bear spread is hoping that the stock price will decline. Then the strike price of anoption purchased is greater than the price of the option sold.

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Profit/Loss

Price0

Profit/Loss

Price0

Figure 1.4: Bull/bear spread created using two call options

A butterfly spread is created by buying a call option with a relatively low strike price X1, buying a call optionwith a relatively high strike price X3 and selling two call options with a strike price X2, half-way between X1

and X3. It provides a modest return when the market is stable. A reversed strategy produces a modest profitif there is a significant change in the stock price.

Profit/Loss

Price0

Figure 1.5: Butterfly spread created using call options

Combinations involve taking a position in both calls and puts on the same stock.

A straddle involves buying a call and a put with the same strike price and expiration date. In a strangle atrader buys a put and a call with the same expiration date and different strike prices. They can be useful ifthe trader is betting that there will be a large price move but is uncertain whether it will be an increase or adecrease.

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Profit/Loss

Price0

Profit/Loss

Price0

Figure 1.6: Straddle and strangle

7.3. Other derivatives (exotic derivatives)

Standard Oil’s Bond (1986). There is no interest, but when the bond matures, the holders will receive asupplement based on the oil price at maturity.

Range forward contract. E.g., a three month sterling contract with dollar exchange rate between 1.6 and 1.64.If the rate lies outside the limits, then the lower/upper limit applies.

Weather derivatives have payoffs depending on the average temperature at particular locations.

Insurance derivatives have payoff dependent on the dollar amount of insurance claims.

Electricity derivatives have payoff dependent on the spot price of electricity, etc. etc. etc.

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2. Stochastic models for stock market prices

1. Single period models

In the simplest case the financial transactions are carried at two time moments: the initial date t = 0 andterminal date t = 1, with trading and consumption possible at these two dates. Further elements of themodel is the probability space Ω which represents possible states of the world and its σ-algebra F of events, aprobability measure P on Ω.

The two other components are

• the bank account (or bond) process Bt, t = 0, 1, such that B0 = 1 and B1 is a random variable whichis almost surely strictly positive. The variable B1 represents the time t = 1 value of the bank accountwhen $1 is deposited at time t = 0. In many cases B1 is assumed to be deterministic and equal to er or rwhere r ≥ 0 is the interest rate. Instead of the bank account it is possible to use any asset with a strictlypositive price (this asset is called a numeraire).

• the price process St, t = 0, 1, where S0 describes the current stock price (more generally vector ofcurrent stock prices) and S1 is a random variable (more generally random vector) that represents theirterminal prices. If there are N securities, then S0 and S1 are N -dimensional vectors.

A trading strategy H = (H0,H1, . . . , HN ) describes an investor’s portfolio as carried forward from tome t = 0to time t = 1. Note that negative values of the H’s correspond to borrowing or selling short.

The value process Vt, t = 0, 1, is the total value of the portfolio, so that

Vt = H0Bt +N∑

i=1

HiSit , t = 0, 1 .

The discounted value process is given by V ∗t = VtB

−1t . The discounted price process is S∗t = B−1

t St.

Recall that a random variable X is a measurable real-valued function on Ω. The expectation of X depends onthe probability measure on Ω. For instance if Ω is discrete, then

EX =∑

X(ω)P(ω) .

Because of this, it is often sensible to write EPX to indicate the dependency of the expectation on P.

A probability measure Q on a discrete space Ω is called a risk neutral probability measure if Q(ω) > 0 for allω ∈ Ω and EQ(S∗1 − S0) = 0. Then

EQ(V ∗t ) = V0 .

A contingent claim is a random variable representing a payoff at time t = 1. Clearly, options are contingentclaims. A contingent claim X is called attainable or marketable if there exists some trading strategy H (calledthe replicating portfolio) such that V1 = X.

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2. Properties of option prices

Factors affecting option prices

• the current stock price and the strike price (call options are more valuable if the stock price increases andless valuable if the strike price increases)

• the time to expiration (put and call American options are more valuable as time to expiration increases;European options are not necessarily more valuable)

• the volatility of the stock price, σ, so that σ√

∆t is the st. deviation of the stock price in a short lengthof time ∆t (the owner of a call benefit from price increases but has limited downside risk; the owner of aput benefits from price decreases but has limited risk if price increases; therefore, the value of both callsand puts increases as volatility increases)

• the risk-free interest rate (if the rate increases, the expected growth rate of the stock price tends toincrease, however, the present value of any future cash flows decreases, therefore the value of a put optiondecreases; for call options the first effect tends to increase the price, while the second tends to decrease it,but the first always dominates, so the value increases)

• the dividends expected during the life of an option (reduce the stock price on the ex-dividend date, decreasethe value of a call option, but increase the value of a put option)

Assumptions: no transaction costs; all trading profits (losses) are subject to the same tax rate; borrowing andlending at the risk-free interest rate is possible.

Notation

S0 current stock priceSt stock price at time tX strike price of optionT time of expiration of optionr risk-free rate of interestC value of American call option to buy one shareP value of American put option to sell one sharec value of European call option to buy one sharep value of European put option to sell one share

2.1. Upper bounds

c ≤ S0 , C ≤ S0 ,

p ≤ Xe−rT , P ≤ X .

2.2. Lower bounds for non-dividend paying stocks

European callsc ≥ S0 −Xe−rT .

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Example. S0 = 20, X = 18, r=10%p.a., T=1 year. Then S0 −Xe−rT = 3.71. Assume that the option costs $3.Then an arbitrageur buys the call and shorts the stock. This provides cash 20 − 3 = 17, invested for one yearrisk-free, it gives 17e0.1 = 18.79. Then the option expires. If the stock price is greater than $18, the arbitrageurexercises the option and makes a profit of 18.79 − 18 = 0.79. If the stock price is less than $18, say $17, thenthe stock is bought in the market and the profit is 18.79− 17 = 1.79.

Proof of the lower bound. Consider two portfolios: A (one European call option and cash Xe−rT ) and B (oneshare). At time T , portfolio A is worth max(ST , X), while portfolio B is worth ST . Thus A must be worthmore than B today, meaning that c + Xe−rT ≥ S0.

European putsp ≥ Xe−rT − S0 .

For the proof consider two portfolios: A (one put option and one share) and B (cash Xe−rT ).

2.3. Put call parity for European options on non-dividend paying stock

Consider two portfolios:

A - one call option plus an amount of cash equal to Xe−rT ;

B - one put option plus one share.

At time T both portfolios are worth max(ST , X), so (because they are European) they must have identicalvalues today:

c + Xe−rT = p + S0 .

This is known as put-call parity.

2.4. Dividends-paying stock

Let D be the present (time zero) value of the future dividends, i.e. the future dividends are discounted.

Consider two portfolios:

A - one European call option and cash D + Xe−rT , where D is the present value of dividends;

B - one share.

Thenc ≥ S0 −D −Xe−rT .

Similarlyp ≥ D + Xe−rT − S0 .

Put-call parity for European optionsc + D + Xe−rT = p + S0 .

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2.5. American options (Early exercise)

Example. A trader owns a call option on a non-dividend paying stock with strike price $40 and the currentstock price $50 with one month to expiry. It is better not to exercise early, because $40 can earn interest fora month, and also the option is a protection against falling price of a stock. It is better to sell the option (orkeep the option and short the stock). Thus

C ≥ S0 −Xe−rT .

For dividends paying stock, it may be optimal to exercise a call option early.

In contrast to call options, it can be optimal to exercise a put option early. For example, let X = 10 and letthe stock price be zero. Then it is better to exercise immediately to get the maximal profit and invest it. Thus

P ≥ X − S0 .

There is no put-call parity for American options.

3. Binomial trees

3.1. Example (a wrong approach to option pricing)

Assume that now 1 Euro=$1.15 and in a month it will be worth either $0.75 or $1.45 with equal probabilities.Assume that interest rates are zero. Let a European call option have strike price $1.15. Then the expected gainon the option will be

V = (1.45− 1.15)× 0.5 + 0× 0.5 = 0.15 .

One might be tempted to think that this is the fair price of an option.

However an arbitrageur may do the following

Buy the option at price 0.15; lend x dollars; borrows y Euros at $1.15. His balance at time zero is−0.15− x + 1.15y.

In case of zero initial investment, the balance at time zero should be zero too, i.e.

−0.15− x + 1.15y = 0 so that x = 1.15y − 0.15 .

Starting with zero investment, in a month there are two possibilities.

• If the Euro has risen, the option is exercised, he buys y Euros at 1.45 and pays back the loan, so that hisbalance is

0.30− 1.45y + x = 0.3− 1.45y + 1.15y − 0.15 = 0.15− 0.3y .

• If the Euro has fallen, option is worthless, he buys y Euros at 0.75 and pays back the loan, so his balanceis

−0.75y + x = −0.75y + 1.15y − 0.15 = 0.4y − 0.15 .

Then there are some values of y, so that the end balance is positive in any case. For this, solve

0.15− 0.3y > 0, 0.4y − 0.15 > 0 ,

i.e. the choice 38 < y < 1

2 leads to a sure positive profit with zero investment.

18


3.2. Riskless portfolios

Assume that the current stock price is $20 and it can be either $22 or $18 at the end of three months. Wewould like to value a European call option to buy the stock for $21 in three months.

Stockprice =$20

Stock price = $22Option price = $1

Stock price = $18Option price = $0

Figure 2.1: Example of a binomial tree

Consider a portfolio

Long: ∆ sharesShort: 1 option

A riskless portfolio should provide the same return in the both cases, so

22∆− 1 = 18∆ ,

whence ∆ = 0.25 shares. Then the return is 18∆ = 4.5. The value of the portfolio today (suppose that therisk-free rate is 12%p.a.)

4.5e−0.12×0.25 = 4.367 .

The value of the stock price today is $20; if the option price is f , then the value of the portfolio today is

20∆− f = 5− f = 4.367 .

Thus, f = 0.633. In other words, this is a price of a bet, that pays $1 if the stock goes up.

3.3. Pricing on one-step binomial trees

In a general case, assume that the stock price may be S0u or S0d (so that u > 1 and d < 1) with option valuesfu and fd (at expiry the option value is the same as the option price).

fu

fd

S0u

S0dfS0

Figure 2.2: A general one-step binomial tree

ThenS0u∆− fu = S0d∆− fd ,

19


whence∆ =

fu − fd

S0u− S0d.

Let r be the risk-free interest rate. Then the present value of the portfolio should be equal to the cost of settingup the portfolio

(S0u∆− fu)e−rT = S0∆− f .

Thenf = S0∆− (S0u∆− fu)e−rT .

After substituting the value of ∆f = e−rT [qfu + (1− q)fd] ,

where

q =erT − d

u− d.

Note that if the value of q lies outside [0, 1], one has an immediate arbitrage opportunity.

This option price f does not refer to the probabilities of the stock moving up or down. Theprobabilities are incorporated into the price of the stock. Then q can be interpreted as theprobability of up movement, (1− q) is the probability of a down movement, and qfu +(1− q)fd

is the expected payoff from the option.

Then the expected stock priceE (ST ) = qS0u + (1− q)S0d = S0e

rT ,

meaning that the price grows, on average, risk-free.

The expectation with respect to the risk-free probabilities will be denoted by EQ, while the expectation in thereal world is EP.

3.4. Risk-neutral valuation

In a risk-neutral world all individuals are indifferent to risk, require no compensation for risk, and the expectedreturn on all securities is the risk-free interest rate. Thus the value of an option is its expected payoff in a risk-neutral world discounted at the risk-free rate.

Risk-neutral valuation principle states that it is valid to assume the world is risk-neutralwhen pricing options. The resulting option prices are correct in the real world as well.

Example. Let q be a probability of up movement in Figure 2.1 (in the risk-neutral world). Then

22q + 18(1− q) = 20e0.12×0.25 ,

so that q = 0.6523. The expected value of the option is then also $0.6523, it should be discounted at the risk-freerate as 0.6523e−0.12×0.25 = 0.633.

20


3.5. Two-step binomial trees

Figure 2.3 shows a two-step tree with two steps of three months long. The value of an option at node B is givenby (u = 1.1, d = 0.9, so that q = 0.6523)

e−0.12×0.25[0.6523× 3.2 + 0.3477× 0] = 2.0257 .

In a similar way it is possible to do calculations backwards.

201.2823

22

2.0257

18

0.0

24.23.2

0.0

16.20.0

A

B

C

D

E

F

19.8

Figure 2.3: Stock and option prices in a two-step tree. The strike price is $21.

f

S0u

fu

S0d

fd

S0u2

fuu

fud

S0d2

fdd

S0

S0ud

Figure 2.4: A general two-step tree

For a general two-step tree shown in Figure 2.4, u and d remain the same, so is q. Then, if each step takes time

21


∆t,

fu = e−r∆t[qfuu + (1− q)fud] ,

fd = e−r∆t[qfud + (1− q)fdd] ,

f = e−r∆t[qfu + (1− q)fd] .

Thusf = e−2r∆t[q2fuu + 2q(1− q)fud + (1− q)2fdd] .

3.6. Put options

Similarly, put options can be priced. For them, there is a difference between American and European options,as early exercise may be profitable, see Figure 2.5. Here the interest rate is 5%, and the strike price is 52$.There are two time steps of one year, and in each step the price goes up or down by 20%.

A

B

C

D

E

F

504.1923(5.0894)

60

1.4147

40

9.4636(12.0)

720

484

3220

Figure 2.5: Two-step tree for European and American put options (if different, American option prices are givenin parentheses)

3.7. Matching volatility with u and d.

Recall that the return on the stock is the logarithm of the ratio of its prices.

Let µ be the expected return of the stock and σ be its volatility. Assume S0 = 1. Then the probability p ofup-movement can be found from

pu + (1− p)d = eµ∆t ,

so that

p =eµ∆t − d

u− d.

The variance of the stock price after time ∆t is σ2∆t. On the other hand, this variance is

pu2 + (1− p)d2 − [pu + (1− p)d]2 .

22


From the equalityeµ∆t(u + d)− ud− e2µ∆t = σ2∆t

(if higher powers of ∆t are ignored),u = eσ

√∆t , d = e−σ

√∆t .

In a risk-neutral world, the probability of upward movement will be

q =er∆t − d

u− d.

If q is used instead of p in the formula for the variance, then the variance remains the same. The change of riskreferences (called also the change of measure) may lead to change of the expected return, but does not changethe variance (this is Girsanov’s theorem of stochastic calculus).

3.8. Delta hedging

Delta is the ratio of the change in the price of the stock option to the change in the price for the underlyingstock, e.g., for Figure 2.1,

∆ =1− 0

22− 18= 0.25 .

Note that ∆ may change over time.

Mathematically,

∆ =∂f

∂S.

Delta hedging requires buying ∆× (the value) shares of the stock.

Example. Stock price $100, ∆ = 0.6, the option price is $10. An investor sells 20 option contracts (i.e. optionsto buy 20×100 = 2000 shares). His position could be hedged by buying 0.6×2000 = 1200 shares. The gain/losson the option position would then tend to be offset by the loss/gain on the shares purchased. For example, ifthe stock price goes up by $1, the option price will tend to go up by $0.6, so loss=gain.

NOTE. Because ∆ changes, the investor’s position remains delta hedged for only a relatively short period oftime. The hedge has to be adjusted (known as rebalancing).

23


4. Stochastic processes on binomial trees

4.1. Main concepts

• The set of possible stock values is a stochastic process S, it depends on time as Si (discrete time) orSt (continuous time).

• The set of probabilities pj or qj associated with nodes of the tree is a measure P or Q. They describehow likely is to jump up or down.

• A filtration Fi is the history of the stock up until tick-time t = i on the tree.

• A claim X is a function of the nodes at a claim time-horizon T ; equivalently, it is an FT -measurablefunction. The claim is only defined on the nodes at time T .

• The conditional expectation operator EP(·|Fi). For example, EP(X|Fi) is the expectation of X alongthe latter position of paths which have initial segment Fi. This expectation depends on Fi and so is arandom variable itself. The claim X can be converted into a process EP(X|Fi) or EQ(X|Fi) if themeasure P or Q is given.

• A previsible process ϕ is a process on the same tree whose value at any given node at time tick i dependsonly on the history up to one time-tick earlier Fi−1. Previsible processes play the part of trading strategieswhere we cannot tell in advance where prices are going to go.

• A process S is a martingale with respect to a measure Q and a filtration Fi if E |Sj | < ∞ for all j and

EQ(Sj |Fi) = Si for all i ≤ j .

This means that S has no drift under Q. Then Q is called a martingale measure for S and S is calleda Q-martingale.NOTE: For any claim X, the process EQ(X|Fi) is a martingale.

4.2. Binomial representation theorem

Theorem. If S is Q-martingale and E is any other Q-martingale, then there exists a previsible process ϕ suchthat

Ei = E0 +i∑

k=1

ϕk∆Sk ,

where ∆Sk = Sk − Sk−1.Proof. Consider a typical node. As there are two values only, the random variables can be transformed to each

Snow

Sup

Sdown

Enow

Eup

Edown

Figure 2.6: Process S on left; process N on right

24


other by scaling and offset∆Ei = ϕi∆Si + k ,

for ϕi and k known by Fi−1. The conditional expectation given Fi−1 of ∆Si and ∆Ei should be zero, as theyare martingales. Thus, k = 0.

4.3. The bond process

• The bond process Bi represents the value of $1 at time i. It is a previsible and positive process, B0 = 1.

• The process B−1i is another previsible process called the discount process.

• Zi = B−1i Si is the discounted stock process.

• B−1T X is the discounted claim.

SummaryS stock price processq probability of up-jumpϕ stock holding strategyψ bond holding strategyB bond price process, B0 = 1X claim payoffT time of claim payoffV claim value at time zero (option price)

q =erT Snow − Sdown

Sup − Sdown

fnow = e−rT [qfup + (1− q)fdown] = EQ(B−1T X)

the expectation taken with respect to q’s

ϕ =fup − fdown

Sup − Sdownas in Delta hedging

ψ = B−1now(fnow − ϕSnow)

4.4. Self-financing strategies

Let the discounted stock process Zi = B−1i Si be a Q-martingale. Another Q-martingale is Ei = EQ(B−1

T X|Fi).Then there exists a previsible process ϕ such that

Ei = E0 +i∑

k=1

ϕk∆Zk .

At time i buy the portfolio Πi with

• ϕi+1 units of stock;

• ψi+1 = (Ei − ϕi+1B−1i Si) units of the cash bond.

25


At time i, the portfolio is worth Vi = ϕi+1Si + ψi+1Bi. At time zero, portfolio Π0 is worth

ϕ1S0 + ψ1B0 = E0 = EQ(B−1T X) .

After one time tick Π0 is worth

ϕ1S1 + ψ1B1 = B1(E0 + ϕ1(B−11 S1 −B−1

0 S0)) = B1(E0 + ϕ1∆Z1) = B1E1

by the binomial representation theorem. On the other hand, the portfolio Π1 scheduled to be acquired at time 1costs also B1E1, so we can cash Π0 to buy Π1. This strategy is called self-financing. At the end the portfoliowill be worth BT B−1

T X = X, as required. Thus the claim price is EQ(B−1T X).

For a general step i, note that

Vi = ϕi+1Si + ψi+1Bi = ϕi+1Si + BiEi − ϕi+1Si = BiEi .

Therefore,

ϕi+1Si+1 + ψi+1Bi+1 = Bi+1(ψi+1 + B−1i+1ϕi+1Si+1)

= Bi+1(Ei − ϕi+1B−1i Si + B−1

i+1ϕi+1Si+1)

= Bi+1(Ei + ϕi+1∆Zi) = Bi+1(Ei + ∆Ei)= Bi+1Ei+1 .

Repeating this until day T , we see that the value of the portfolio will be equal to BT ET .

Self-financing hedging strategy (ϕi, ψi)

• both ϕ and ψ are previsible

• the change in value V of the portfolio defined by the strategy obeys the difference equation Vi+1 − Vi =∆Vi = ϕi+1∆Si + ψi+1∆Bi

• ϕT ST + ψT BT is identically equal to the claim X

Option price formula (discrete case)The value at time i of a claim X maturing at date T is

BiEQ(B−1T X|Fi) .

A martingale measure Q always exists in the binomial model.The real measure P (which S follows) is irrelevant.

26


5. Continuous processes

5.1. Heuristic arguments

Let Bt = ert and let S0 be the initial stock price.

Single small step. Assume that over a small time interval ∆t the stock moves to either value S0eµ∆t+σ

√∆t

(if up) or to S0eµ∆t−σ

√∆t (if down). Here µ denotes the return on the stock, so that the expected price after

time ∆t is Seµ∆t. Further σ is the volatility, see Section 3.7.

The expression for the probability of up-movement from Section 3.7 yields that

p =S0e

µ∆t − Sdown

Sup − Sdown=

S0eµ∆t − S0e

µ∆t−σ√

∆t

S0eµ∆t+σ√

∆t − S0eµ∆t−σ√

∆t=

1− e−σ√

∆t

eσ√

∆t − e−σ√

∆t

≈ 12

(1− 1

2σ√

∆t)

,

where the exponentials is decomposed using the Taylor expansion up to the second order.

Many steps. Real world. If n = t/∆t, then

St = S0 exp

µt + σ√

t

(2Xn − n√

n

),

where Xn is the total number of up-jumps. The random variable Xn has the binomial distribution Bi(n, p).Thus, (2Xn − n)/

√n has mean

2np− n√n

= − 12σ√

n∆t = − 12σ√

t .

and the variance4np(1− p)

n= (1− 1

2σ√

∆t)(1 + 12σ√

∆t) = (1− 12σ

2∆t) ≈ 1 .

By the central limit theorem, (2Xn − n)/√

n converges to N(− 12σ√

t, 1), which can be represented as Z − 12σ√

tfor the standard normal random variable Z. Thus, St can be expressed as

St = S0 exp

µt + σ√

t(Z − 12σ√

t)

= S0 exp

(µ− 12σ

2)t + σ√

tZ

Thus log(St) has the normal distribution and so St has the log-normal distribution.

Stock prices are widely believed to be log-normally distributed.

Many steps. Risk-neutral world. In the risk-neutral world, the probability of up-movement is

q =S0e

r∆t − Sdown

Sup − Sdown≈ 1

2

(1−

√∆t

(µ + 1

2σ2 − r

σ

)).

Then Xn ∼ Bi(n, q), and (2Xn − n)/√

n has mean −√t(µ + 12σ

2 − r)/σ and the variance 1, and so converges toN(−√t(µ + 1

2σ2 − r)/σ, 1). Finally, log St ∼ N(log S0 + (r − 1

2σ2)t, σ2t) or

St = S0 expσ√

tZ + (r − 12σ

2)t ,

where Z ∼ N(0, 1) under the risk-neutral probability measure Q.

27


5.2. Stock price process

Assume that the volatility vanishes, i.e. σ = 0. In this case

dS = µSdt ,

where µ is the expected return on the stock. Then

dS

S= µdt

andSt = S0e

µt .

The volatility of stock prices is modelled using the Brownian motion (also called Wiener process). Thestandard Brownian motion Wt is a stochastic process, such that

• W0 = 0;

• Wt −Ws is normally distributed with mean zero and the variance t− s for t ≥ s;

• Wt has independent increments, i.e. Wtn − Wtn−1 , . . . , Wt2 − Wt1 are jointly independent for any timemoments t1 ≤ t2 ≤ · · · ≤ tn.

The Brownian motion has continuous but nowhere differentiable paths.

Let Wt be the standard Brownian motion. Then

dS = µSdt + σSdW

or with time as subscriptdSt = µStdt + σStdWt .

5.3. Stochastic differential equations

In a more general case Xt is a general stochastic process (not necessarily stock price process) such that

dXt = µtdt + σtdWt .

The drift µt and the volatility σt may be random, but they must be adapted to the same filtration, so dependon the the events up to the current time (i.e. Ft), but not on the future. The drift and volatility determineuniquely the underlying process and can be determined uniquely from Xt.

Stochastic differential equation for Xt

dXt = µ(Xt, t)dt + σ(Xt, t)dWt .

Example: A stock pays dividends at 15%p.a. with continuous compounding and has a volatility of 30%p.a.Then

dS = 0.15Sdt + 0.30SdW .

In other words,∆S

S= 0.15∆t + 0.30Z

√∆t

where Z ∼ N(0, 1).

28


5.4. Ito’s formula

If dXt = µtdt + σtdWt and f is a deterministic twice continuously differentiable function, then Yt = f(Xt)satisfies

dYt = (µtf′(Xt) + 1

2σ2t f ′′(Xt))dt + (σtf

′(Xt))dWt .

More generally, let Yt = F (Xt, t) for a twice differentiable function F . Then

dYt =∂F

∂XtdXt +

∂F

∂tdt +

12

∂2F

∂X2t

σ2t dt

=[

∂F

∂Xtµt +

∂F

∂t+

12

∂2F

∂X2t

σ2t

]dt +

∂F

∂XtσtdWt .

This formula transforms a stochastic differential equation for Xt into a stochastic differential equation for Yt.

Example. X = W , Y = X2. ThendYt = d(W 2

t ) = dt + 2WtdWt .

This can be used to find∫ T

0WtdWt, since

T∫

0

dYt =

T∫

0

dt +

T∫

0

2WtdWt ,

whenceT∫

0

WtdWt = 12W

2T − 1

2T .

Exercise. Let dSt = µSdt + σSdWt for the (non-dividend paying) stock price. Find the differential for theforward price Ft = Ste

r(T−t).

5.5. Geometric Brownian motion

ConsiderdSt = µStdt + σStdWt .

Its solution (prove it using Ito’s formula) is

St = S0 expσWt + (µ− 12σ

2)t

called the geometric (or exponential) Brownian motion.

This formula gives the stock price in the real world if the stock follows the Geometric Brownianmotion with given µ and σ.

Note: Wt has the normal distribution with mean zero and the variance t, so that

St = S0eη ,

where η has the normal distribution with mean (µ− 12σ

2)t and the variance σ2t.

29


5.6. Estimating volatility

Take a sample of daily returnsui = log(Si/Si−1)

for i = 1, . . . , n and then the standard deviation of this sample will estimate σ√

τ , where τ = 1/252 is thelength of time interval in years (a year is assumed to have 252 trading days). The choice of n is crucial, moreobservations do not necessarily mean a better estimate as the volatility may change over time.

When dealing with ui it is appropriate to give latter observations more weights, for example, use exponentiallydecreasing weights. It is possible to use various models from Time Series.

6. Martingales

A process Mt is called a Q-martingale if Mt is Ft measurable for all t,

• EQ|Mt| < ∞ for all t;

• EQ(Mt|Fs) = Ms for all s ≤ t.

Examples:(1) A constant process is a martingale.(2) Q-Brownian motion is a Q-martingale.(3) Mt = expσWt − 1

2σ2t is a martingale.

(4) For any claim X (with EP|X| < ∞), the process Nt = EP(X|Ft) is a P-martingale.

For pricing options, the central point was to ensure that the process is a martingale with respect to somemeasure. The price of derivatives then becomes the expectation with respect to this martingale measure.

6.1. Self-financing portfolios

Portfolio is a pair of processes ϕt and ψt which describe the number of units of security and of the bond whichwe hold at time t. The processes can take positive or negative values (short-selling is allowed). The securitycomponent ϕ should be F-previsible. The value of the portfolio is Vt = ϕtSt + ψtBt.

A portfolio is self-financing if and only if the change in its value only depends on the change of the assetprices, i.e.

dVt = ϕtdSt + ψtdBt .

Examples (assume that St = Wt and Bt = 1).(1) ϕt = ψt = 1 is self-financing(2) ϕt = 2Wt, ψt = −t−W 2

t is self-financing (use Ito’s formula).

If X is a claim that depends on events up to time T , then a replicating strategy for X is a self-financing

portfolio (ϕ,ψ) such that∫ T

0σ2

t ϕ2t dt < ∞ and X = VT = ϕT ST + ψT BT . Then the price of X at time t must

beVt = ϕtSt + ψtBt .

Idea: take the stock model, use the change of measure to change it into a martingale and thenuse the martingale representation theorem to create a replicating strategy for each claim.

30


6.2. Change of measure

If γ is a constant, and Wt is a P-Brownian motion, then there exists a measure Q such that

Wt = Wt + γt

is a Q-Brownian motion. This change of measure theorem is a particular case of the Cameron-Martin-Girsanov theorem from stochastic calculus.

The change of measure changes only the drift; the volatility remains the same.

Example. Xt = σWt + µt, where Wt is a P-Brownian motion. Then, with γt = µ/σ, there exists a measure Qsuch that Wt = Wt + (µ/σ)t is a Q-Brownian motion up to time T , i.e. Xt = σWt.

6.3. Black-Scholes model

Published in 1973, later Black, Scholes and Merton received the Nobel Prise in Economics.

Assumptions:Bt = ert , dSt = µStdt + σStdWt .

Short-selling is permitted, no transaction costs or taxes, all securities are perfectly divisible, there are nodividends, there are no riskless arbitrage opportunities, security trading is continuous, the risk-free interest rateis constant and the same for all maturities.

Take the discount process B−1t and form a discounted stock Zt = B−1

t St and a discounted claim B−1T X.

Step 1. Make Zt into a martingale.

Zt = B−1t St = S0 expσWt + (µ− r − 1

2σ2)t

= S0 expσ(Wt +µ− r

σ)− 1

2σ2t

= S0 expσWt − 12σ

2t

for a Q-Brownian motion Wt, so that Zt is a Q-martingale. Then St = BtZt = ertZt and so

St = S0 expσWt + (r − 12σ

2)t

determines the stock price process in the risk-neutral world. The corresponding stochastic differential equationis

dSt = rStdt + σStdWt .

Step 2. Et = EQ(B−1T X|Ft) is the value of the discounted claim at time t.

Step 3. By martingale representation theorem, dEt = ϕtdZt.

31


Replicating strategy.

• hold ϕt units of the stock at time t

• hold ψt = Et − ϕtZt units of the bond

Let us check that so defined portfolio is self-financing. Its value at time t is

Vt = ϕtSt + ψtBt = BtEt .

Then

dVt = BtdEt + EtdBt = ϕtBtdZt + EtdBt

= ϕtBtdZt + (ϕtZt + ψt)dBt

= ϕt(BtdZt + ZtdBt) + ψtdBt

= ϕtd(BtZt) + ψtdBt

= ϕtdSt + ψtdBt .

Thus, (ϕt, ψt) is self-financing. Note that we have used the following fact: if Bt is zero volatility process, thend(BtXt) = XtdBt + BtdXt.

A strategy (ϕt, ψt) of holding in a stock St and a non-volatile cash bond Bt has value Vt =ϕtSt + ψtBt and discounted value Et = ϕtZt + ψt.The strategy is self-financing if dVt = ϕtdSt + ψtdBt or, equivalently, dEt = ϕtdZt.

All claims X, knowable up to some horizon, T have associated replicating strategies. Thearbitrage price of such a claim at time t is given by

Vt = BtE Q(B−1T X|Ft) = e−r(T−t)E Q(X|Ft)

where Q is a measure that makes the discounted stock a martingale.

6.4. Example: pricing of a call option

Consider a call option with the exercise date T and strike price k. Then the claim is X = max(ST − k, 0) .Find V0, the value of the replicating strategy (and thus the option) at time zero as

V0 = e−rT EQ(max(ST − k, 0)) .

We should find the marginal distribution of ST under Q

ST = S0 expσWT + (r − 12σ

2)T .

Since WT ∼ N(0, T ) under Q,σWT + (r − 1

2σ2)T = Z + rT ,

where Z ∼ N(− 12σ

2T, σ2T ). ThenST = S0e

Z+rT ,

32


so that the claim price is

V0 = e−rT E ((S0eZ+rT − k)+) =

1√2πσ2T

∞∫

log(k/S0)−rT

(S0ex − ke−rT ) exp

− (x + 1

2σ2T )2

2σ2T

dx .

Change of variables v = −(x + 12σ

2T )/σ√

T leads to

V0 =1√2π

a∫

−∞

(S0e

−σ√

Tv− 12 σ2T − ke−rT

)e−

12 v2

dv ,

where a = (log(S0/k) + (r − 12σ

2)T )/σ√

T . Then writing

(−σ√

Tv − 12σ

2T − 12v

2) = − 12 (v + σ

√T )2

yields

V0 =S0√2π

a+σ√

T∫

−∞e−

12 v2

dv − ke−rT

√2π

a∫

−∞e−

12 v2

dv .

This leads to Black-Scholes formula

V0 = S0Φ

(log S0

k + (r + 12σ

2)T

σ√

T

)− ke−rT Φ

(log S0

k + (r − 12σ

2)T

σ√

T

).

We only need that the drift of the stock price is constant, its exact value is immaterial.

The Black-Scholes formula may be written as

c = S0Φ(d1)− ke−rT Φ(d2) .

Note that d2 = d1 − σ√

T .

The corresponding price formula for a European put is

p = ke−rT Φ(−d2)− S0Φ(−d1) .

Implied volatility can be derived from option prices for a particular stock. They can be used to monitor themarket’s opinion about the volatility of a particular stock.

NOTE. A variant of Black-Scholes formula for call price on a general asset with futures price F

V0 = e−rT FΦ(d1)− kΦ(d2) ,

where

d1 =log(F/k) + 1

2σ2T

σ√

T.

This formula has wider applicability, also for indices, dividend paying stocks and currencies.

33


6.5. Foreign exchange

For foreign currency options consider three instruments

• dollar bond Bt = ert (domestic)

• sterling bond Dt = eut, where u is the sterling interest rate (foreign)

• exchange rate Ct = C0 expσWt + (µ− 12σ2)t

Sterling cash bond is not directly tradable, what is tradable is St = CtDt. Then we can use the Black-Scholesapproach, find Q that makes Zt = B−1

t St = B−1t CtDt a martingale.

Implementing this:Zt = C0 expσWt + (µ + u− r − 1

2σ2)t .

By the change of measure theoremZt = C0 expσWt − 1

2σ2t

and Ct = BtD−1t Zt, whence

Ct = C0 expσWt + (r − u− 12σ

2)t .

Then the forward price is F = e(r−u)T C0 and the price for a European call is

V0 = e−rT

FΦ

(log(F/k) + 1

2σ2T

σ√

T

)− kΦ

(log(F/k)− 1

2σ2T

σ√

T

).

6.6. Dividend paying stock

Example (European call option on a dividend paying stock).Ex-dividend dates in two months and five months. The dividend on each ex-dividend date is expected to be$0.50. The current share price is $40, the volatility is 30%p.a., the risk-free rate of interest is 9%p.a. The timeof maturity is 6 months. The present rate of the dividends is

0.5e−2/12×0.09 + 0.5e−5/12×0.09 = 0.9741 .

The option price can be calculated from the Black-Scholes formula with S0 = 39.0259, k = 40, r = 0.09, σ = 0.3,and T = 0.5. The result is $3.67.

Stocks paying dividends continuously. If the stock is paying dividends at rate q (continuously), then theBlack-Scholes formula is applicable with S0 replaced by the dividend-adjusted stock price S0e

−qT . This is alsoapplicable for options on stock indices.

Alternatively, options on shares paying dividends continuously can be regarded as options on foreign currencywith the foreign risk-free interest rate u replaced by the dividends rate q.

Then the forward price is F = e(r−q)T S0 and the price for a European call is again

V0 = e−rT

FΦ

(log(F/k) + 1

2σ2T

σ√

T

)− kΦ

(log(F/k)− 1

2σ2T

σ√

T

).

NOTE. The same result is obtained if one adjusts S0 for the dividends and uses the standard Black-Scholesformula on non-dividend paying stock with S0 replaced by S0e

−qT .

34


6.7. Finding the replicating strategy

The value of the claim X at time t is equal to V (St, t) as in Black-Scholes formula with St (stock price at timet) instead of S0 and (T − t) (the remaining time) instead of T .

Since dSt = σStdWt + rStdt, Ito’s formula yields

dV (St, t) =∂V

∂StdSt +

(∂V

∂t+ 1

2σ2S2

t

∂2V

∂S2t

)dt .

On the other hand, the self-financing condition gives

dVt = ϕtdSt + ψtdBt .

By equalling coefficients of dSt, we get

ϕt =∂V

∂St(St, t)

meaning that the amount of stock at any stage is the derivative of the option price with respect to the stockprice.

Compare the values of the replicating strategy V (St, t) = ϕtSt + ψtBt with option price given by the Black-Scholes formula

V (St, t) = StΦ(

log(St/k) + (r + 12σ

2)(T − t)σ√

T − t

)− ke−r(T−t)Φ

(log(St/k) + (r − 1

2σ2)(T − t)

σ√

T − t

)

to see that the replicating strategy is given by

ϕt = Φ(

log(St/k) + (r + 12σ

2)(T − t)σ√

T − t

).

and

Btψt = −ke−r(T−t)Φ(

log(St/k) + (r − 12σ

2)(T − t)σ√

T − t

).

The value of ϕt is always between zero and one. The amount of borrowing is bounded by the exercise price k.

6.8. Black-Scholes differential equation

Consider a portfolio with one long option and short position in the asset

Π = V (St, t)−∆S .

Then dΠ = dV −∆dS and using Ito formula for Vt = V (St, t)

dV =∂V

∂tdt +

∂V

∂SdS +

12σ2S2 ∂2V

∂S2dt .

Therefore

dΠ =∂V

∂tdt +

∂V

∂SdS +

12σ2S2 ∂2V

∂S2dt−∆dS .

The coefficient before dS is zero if∆ =

∂V

∂S

35


as in Delta hedging. With such ∆,

dΠ =(

∂V

∂t+

12σ2S2 ∂2V

∂S2

)dt .

This change is riskless and must be equal to the risk-free growth (to avoid arbitrage possibilities)

dΠ = rΠdt .

Thus, (∂V

∂t+

12σ2S2 ∂2V

∂S2

)dt = r

(V − S

∂V

∂S

)dt ,

and we obtain the Black-Scholes equation

12σ2s2 ∂2V

∂s2+ rs

∂V

∂s− rV +

∂V

∂t= 0 .

This is a partial differential equation that must be solved under a boundary condition V (ST , T ) = f(ST , T ).

6.9. The Greek letters of a portfolio

Π is a portfolio dependent on a single asset S.

∆ =∂Π∂S

=n∑

i=1

wi∆i ,

where ∆i =∂ci

∂Sis the Delta of particular options ;

Θ =∂Π∂t

;

Γ =∂2Π∂S2

.

They satisfy the Black-Scholes differential equation

12σ

2S2Γ + rS∆ + Θ = rΠ .

The following characteristics are also often used:

V =∂Π∂σ

Vega ,

ρ =∂Π∂r

rho .

6.10. American options

The buyer of an American option has the choice when to stop and this choice can only use price informationup to the present moment. Such a random time τ is called a stopping time. Then the payoff for a call optionis (Sτ − k)+. The cost of hedging that payoff will be

EQ(e−rτ (Sτ − k)+) .

But the writer does not know which time τ will be used, so he has to prepare for the worst and charge

V0 = maxτ

EQ(e−rτ (Sτ − k)+) .

36


Pricing derivatives with optionalityIf A is a set of possible choices and Xa is the payoff under a ∈ A, then the option writer should charge

V0 = maxa∈A

EQ(e−rT Xa) .

If the purchaser does not exercise the option optimally, then the issuer’s hedge will produce a surplus by dateT .

American calls on non-dividend paying stock should not be exercised early.

6.11. Exotic options

It is often possible to represent some complicated derivatives as combinations of call and put options.

Example. A five-year contract pays out 90% times the ratio of the terminal and initial values of FTSE. Or itpays out 130% if otherwise it would be less or 180% if otherwise it would be more. Assume FTSE drift µ = 7%,FTSE volatility σ = 15%, FTSE dividend rate δ = 4% (the compound dividends payments from 100 stocksused to compose the FTSE), UK interest rate r = 6.5%. The claim

X = min(max(1.3, 0.9ST ), 1.8) ,

where T = 5 and S0 = 1. Rewrite X as

X = 1.3 + 0.9((ST − 1.44)+ − (ST − 2)+) .

Thus, X is the difference of the two FTSE calls plus some cash. The forward price F = e(r−δ)T S0 = 1.133. Thetwo calls can be valued at 0.0422 and 0.0067, so that

V0 = 1.3e−rT + 0.9(0.0422− 0.0067) = 0.9712 .

• forward start options (start some time in the future; used in employee incentive schemes)

• compound options (e.g., a call on a call with two strike prices)

• chooser options (the holder can choose whether the option is a call or a put)

• barrier options (the payoff depends on whether the asset’s price reaches a certain level during a certainperiod of time)

• binary options (pays nothing if the stock price ends up below the strike price and pays a fixed amount ifit ends above the strike price)

• lookback options (payoff depends on the maximum or minimum stock price during the life of an option)

• a shout option (a European option where the holder can “shout” to the writer once; at the end the optionholder receives either the usual payoff or the intrinsic value at the time of the shout, whichever is greater)

• Asian options (the payoff depends on the average price of the asset during at least some part of the life ofan option)

• basket options (the payoff depends on the value of a portfolio of assets), etc.

37

3. Optimal portfolios

If properly hedged, derivatives provide riskless returns. This is used by banks who sell their products for a bitmore than it is worth. Fund managers buy and sell assets (and also derivatives) with the aim of beating thebank’s rate of return. This often involves taking a degree of risk.

1. Mean-variance approach

The mean-variance approach was developed by H. Markowitz (Nobel Prise winner in economics).

Let Pi(t) be the price of security i at time t, so that Pi(0) its initial price.The returns Ri = (Pi(t)− Pi(0))/Pi(0) are modelled as random variables with

µi = E (Ri) , σij = Cov(Ri, Rj) .

The variance-covariance matrix of returns is denoted by Σ. If the first asset is risk-free, then the correspondingvariance and covariances are zeros.

Let πi be the fraction of initial wealth X of the investor invested in security i at time t = 0

πi =ψiPi(0)

X,

where ψi is the number of shares of security i held by investor at time t = 0.π = (π1, . . . , πd)> is called the portfolio vector. Note that

π1 + π2 + · · ·+ πd = 1 .

If short-selling is not allowed, portfolios should have all non-negative components. Such portfolios are calledadmissible. The total return is given by

R =d∑

i=1

πiRi .

Then

E(R) = E (R) =d∑

i=1

πiµi ,

V (E) = Var(R) =d∑

i,j=1

πiσijπj .

1.1. Mean-variance principle

A For a given upper bound σ2max on σ2, choose an admissible portfolio π such that µ(π) is maximal.

38


B For a given lower bound µlow on µ for the mean of the portfolio, choose an admissible portfolio with thesmallest variance.

Mathematical formulations of this principle is possible as a quadratic programming problem.

risk

return

σ2σ2min

µmax

µ

Figure 3.1: Mean-variance efficient set (shown as a thick line)

1.2. Two assets portfolios

Then π = (π1, π2)> with π1 + π2 = 1. For simplicity, denote π1 = x and π2 = 1− x. Then

E(R) = xµ1 + (1− x)µ2 ,

V (R) = x2σ21 + (1− x)2σ2

2 + 2x(1− x)ρσ1σ2 ,

where ρ is the correlation coefficient between the two individual returns. If short sales are forbidden, then0 ≤ x ≤ 1, otherwise x can be an arbitrary number.

Then V (R) is a quadratic function of E(R), namely

V (R)(µ1 − µ2)2 = (E(R)− µ2)2σ21 + (µ1 − E(R))2σ2

2 + 2(E(R)− µ2)(µ1 − E(R))σ1σ2 .

If the short sales are allowed, then the minimum variance is achieved at

x∗ =σ2

2 − ρσ1σ2

σ21 + σ2

2 − 2ρσ1σ2

(derive it!). Note that the corresponding x may be negative or greater than 1. If short sales are forbidden, thenthe optimal x can be found as either 0 or 1 or x∗ given above (if x∗ ∈ (0, 1)).

The objective function may be written as

f(π) = −AE(R) + V (R) ,

where A is a risk aversion index. If A = 0, the portfolio with the lowest variance of return will be selected. AsA increases, the investor becomes more willing to accept risk in order to achieve a higher expected return, and

39


risk

return

σ2σ2min

µ

Figure 3.2: Mean-variance relationship and the efficient frontier for a two-asset portfolio

if A = ∞ the portfolio with the highest expected return will be optimal. In application to two asset portfolios,this gives

f(x) = −A(xµ1 + (1− x)µ2) + x2σ21 + (1− x)2σ2

2 + 2x(1− x)ρσ1σ2 .

Thendf

dx= −A(µ1 − µ2) + 2xσ2

1 − 2(1− x)σ22 + 2(1− 2x)ρσ1σ2 .

The derivative is zero if

x∗ =12A(µ1 − µ2) + σ2

2 − ρσ1σ2

σ21 + σ2

2 − 2ρσ1σ2.

If short sales are forbidden, then the optimal value of x is either 0 or 1 or x∗ above (if x∗ ∈ (0, 1)).

2. Risk-free assets and the capital market line

Assume the the portfolio Rp has two assets, and one is risk-free. The other (risky asset) may be interpreted as aportfolio R of several risky assets with return E(R) and variance V (R). Let x be the share of the fund investedin the first (risky) asset. Assume that the risk-free asset provides return r. Then the asset of the combinedportfolio Rp will be

E(Rp) = xE(R) + (1− x)r ,

V (Rp) = x2V (R) .

Then, if σ(R) =√

V (R) is the standard deviation of the risky portfolio and σ(Rp) =√

V (Rp),

E(Rp) = r +E(R)− r

σ(R)σ(Rp) .

Thus, the set of portfolios comprising a combination of the risk-free asset and a risky portfolio is a straight line.The tangent point M to the set of admissible portfolios represents the market portfolio or the best combinationof risky assets. The optimal position on the capital market line corresponds to the degree of risk preferred. Ifcontinued beyond M , it represents a ‘borrowing’ portfolio.

40


RF

capital market line

σM

ME (RM)

Attainableportfolios

Figure 3.3: Capital market line

If the borrowing rate RB is different from the lending rate r, then two tangent points are to be determined, seeFigure 3.4.

Attainableportfolios

RF

RB

PL

PB

Figure 3.4: Different borrowing and lending rates

41


3. Capital Asset Pricing Model (CAPM)

Assume that the return of a security is determined entirely by the market index and random factors

Rj = αj + βjRM + εj ,

where RM is the return of the market portfolio and εj is a random error term.

Note that,

Cov(Rj , RM ) = E (αjRM + βjR2M + εjRM )−E (αj + βjRM + εj)E (RM )

= βj(E (R2M )− (E (RM ))2) = βj Var(RM ) .

Therefore, the beta of an asset Rj can be also defined as βj = Cov(Rj ,RM )Var(RM ) .

The following is a very important representation of the expected returns

E (Rj) = r + βj [E (RM )− r] ,

where r is the risk-free rate; E (Rj) is the expected return of asset j; E (RM ) is the expected return on themarket portfolio; βj is the “beta” of asset j.

In order to prove this representation, consider a portfolio M ′ with the asset Aj taken out of it. The convexcombinations of Aj and M ′ can be drawn on (σ, µ)-plane as a curve that joints the points corresponding to Aj

and M ′. This curve passes through point M such that the tangent to the curve at M coincides with the capitalmarket line. This slope equals (E (RM )− r)/σM . The part of this curve that joins M and Aj has the followingparametric representation as

rt = (1− t)E (RM ) + tE (Rj)

σt =√

(1− t)2σ2M + t2σ2

j + 2(1− t)tCov(Rj , RM ) .

After differentiation at point t = 0 we find

r′t = −E (RM ) + E (Rj)σ′t = −σM + βjσM .

Sincer′tσ′t

=E (RM )− r

σMat t = 0 ,

we immediately obtain that E (Rj) = r + βj [E (RM )− r].

The beta’s can be estimated from the following linear model:

rj,t = αj + βjrM,t + εj,t ,

where rj,t is the return of asset j in the tth period, rM,t is the return of the market portfolio in the tth period,αj is the intercept, εj,t is the residual error for period t.

Further,Var(Rj) = β2

j Var(RM ) + Var(εj)

and, for i 6= j, Cov(Ri, Rj) = βiβj Var(RM ) .

The beta of a portfolio can be found as the weighted sum of beta’s of individual assets

βΠ =d∑

i=1

πiβi .

42


4. Hedging the value of a portfolio

β describes relationship between the expected return of a portfolio and market performance. The correct numberof contracts to hedge the value of a portfolio is

βP

A

where P is the value of a portfolio and A is the value of assets underlying one futures contract (on stock indices).

Example. A company hedges a portfolio worth $2,100,000 over the next three months using S&P 500 indexfutures with 4 months to maturity. The current level of S&P is 900 and β = 1.5. The value of assets underlyingone futures contract is 900x250=$225,000. The number of contracts to short

1.5× 2, 100, 000225, 000

= 14 .

Suppose the risk-free rate is r=4% per year (so 1% per three months), while the market provides return of-7% in the course of the next three months. Thus, the return on portfolio is 1.5×8%=12% below the risk-freerate or -11%. Assume that the stocks underlying the index provide a dividend of 2% per annum (0.5% perthree months). This means that the index declined by 7.5%, from 900 to 832.5. The initial futures price is900e(0.04−0.02)×1/3 = 906.02, the final futures price is 832.5e(0.04−0.02)×1/12 = 833.89. The gain on the futuresposition is

(906.02− 833.89)× 250× 14 = 252, 455 .

The total gain/loss on the portfolio is −0.11× 2, 100, 000 = 231, 000. The net gain from the hedged position is21,455 or about 1% of the value of the portfolio (the risk-free rate).

Why hedge and not to invest risk-free? Because the hedger feels that the stocks in the portfolio have been chosenwell. A hedge removes the risk from market moves and leaves the hedger exposed only to the performance ofthe portfolio relative to the market.

5. Risk-assessment of a portfolio

The Value-at-Risk (VaR) at level α is the maximal loss, which is not exceeded at a given time point withprobability α. If X is the random variable representing the loss, then

V@Rα(X) = infx : P X > x ≤ 1− α .

The value of α is typically chosen to be large, e.g. 0.99.

Note that if X represents a gain, then one should consider its lower tail.

If X is normally distributed with mean zero, then its Value-at-Risk is proportional to the standard deviation,i.e. the volatility. Assume further that the loss is normally distributed. Let σyr be the volatility per annum.Then the daily volatility (assuming 252 trading days) is

σday = σyr/√

252 ≈ 0.06σyr .

Here all volatilities are daily. They correspond to the standard deviation of the price change in a day.

In the geometric Brownian motion model we have

dS = µSdt + σSdWt .

The first return term is much smaller than the volatility contribution ans is typically neglected. Thus, thechange δS in the value of the asset is approximated by σSZ

√δt, so that dWt is written as the product of the

43


standard normal variable Z and the time interval δt. If the daily volatility is used, then the time should be alsomeasured in days. Thus, the Value-at-Risk of the price change is the product of σS

√δt and the corresponding

quantile of the standard normal distribution.

V@R calculationExample. Portfolio $10 million shares of IBM, σday = 0.02. We can ignore expected returns of shares as theyare smaller than the volatility (say 13%p.a.=0.05%p.day.). The standard deviation of the value of a portfolioover N = 10 days is σ

√10× 10 · 106=$632,456. As Φ(−2.33) = 0.01 (c.d.f. of N(0, 1)), the value at risk 99% of

the portfolio is 2.33× 632, 456 = $1, 473, 621.

For two-asset portfolio

σX+Y =√

σ2X + σ2

Y + 2ρσXσY .

This shows that the diversification of normally distributed assets reduces V@R.

Linear model. Recall that the change of value for an option satisfies

δf

δS= ∆ ,

where ∆ if called the Delta of the option f . Thus,

δf = ∆δS = ∆σSZ√

δt .

If the portfolio Π contains various options, then the linear model involves Delta of options:

δΠ =n∑

i=1

∆iSiδxi ,

where ∆i is the delta of the portfolio with respect to the ith market variable (the partial derivative of thevalue with respect to the ith stock price), Si is the value of the ith market variable, and δxi = σiZi

√δt is the

(random) relative change of value of the ith variable. Note that δxi may be dependent.

Example. Consider portfolio that consists of options on IBM shares with ∆ = 1, 000 at $120 per share andoptions on AT&T shares with ∆ = 20, 000 at $30 per share. Then

δΠ = 120× 1, 000× δx1 + 30× 20, 000× δx2 = 120, 000δx1 + 600, 000δx2

assuming that the daily volatility of IBM is 2%, and AT&T is 1% and the correlation is 0.7, the st. deviationof δΠ (in thousands of USD)

√(120× 0.02)2 + (600× 0.01)2 + 2× 120× 0.02× 600× 0.01× 0.7 ≈ 7.869 .

The 5-day 95% V@R is1.65×

√5× 7, 869 = $29, 033 .

Quadratic model involves Γ’s of options and their cross-gammas γij = ∂2Π/∂Si∂Sj

δΠ =n∑

i=1

Si∆iδxi +n∑

i=1

n∑

j=1

12SiSjγijδxiδxj .

In case of options on a single-stock we obtain

δΠ = S∆δx +12S2Γ(δx)2 .

44


6. Risk measures

It should be noted that the diversification reduces the Value-at-Risk not for all underlying distributions of theloss.

Example. Consider 100 independent bonds, which all have the same default probability q = 0.02 and thecurrent price 100. In case of no default until maturity, each of these bonds will pay back 105 and nothingotherwise. The loss of bond i is thus

Li =

100 with probability 0.02−5 with probability 0.98

Two portfolios, each with an initial value 10′000 are compared. Portfolio A is fully concentrated in one singlebond. Portfolio B is diversified and invests the same amount in each of the 100 bonds. The Value-at-Risk atlevel α = 0.95 for A is given by

V@R(LA) = V@R(100L1) = 100V@R(L1) = −500,

meaning that at level 0.95 we do not see any loss, but only a gain from A.

In order to calculate the loss on B, let Yi be one if bond i defaults and zero otherwise. Then

Li = 100Yi − 5(1− Yi) = 105Yi − 5

and the loss on B is given by

LB =100∑

i=1

Li = 105100∑

i=1

Yi − 500 .

Thus,V@R(LB) = 105V@R(S)− 500 ,

where S =∑100

i=1 Yi has the binomial distribution Bi(100, 0.02). Thus, V@R(S) = 5, so that

V@R(LB) = 525− 500 = 25 .

If the Value-at-Risk is used to assess the risks of A and B, then it would prefer A over B, which contradictsfinancial realities.

It is possible to define risk measures axiomatically as follows. Let ρ(X) be a real-valued function that definesrisk of a random variable X, which represents a financial gain. This function ρ is called a risk measure. A riskmeasure is called monetary if

• X ≤ Y implies ρ(X) ≥ ρ(Y );

• ρ(X + a) = ρ(X)− a for all a ∈ R.

It is normalised if

• ρ(0) = 0 .

A risk measure is called coherent if

• ρ(λX) = λρ(X) for all λ ∈ R;

• ρ(X + Y ) ≤ ρ(X) + ρ(Y ) for all X and Y .

45

4. Risk and Insurance

1. Credit risk

Bonds may be issued by various corporations that may default on their obligations. The default loss is givenin percents of no-default value. Value of T-bond minus value of corporate bond is the present value of cost ofdefaults.

Maturity Treasury Corp bond Default loss

1 5.00 5.25 0.2497

2 5.00 5.50 0.9950

3 5.00 5.70 2.0781

4 5.00 5.85 3.3428

Table 4.1: Zero-coupon yields on T-bonds and on bonds issued by a corporation (example)

Example. See Table 4.1. The value of a one-year T-bond with a principal of 100 is 100e−0.05 = 95.1229. For acorporate bond, 100e−0.525 = 94.8854. Then 95.1229− 94.8854 = 0.2375 and

0.2375/95.1229 = 0.2497% .

Credit rating by credit rating agencies, e.g. Moody’sAaa, Aa, A, Baa, Ba, B, Caa, Ca, CS&P ratings: AAA, AA, A, BBB, BB, B, CCC

Term (years) 1 2 3 4 5 10 15

AAA 0.00 0.00 0.07 0.15 0.24 1.12 1.40

AA 0.00 0.02 0.12 0.25 0.43 1.29 1.48

A 0.06 0.16 0.27 0.44 0.67 1.40 3.00

BBB 0.18 0.44 0.72 1.27 1.78 4.34 4.70

BB 1.06 3.48 6.12 8.68 10.97 17.73 19.91

B 5.20 11.00 15.65 19.40 21.88 29.02 30.65

CCC 19.79 26.92 31.63 35.97 40.15 45.10 45.10

Table 4.2: Cumulative default rates (%) (quoted by S&P Credit Week, 1996)

A change of rating is called a migration. The price of risk bonds normally depends on credit rating and shouldbe adjusted if the rating changes. It can be modelled by discrete time or continuous time Markov chains.

46


2. Non-life insurance

2.1. The claims

A risk is described by a pair (Pt, Xt), where

• Pt = premium earned in the time interval (0, t] (usually regarded as a deterministic process);

• Ct = sum of claim amounts incurred in (0, t].

The claim process, Nt, is the number of claims incurred in time interval (0, t]. We assume that Nt is a non-homogeneous Poisson process with intensity function λ(t). This means that

• N0 = 0;

• Nt −Ns has the Poisson distribution with mean∫ t

sλ(x)dx;

• Nt has independent increments, i.e. Ntn − Ntn−1 , . . . , Nt2 − Nt1 are jointly independent for any timemoments t1 ≤ t2 ≤ · · · ≤ tn.

The claim times are denoted by S1, S2, . . . . The inter-occurrence time (or gaps) are Tn = Sn+1 − Sn, n ≥ 1. Ifλ(t) = λ is constant, then Tn ∼ Ex(λ).

It is possible to consider non-homogeneous Markov process Nt where all λn dependent on t and then it may bepossible to transform it to a homogeneous by time distortion. The corresponding time transformation is calledoperational time.

2.2. The risk process

The accumulated claim process Ct is the sum of claim amounts incurred before time t, so that

Ct =Nt∑

j=1

Zj ,

where Z1, Z2, . . . are i.i.d. random variables (claim sizes). The process Ct is called a compound Poisson process.More generally, claim amounts may be dependent on the accumulated claim, i.e. Zj may depend on CSj−, theamount claimed just before claim number j has been made at time Sj . The financial standing of an insurancecompany is described by a process Xt = u + Pt − Ct, see Figure 4.1. Note that u is the initial capital.

It is possible to express the expected total claim amount as

E (Ct) = E (Z)

t∫

0

λ(s)ds = E (Z)λt ,

where Z is the amount of the typical claim, and assuming that the rate of claims λ remains constant. Theabove relation can be proved by the total probability formula

ECt =∞∑

n=0

E (Nt∑

j=0

Zj |Nt = n)P(Nt = n)

=∞∑

n=0

nE (Z)P(Nt = n) = E (Z)E (Nt) .

47


Timeyear 1 year 2

Figure 4.1: Premiums minus claims. A ruin occurs just before the end of year 2.

Reminder: If X is a random variable, its moment generating function (m.g.f.) is given by ψX(a) = E eaX . IfN is a discrete random variable, its probability generating function (p.g.f.) is πN (z) = E (zN ). The followingtheorem is also proved by the total probability formula.

Theorem. If πNt is the p.g.f. for the number of claims Nt and ψZ is the m.g.f. of a typical claim, then

ψCt(a) = πNt(ψZ(a)) .

2.3. Ruin probabilities

A ruin occurs if Ct − Pt > u for some t, where u is the amount of initial reserves. Assume that Pt = tc, c > 0,so that the rate of premiums is constant. By Ψ(u) denote the ruin probability if the initial reserves amount tou. Also consider the non-ruin probability Φ(u) = 1−Ψ(u). Note that Φ(u) = 0 if u < 0.

Let S1 = t be the time of the first claim and Z1 = y its amount. Then a ruin does not occur if and only if

y ≤ u + ct no ruin at the first claim

and the ruin does not occur given that we start (at time t) from the capital u + ct− y. If F is the cumulativedistribution function of the typical claim, then integrating over all possibilities for the first jump yields that

Φ(u) = EΦ(u + cS1 − Z1) =

∞∫

0

λe−λs

u+cs∫

0

Φ(u + cs− z)dF (z)ds .

The change of variables x = u + cs leads to

Φ(u) =λ

ceλu/c

∞∫

u

e−λx/c

x∫

0

Φ(x− z)dF (z)dx .

Since Φ is representable as an integral, it is differentiable, and

Φ′(u) =λ

cΦ(u)− λ

c

u∫

0

Φ(u− z)dF (z) . (4.1)

48


Integration of the both sides of (4.1) yields that

Φ(t)− Φ(0) =λ

c

t∫

0

Φ(u)du +λ

c

t∫

0

u∫

0

Φ(u− z)d(1− F (z))du

=λ

c

t∫

0

Φ(u)du +λ

c

t∫

0

Φ(0)(1− F (u))− Φ(u) +

u∫

0

Φ′(u− z)(1− F (z))dz

du

=λ

cΦ(0)

t∫

0

(1− F (u))du +λ

c

t∫

0

(1− F (z))dz

t∫

z

Φ′(u− z)du

=λ

cΦ(0)

t∫

0

(1− F (u))du +λ

c

t∫

0

(1− F (z))dz[Φ(t− z)− Φ(0)] .

Finally, the equation simplifies to

Φ(u) = Φ(0) +λ

c

u∫

0

Φ(u− z)(1− F (z))dz . (4.2)

Assume that µ = EZ1 is the mean value of the typical claim. By the monotone convergence if u →∞, we have

Φ(∞) = Φ(0) +λ

cΦ(∞)µ .

If c ≤ λµ, the ruin happens with probability 1. So assume that c > λµ, i.e.

c = λµ(1 + δ) ,

where δ > 0 is called the security (or premium) loading. Note that λµ is called pure risk premium (amountnecessary to pay the average claim load).

The law of large numbers implies that Xt/t → c − λµ > 0. Thus, there exists a random time T such thatXt/t > 0 for all t > T . Since only a finite number of claims may occur before T , we see that inft>0 Xt > −∞almost surely. Thus, the infinite initial capital secures that the ruin never happens, i.e.

Φ(∞) = 1 .

This implies

Ψ(0) =λµ

c=

11 + δ

.

Note that this result does not rely on a particular distribution of the claim size, only on its mean value.

Now assume that the claims are exponentially distributed, i.e.

1− F (z) = e−z/µ .

Then (4.1) turns into

Φ′(u) =λ

cΦ(u)− λ

cµ

u∫

0

Φ(u− z)e−z/µdz

=λ

cΦ(u)− λ

cµ

u∫

0

Φ(z)e−(u−z)/µdz .

49


By differentiating again we arrive at

Φ′′(u) =λ

cΦ′(u)− λ

cµΦ(u)− λ

cµ

u∫

0

Φ(z)e−(u−z)/µ(−1/µ)dz

=λ

cΦ′(u)− λ

cµΦ(u) +

1µ

(λ

cΦ(u)− Φ′(u))

= (λ

c− 1

µ)Φ′(u)

=λµ− c

cµΦ′(u) .

ThusΦ′′(u) = − δ

(1 + δ)µΦ′(u) .

The solution of this differential equation is given by

Φ(u) = C1 − C2e−δu/((1+δ)µ) .

The given values of Φ(∞) = 1 and Φ(0) = 1− 1/(1+ δ) make it possible to identify the unknown constants andto arrive at the solution

Ψ(U) =

1

1+δ e−δ

1+δUµ ; if λµ ≤ c ,

1 otherwise .

2.4. Cramer-Lundberg inequality

For a rather general claim distribution the following Cramer-Lundberg inequality holds:

Ψ(u) ≤ e−ru ,

where r (called the Lundberg exponent) is the solution of

c

λr = E (erZ)− 1 .

Denote V = Z − cS, where Z is the claim’s amount and S is the time until the first claim. Noticing that S isexponentially distributed with parameter λ, we see that

E (ercS) =λ

λ− rc.

Thus, the Lundberg exponent also solves the equation

E erV = 1 .

The risk process starts at u and after paying the kth claim takes value u− Yk, where

Yk = V1 + · · ·+ Vk

and V1, . . . , Vk are i.i.d. random variables, all distributed as V = Z − cS. The ruin means that Yk exceeds ufor some k, so that the ruin probability satisfies

Ψ(u) = limn→∞

Ψn(u) ,

50


whereΨn(u) = P( max

1≤k≤nYk > u) .

It suffices to show that Ψn(u) ≤ e−ru for all n. We will do this by an induction argument.

If n = 1, then the Markov inequality yields that

Ψ1(u) = P(Y1 > u) ≤ E e(Z−cS)r

eur= e−ru ,

since the numerator is one by the choice of r. Now assume that the statement holds for n = k and check it fork = 1

Ψk+1(u) = P( max1≤n≤k+1

Yn > u)

= P(V1 > u) + P( max2≤n≤k+1

(V1 + (Yn − V1)) > u, V1 ≤ u)

= p1 + p2 ,

where

p2 =∫

−(∞,u]

P( max1≤n≤k

(x + Yn) > u)dFV1(x)

=∫

−(∞,u]

P( max1≤n≤k

Yn > u− x)dFV1(x)

=∫

−(∞,u]

Ψk(u− x)dFV1(x)

≤∫

−(∞,u]

er(x−u)dFV1(x) ,

where the last inequality relies on the induction assumption. Furthermore,

p1 = P(V1 > u) =∫

(u,∞)

dFV1(x) ≤∫

(u,∞)

er(x−u)dFV1(x) .

Thus,

Ψk+1(u) = p1 + p2 ≤∫

er(x−u)dFV1(x) = e−ruE erV1 = e−ru .

2.5. Premiums

If C is the total amount of claims, then total premiums can be determined as

P = E (C)

(often used in life insurance). It is called pure risk premium. They can be of use as the planning horizon islimited and also there are indirect ways of loading the premium, e.g., interests on premiums.

Security loaded premiums P = (1 + δ)E (C) .

A loading may depend on the degree of fluctuation of C and so can be determined from

P = E (C) + βσ(C) or P = E (C) + γ Var(C)

51


called the σ- and σ2-loading respectively. Sometimes mixtures of the two are used. Different formulae can beused for different tariff groups.

The principle of zero utility determines P from

E (h(P − C)) = h(0) ,

where h is a non-decreasing function with h′′ ≤ 0.

Examples: exponential utility, quadratic utility h(x) = x− x2/(2c) for x ≤ c.

The credibility premium takes into account the history of actual claims in the past, it corresponds to takingconditional expectations given past claims.

2.6. Reinsurance

C - total amount of claimsC - reinsured amount of claimsC = C − C - net amount of claims

Then the net premium is P = P − P , where P = (1 + δ)E (C), P = (1 + δ)E (C).

3. Life insurance

3.1. Lifetime distribution

Consider a person aged x, also called a life aged x. His future life time is denoted by T or T (x). Denote

tqx = G(t) = P(T ≤ t) , tpx = 1− tqx ,

ands|tqx = G(s + t)−G(s) .

Furthermore,tpx+s = P(T > s + t|T > s) , tqx+s = P(T ≤ s + t|T > s) .

The expected remaining lifetime

oex = E (T ) =

∞∫

0

tpxdt

The force of mortality

µx+t =g(t)

1−G(t)= − d

dtlog(1−G(t))

Life tables contain one-year death probabilities qx. It is possible to derive from them tqx and tpx.

3.2. Basic ideas of life insurance

Whole life insurance provides for payment of 1 unit at the end of the year of death. Let v be the annualiseddiscount factor. Then the current value is

Z = vK+1 ,

52


where K is the number of completed future years lived by a life aged x (K is the integer part of T (x)). Thenthe net single premium

E (vK+1) =∞∑

k=0

vk+1kpx qx+k .

For the term insurance of duration n years the above sum is taken up to and including k = n− 1.

A pure endowment of duration n provides for payment of the sum insured only if the insured is alive at theend of n years

E (Z) = vnnpx .

All these formulae can be evaluated from a life table.

More general types of life insurance are

Z = cK+1vK+1 or Z = c(T )vT .

A whole life annuity-due has the present value

Y = 1 + v + v2 + · · ·+ vK .

It may incorporate annual increments, multiple payments per year, etc. It is possible also to incorporatepayments made more frequently than once a year.

The perpetuity consists of annual payments of 1 unit. If the first payment occurs at time 0, then the presentvalue of the perpetuity is

1 + v + v2 + · · · = 11− v

.

53

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