Introductory Concepts of Financial Risk Management and Related Mathematical Tools

1.1 Introductory concepts of the securities market 

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The notion of an asset (anything of value) is one of the fundamental notions in mathematical finance. Assets can be risky and non-risky. 

Here risk is understood to be an uncertainty that can cause loss (e.g., of wealth). The most typical representatives of such assets are the basic securities: stocks S and bonds (bank accounts) B. 

These securities constitute the basis of a financial market that can be understood as a space of assets equipped with a structure for their trading. 

Stocks are share securities that accumulate capital required for a company’s successful operation. 

The stockholder holds the right to participate in the control of the company and to receive dividends. 

Bonds are debt securities issued by a government or a company for accumulating capital, restructuring debts, and so forth. In contrast to stocks, bonds are issued for a specified period of time. 

The essential characteristics of a bond include their exercise and maturity time, face value (principal or nominal), coupons (payments up to maturity), and yield (return up to maturity). 

The zero-coupon bond is similar to a bank account, and its yield corresponds to a bank interest rate. An interest rate r ≥ 0 is typically quoted by banks as an annual percentage. 

Suppose that a client opens an account with a deposit of B0, then at the end of a 1-year period the client’s non-risky profit is ΔB1 = B1 −B0 = rB0. After n years, the balance of this account will be Bn = Bn−1 +rB0, given that only the initial deposit B0 is reinvested every year. 

In this case, r is referred to as a simple interest. Alternatively, the earned interest also can be reinvested (compounded), and then, at the end of n years, the balance will be Bn = Bn−1(1 + r) = B0(1 + r) n. Note that here the ratio ΔBn/Bn−1 reflects the profitability or return of the investment as it is equal to r, the compound interest. 

1 2 Risk Analysis in Finance and Insurance Now suppose that interest is compounded m times per year, then Bn = Bn−1 1 + r(m) m m = B0 1 + r(m) m mn . Such rate r(m) is quoted as a nominal (annual) interest rate, and the equivalent effective (annual) interest rate is equal to r =  1 + r(m) m m − 1. Let t ≥ 0, and consider the ratio Bt+ 1 m − Bt Bt = r(m) m , where r(m) is a nominal annual interest rate compounded m times per year. 

Then another rate r = lim m→∞ Bt+ 1 m − Bt 1 mBt = lim m→∞ r(m) = 1 Bt dBt dt is called the nominal annual interest rate compounded continuously. Clearly, Bt = B0ert. 

Thus, the concept of interest is one of the essential components in the description of money value time evolution. Now consider a series of periodic payments (deposits) f0, f1,...,fn (annuity). 

It follows from the formula for compound interest that the present value of k-th payment is equal to fk  1+r −k , and therefore the present value of the annuity is n k=0 fk  1+r −k . Worked Example 1.1 Let an initial deposit into a bank account be $10, 000. 

Given that r(m) = 0.1, find the account balance at the end of 2 years for m = 1, 3, and 6. Also find the balance at the end of each of years 1 and 2 if the interest is compounded continuously at the rate r = 0.1. Solution Using the notion of compound interest, we have B(1) 2 = 10, 000 1+0.1 2 = 12, 100 for interest compounded once per year; B(3) 2 = 10, 000 1 + 0.1 3 2×3 ≈ 12, 174 for interest compounded three times per year; B(6) 2 = 10, 000 1 + 0.1 6 2×6 ≈ 12, 194 for interest compounded six times per year. 

Financial Risk Management and Related Mathematical Tools 3 For interest compounded continuously we obtain B(∞) 1 = 10, 000 e0.1 ≈ 11, 052 , B(∞) 2 = 10, 000 e2×0.1 ≈ 12, 214 . 

Stocks are significantly more volatile than bonds, and therefore they are characterized as risky assets. Similarly to bonds, one can define their profitability or return ρn = ΔSn/Sn−1, n = 1, 2,..., where Sn is the price of a stock at time n.

 Then we have the following discrete equation for stock prices: Sn = Sn−1(1 + ρn), S0 > 0. The mathematical model of a financial market formed by a bank account B (with an interest rate r) and a stock S (with profitabilities ρn) is referred to as a (B,S)-market.     The volatility of prices Sn is caused by a great variety of sources, some of which may not be easily observed. In this case, the notion of randomness appears to be appropriate, so that Sn, and therefore ρn, can be considered as random variables. Since at every time step n the price of a stock goes either up or down, then it is natural to assume that profitabilities ρn form a sequence of independent random variables (ρn)n=1,2,... that take values b and a (b>a) with probabilities p and q, respectively (p + q = 1). Next, we can write ρn as a sum of its mean μ = bp + aq and a random variable wn = ρn − μ, which has the expectation equal to zero. Thus, profitability ρn can be described in terms of an independent random deviation wn from the mean profitability μ. When the time steps become smaller, the oscillations of profitability become more chaotic. Formally, the limit continuous model can be written as S˙ t St ≡ dSt dt 1 St = μ + σW˙ t , where μ is the mean profitability, σ is the volatility of the market, and W˙ t is the Gaussian white noise. The formulas for compound and continuous interest rates together with the corresponding equation for stock prices define the binomial (Cox-RossRubinstein) and the diffusion (Black-Scholes) models of the market, respectively. A participant in a financial market usually invests free capital in various available assets that then form an investment portfolio. The effective management of the capital is realized through a process of building and managing the portfolio. The redistribution of a portfolio with the goal of limiting or minimizing the risk in various financial transaction is usually referred to as hedging. The corresponding portfolio is then called a hedging portfolio. An investment strategy (portfolio) that may give a profit even with zero initial investment is called an arbitrage strategy. The presence of arbitrage reflects the instability of a financial market. The development of a financial market offers the participants derivative securities, that is, securities that are formed on the basis of the basic securities – stocks and bonds. The derivative securities (forwards, futures, options, 4 Risk Analysis in Finance and Insurance etc.) require smaller initial investment and play the role of insurance against possible losses. They also increase the liquidity of the market. For example, suppose company A plans to purchase shares of company B at the end of the year. To protect itself from a possible increase in shares prices, company A reaches an agreement with company B to buy the shares at the end of the year for a fixed (forward) price F. Such an agreement between the two companies is called a forward contract (or simply, forward). Now suppose that company A plans to sell some shares to company B at the end of the year. To protect itself from a possible fall in price of those shares, company A buys a put option (seller’s option), which confers the right to sell the shares at the end of the year at the fixed strike price K. Note that, in contrast to the forwards case, a holder of an option must pay a premium to its issuer. Futures contract is an agreement similar to the forward contract, but the trading takes place on a stock exchange, a special organization that manages the trading of various goods, financial instruments, and services. Finally, we reiterate here that mathematical models of financial markets, methodologies for pricing various financial instruments and for constructing optimal (minimizing risk) investment strategies, are all subject to modern mathematical finance. 1.2 Probabilistic foundations of financial modeling and pricing of contingent claims Suppose that a non-risky asset B and a risky asset S are completely described at any time n = 0, 1, 2, ... by their prices. Therefore, it is natural to assume that the price dynamics of these securities are the essential component of a financial market. These dynamics are represented by the following equations: ΔBn = rBn−1 , B0 = 1 , ΔSn = ρnSn−1 , S0 > 0 , where ΔBn = Bn − Bn−1 , ΔSn = Sn − Sn−1 , n = 1, 2, ... ; r ≥ 0 is a constant rate of interest and ρn represent the level of risk. Quantities ρn will be specified later in this section. Another important component of a financial market is the set of admissible actions or strategies that are allowed in dealing with assets B and S. A sequence π = (πn)n=1,2,... ≡ (βn, γn)n=1,2,... is called an investment strategy (portfolio) if for any n = 1, 2, ..., the quantities βn and γn are determined by prices S1,...,Sn−1. In other words, βn = βn(S1,...,Sn−1) and γn = γn(S1,...,Sn−1) are functions of S1,...,Sn−1, and they are interpreted Financial Risk Management and Related Mathematical Tools 5 as the amounts of assets B and S, respectively, at time n. The value of a portfolio π is Xπ n = βnBn + γnSn , where βnBn represents the part of the capital deposited in a bank account and γnSn represents the investment in shares. If the value of a portfolio can change only because of changes in assets prices ΔXπ n = Xπ n−Xπ n−1 = βnΔBn+ γnΔSn , then π is said to be a self-financing portfolio. The class of all such portfolios is denoted SF. A common feature of all derivative securities in a (B,S)-market is their potential liability (payoff) fN at a future time N. For example, for forwards, we have fN = SN −F and for call options fN = (SN −K)+ ≡ max{SN −K, 0}. Such liabilities inherent in derivative securities are called contingent claims. One of the most important problems in the theory of contingent claims is their pricing at any time before the expiry date N. This problem is related to the problem of hedging contingent claims. A self-financing portfolio is called a hedge for a contingent claim fN if Xπ N ≥ fN for any behavior of the market. If a hedging portfolio is not unique, then it is important to find a hedge π∗ with the minimum value Xπ∗ n ≤ Xπ n for any other hedge π. Hedge π∗ is called the minimal hedge. The minimal hedge gives an obvious solution to the problem of pricing a contingent claim: the fair price of the claim is equal to the value of the minimal hedging portfolio. Furthermore, the minimal hedge manages the risk inherent in a contingent claim. Next, we introduce some basic notions from probability theory and stochastic analysis that are helpful in studying risky assets. We start with the fundamental notion of an “experiment” when the set of possible outcomes of the experiment is known, but it is not known a priori which of those outcomes will take place (this constitutes the randomness of the experiment). Example 1.1 (Trading on a stock exchange) A set of possible exchange rates between the dollar and the euro is always known before the beginning of trading but not the exact value. Let Ω be the set of all elementary outcomes ω and let F be the set of all events (non-elementary outcomes), which contains the impossible event ∅ and the certain event Ω. Next, suppose that after repeating an experiment n times, an event A ∈ F occurred nA times. Let us consider random experiments that have the following property of statistical stability: for any event A, there is a number P(A) ∈ [0, 1] such that nA/n → P(A) as n → ∞. This number P(A) is called the probability of event A. Probability P : F → [0, 1] is a function with the following properties: 1. P(Ω) = 1 and P(∅) = 0; 2. P  ∪k Ak  =  k P(Ak) for Ai ∩ Aj = ∅.