ACTS 4302/ACTS 4309 IFM FORMULA SUMMARY IFM Lesson 1: Introduction to derivatives

Measures of market size and activity: 1. Trading volume; 2. Market value; 3. Notional value; 4. Open interest. Derivatives serve the following purposes: 1. Risk management; 2. Speculation; 3. Reduced transaction costs; 4. Regulatory arbitrage. 2

IFM Lesson 2: Project Analysis

1. The Net (NPV) of a project is: ∞ X FCFn NPV = , (1 + r)n n=0 where FCFn is the free cash flow at time n, r is the . 2. If FCC’s are k at time 1 and grow at constant rate g < r, then k NPV = r − g 3. Sample mean: n 1 X R¯ = R n i i=0 4. Sample variance, biased estimate: n 1 X 2 σˆ2 = R − R¯ B n i i=0 5. Sample variance, unbiased estimate: n 1 X 2 σˆ2 = R − R¯ UB n − 1 i i=0 6. Sample confidence interval using the unbiased estimate for the sample variance:  1 1  R¯ − √ · σˆ , R¯ + √ · σˆ n UB n UB 7. Downside semi-variance: 2 h  2i σSV = E min 0, (R − µ) 8. Sample downside semi-variance: n 1 X  2 σˆ2 = min 0, R − R¯ SV n i i=0 9. Value-at-risk of a random variable X at level α: −1 V aRα(X) = FX (α) 10. Tail value-at-risk of a random variable X at security level α for downside risk: 1 Z V aRα(X) T V aRα(X) = E [X|X < V aRα(X)] = xf(x) dx α −∞ 11. Tail value-at-risk of a random variable X at security level α for downside risk: 1 Z ∞ T V aRα(X) = E [X|X > V aRα(X)] = xf(x) dx 1 − α V aRα(X) 3

IFM Lesson 3: Monte Carlo Simulation

Monte Carlo methods generate pseudorandom numbers from a distribution.

1. For a continuous distribution, if F (x) is a cumulative distribution function and u1, u2, . . . , un are −1 randomly generated uniform numbers between 0 and 1, then xi = F (ui), i = 1, 2, . . . , n will be pseudorandom numbers generated from the distribution F .

2. For a discrete distribution, if p0, p1, . . . , pn,... are probabilities P r(N = k) = pk, and u1, u2, . . . , un are randomly generated uniform numbers between 0 and 1, then for each u, the corresponding k is generated based on the following table:

u k

0 ≤ u < p0 0

p0 ≤ u < p0 + p1 1 ...... Pk−1 Pk i=0 pi ≤ u < i=0 pi k ...... 4

IFM Lesson 4: Efficient Markets Hypothesis (EMH)

Based on EMH, price takes into account: 1. Weak form of market efficiency states that stock price takes into account past stock prices. 2. Semi-strong form of market efficiency states that stock price takes into account past stock prices and public/easily accessible information. 3. Strong form of market efficiency states that stock price takes into account past stock prices, public/easily accessible information and private/hard-to-get information. Evidence for EMH: 1. Evidence for weak form of market efficiency reveals itself in stock prices following random walk. 2. Evidence for semi-strong form of market efficiency reveals itself in announcements being immediately reflected in stock price. 3. Evidence for strong form of market efficiency reveals itself in the majority of the fund managers not being able to beat the market. 5

IFM Lesson 5: Mean-Variance Portfolio Theory

1. Variance on return on portfolio is: n n X X V ar(R) = xixjCov(Ri,Rj) i=1 j=1 2. Variance on return on equally-weighted portfolio is: 1 n − 1 V ar(R) = AveV ar + AveCov n n 3. Volatility of return on portfolio in terms of its components: n X σP = xi · ρP,i · σi i=1 4. Sharpe Ratio: α − r φ = f σ 6

IFM Lesson 6: Capital Asset Pricing Model (CAPM)

1. Beta of investment i with respect to portfolio P :

P Corr(RP ,Ri)SD(Ri) Cov(RP ,Ri) βi = = SD(RP ) V ar(RP ) 2. Required return for investment i with respect to portfolio P : P E [Ri] − rf > φP · SD(Ri) · Corr(RP ,Ri) = βi (E [RP ] − rf ) 3. Beta: Corr(RMkt,Ri)SD(Ri) Cov(RMkt,Ri) βi = = SD(RMkt) V ar(RMkt) 4. Beta of portfolio as a function of betas of its assets: n X βP = xi · βi i=1 5. Capital Market Line is the line relating expected return of efficient portfolio to its volatility. 6. Security Market Line is the line relating expected return of investment to beta. 7

IFM Lesson 7: Cost of Capital

1. Fundamental approach to calculating market return: if P0 is the value of the market, then Div1 RMkt = + g P0 2. Debt cost of capital: the relationship between ’s and its expected return,

rd = y − pL, where y is the annual effective yield to maturity on the bond, p is the annual probability of default, and L is the proportion of debt that is lost when a default occurs. 3. The unlevered or the asset cost of capital is calculated the weighted average of the equity and debt costs of capital: E D r = · r + · r U E + D E E + D D 4. In the presence of cash C, the unlevered or the asset cost of capital is calculated the weighted average of the equity and debt costs of capital and the risk-free rate: E D C r = · r + · r − · r U E + D − C E E + D − C D E + D − C f 5. The unlevered beta is the weighted average of the equity and debt betas: E D β = · β + · β U E + D E E + D D 8

IFM Lesson 8: Behavioral Finance and Multifactor Models

1. Arbitrage Pricing Theory (APT):

X Fn E[Rs] − rf = n = 1Nβs (E[RFn ] − rf ) 2. APT with self-financing portfolios:

X Fn E[Rs] = n = 1Nβs E[RFn ],

Fn where βs is the beta of the stock with respect to the portfolio Fn:

Fn Corr(RFn ,Rs)SD(Rs) Cov(RFn ,Rs) βs = = SD(RFn ) V ar(RFn ) 3. Fama-French-Carhart (FFC) factor specification:

Mkt SMB HML PR1YR E[Rs] = rf + βs (E[RMkt] − rf ) + βs E[RSMB] + βs E[RHML] + βs E[RPR1YR] 9

IFM Lesson 9:

1. Definition of perfect market: (1) Competitive prices are available to all. (2) Transactions are efficient. (3) Capital structure provides no information. 2. Modigliani and Miller Propositions: I. Capital structure does not affect firm value. II. Cost of equity capital rises with leverage: D r = r + (r − r ) E U E U D 3. Equity beta: D β = β + (β − β ) E U E U D 10

IFM Lesson 10: The Effect of Taxes on Capital Structure

1. V L = V U + PV (Interest ) 2. Present value of interest tax shield for permanent debt: Dτc, where τc is the company’s tax rate. 3. Pre-tax weighted average cost of capital: E D r = · r + · r pre−tax W ACC E + D E E + D D 4. After-tax weighted average cost of capital or weighted average cost of capital (WACC): E D D r = · r + · r (1 − τ ) = r − · r · τ W ACC E + D E E + D D c pre−tax W ACC E + D D c 5. If debt-equity ratio is constant, the interest tax shield is V L − V U , where V L is computed at the WACC and V U is computed at the pre-tax WACC. 11

IFM Lesson 11: Other Factors Affecting Optimal Debt-Equity Ratio

1. Indirect costs of bankruptcy: 1) Loss of customers 2) Loss of suppliers 3) Loss of employees 4) Loss of receivables 5) Fire sale of assets 6) Insufficient liquidation 7) Costs to creditors 2. Trade-off theory: V L = V U +PV (Interest Tax Shield)−PV (F inancialDistressCosts)+PV (Agencybenefits)−PV (Agencycosts) 3. Asset substitution problem: companies in distress substitute risky assets for non-risky assets. 4. Debt overhang: companies do not make positive-NPV investments because only creditors will benefit. 5. Approximate required NPV for equity holders to benefit: NPV β D > D · , where I βE E D is debt, E is equity, I is the amount invested, βD and βE are the betas for debt and equity. 6. Leverage ratchet effect: presence of debt leads to issuing more debt. 7. Agency benefits: 1) Control of company in fewer hands. 2) Management has greater share of equity, discouraging waste. 3) No empire building. 4) Management more likely to be fired in financial distress. 5) may lead to wage concessions. 6) More incentive to compete. 8. Credibility principle: actions speak louder than words, when the words are in self-interest. 9. Adverse selection: sellers with private information sell the least desirable items. 10. Lemons principle: buyers discount price when seller has private information. 11. Pecking order hypothesis: management prefers to finance first with retained earnings, then with debt, and only finally with equity. 12

IFM Lesson 12: Equity Financing

I. Sources of Equity for Private Companies: 1. All companies start out as private. 2. Founders are individuals who invest their own money to start a company. 3. Angel investors are individuals who supplement the funds of the founders. 4. Convertible notes are the certificates that angel investors or angel groups get instead of a stock since at the beginning it is difficult to estimate the value of the company. 5. Venture capital firms are the companies that help companies raise equity when they start to expand. 6. Existing private companies (not start-ups) can raise capital from: a. firms ; b. Institutional investors ; c. Corporate investors. 7. Leveraged is buying out a public company by a private equity firm that uses borrowed money to make it private. II. Allocation of company value among investors: 1. Seed money (Series A stock) is the initial amount invested in a company by the founders. 2. Funding round is each subsequent event when additional stock is issued (Series B, C, etc stock). 3. Pre-money is the value of the company before the funding round based on the price per share of the new series. 4. Post-money valuation is the value of the company after the funding round based on the price per share of the new series. 5. Venture capitalists require favorable terms in their financial aggreements : a. Liquidation preference ; b. (); c. Participation rights ; d. Anti-dilution protection : i. Down round ; ii. Full ratchet protection ; iii. Broad-based weighted average protection. e. Board membership. 6. Shareholders in a private company can cash in their ownership only if : a. the company liquidates ; b. is purchased by another company ; c. goes public through an IPO. III. Going public : the Initial Public Offering (IPO): 1. A private company goes public with an IPO. a. Advantages liquidity and greater access to capital ; b. Disadvantages spreading of ownership. 2. Primary offering when an IPO sells new shares. 3. Secondary offering when current stockholders sell their shares to the public. 4. An underwriter may sell an IPO on three different basis : a. Best-efforts common for small IPOs ; b. Firm commitment ; c. At an auction. 5. Syndicate is a group of underwriters that manage IPO. 6. Red herring preliminary prospectus distributed to potential investors. 7. Road show company management and underwriters travel around the country to gauge interest. 13

8. the process of gauging demand and coming up with an offer price. 9. provision – allows the underwriters to sell additional shares (over-allotment) after the company goes public above the amount the company is offering (10-15%). 10. Over-allotment are the additional shares specified in the greenshoe provision. IV. IPO Puzzles: 1. Underpricing. 2. Cyclicality. 3. Cost. 4. Long-run underperformance. 14

IFM Lesson 13: Debt Financing

I. Corporate Debt: 1. Public debt debt available to the public. 2. Private debt debt arranged privately. 3. Public debt consists of securities issued by corporations. 4. Prospectus is a document that describes the debt offering. 5. Indenture is a formal contract between the bond issuer and a trust company. 6. Original issue discount is the difference between the face amount and the sales price (assuming P < F ). 7. Bonds: a. Bearer similar to currency; b. Registered belong to the registered owner. This is more typical. 8. Types of public debt: a. Notes unsecured, less than 10 years; b. Debentures unsecured, longer than 10 years; c. Mortgage bonds secured by real property; d. Asset-based bonds secured by assets other than real property. 9. Debt can be: a. Senior; b. Subordinate. 10. Types of International bonds: a. Domestic bonds; b. Foreign bonds (Yankee bonds); c. Eurobonds; d. Global bonds; 11. Types of private debt: a. Term loans (with a revolving line of credit) issued by banks; b. Private placements issued by a small group of investors easier to get, not as liquid.

II. Other Debt: 1. Sovereign debt is debt issued by national governments. 2. Four types of debt issued by the US government: a. Treasury bills less than one year, no coupons; b. Treasury notes between 1 and 10 years, semi-annual coupons; c. Treasury bonds greater than 10 years (typically 30), semi-annual coupons; d. Treasury inflation protected securities (TIPS)- fixed rate, principal is adjusted with inflation, issued for 5, 10, and 30 years. 3. Other bonds include: a. Municipal bonds; b. General obligation bonds; c. Revenue bonds; d. Double-barreled bond (mix of b. and c.) 4. Asset-based securities (ABS) pay interest and principal from cash flows generated by specific assets. 5. Mortgage based securities (MBS) the largest sector of the asset-based securities. 6. MBS are offered by: a. Government National Mortgage Association (GNMA), or Ginnie Mae . b. Federal National Mortgage Association (FNMA), or Fannie Mae . c. Federal Home Mortgage Corporation (FHLMC), or Freddie Mac . d. Student Loan Marketing Association (SLMA), or Sallie Mae . 15

7. Collateralized debt obligations (CDOs) debts packaged from asset-based loans such as mortgages, auto-loans, etc. 16

IFM Lesson 14: Forwards.

1. Price to be paid at time T in a forward agreement made at time t to buy an item at time T : Ft,T 2. Cost of carry = r − δ 3. Implied fair price: the implied value of S0 when it is unknown based on an equation relating S0 to F0,T 4. Implied repo rate: implied value of r based on the price of a stock and a forward   5. Annualized forward premium = 1 ln F0,T T S0 6. Payoff and Profit of a Forward: a. Payoff(Purchased Forward)= ST − Ft,T (S0) b. Profit(Purchased Forward)= ST − Ft,T (S0) c. Payoff(Short Forward)= Ft,T (S0) − ST d. Profit(Short Forward)= Ft,T (S0) − ST 7. Price of a forward on a non-dividend paying stock: rT F0,T = S0e 8. Price of a forward on a dividend paying stock with discrete dividends: rT F0,T = S0e − CumV alue(Div) 9. Price of a forward on a dividend paying stock with continuous dividends: (r−δ)T F0,T = S0e

10. Price of a forward expressed in domestic currency to deliver foreign currency at x0 exchange rate:

(rd−rf )T F0,T = x0e 11. When dealing with futures, we are always given an initial margin and a maintenance margin. If the margin balance drops below the maintenance margin, a margin call is made requiring the investor deposit enough to bring the margin account balance to the initial margin. 17

IFM Lesson 15: Variations of the Forward Concept.

P −rT 1. The prepaid forward is the present value of the forward price: F0,T = F0,T e P 2. If an asset pays no dividends, then the prepaid forward price is F0,T = S0. 3. If an asset pays dividends with amounts D1,...,Dn at times t1, . . . , tn, then the prepaid forward price is n P X −rti F0,T = S0 − Die i=1 4. If an asset pays dividends as a percentage of the stock price at a continuous (lease) rate δ, then P −δT the prepaid forward price is F0,T = S0e . 18

IFM Lesson 16: Options

1. Assets (a) S0 = asset price at time 0; it is assumed the purchaser of the stock borrows S0 and must repay the loan at the end of period T (b) Payoff(Purchased Asset)= ST rT (c) Profit(Purchased Asset)= ST − S0e (d) Payoff(Short Asset)= −ST rT (e) Profit(Short Asset)= −ST + S0e 2. Calls (a) C0 = premium for call option at time 0; it is assumed the purchaser of the call borrows C0 and must repay the loan when the option expires (b) Payoff(Purchased Call)= max {0,ST − K} rT (c) Profit(Purchased Call)= max {0,ST − K} − C0e (d) Payoff(Written Call)= − max {0,ST − K} rT (e) Profit(Written Call)= − max {0,ST − K} + C0e 3. Puts (a) P0 = premium for put option at time 0; it is assumed the purchaser of the put borrows P0 and must repay the loan when the option expires (b) Payoff(Purchased Put)= max {0,K − ST } rT (c) Profit(Purchased Put)= max {0,K − ST } − P0e (d) Payoff(Written Put)= − max {0,K − ST } rT (e) Profit(Written Put)= − max {0,K − ST } + P0e 19

IFM Lesson 17: Option Strategies

1. Floor: buying an asset with a purchased put option 2. Cap: short selling an asset with a purchased call option 3. Covered put (short floor): short selling an asset with a written put option 4. Covered call (short cap): buying an asset with a written call option 5. Synthetic Forward: combination of puts and calls that acts like a forward; purchasing a call with strike price K and expiration date T and selling a put with K and T guarantees a purchase of the asset for a price of K at time T 6. Short synthetic forward: purchasing a put option with strike price K and expiration date T and writing a call option with strike price K and expiration date T 7. No Arbitrage Principle: If two different investments generate the same payoff, they must have the same cost. 8. C(K,T ) = price of a call with an expiration date T and strike price K 9. P (K,T ) = price of a put with an expiration date T and strike price K T 10. Put-Call parity: C0 − P0 = (F0,T − K)v −rT a. If the asset pays no dividends: C0 − P0 = S0 − Ke −δT −rT b. For a dividend paying stock: C(K,T ) − P0(K,T ) = S0e − Ke 11. Spread: position consisting of only calls or only puts, in which some options are purchased and some written 12. Bull: investor betting on increase in market value of an asset 13. Bull spread: purchased call with lower strike price K1 and written call with higher strike price K2 14. A bull spread can also be created by a purchased put with strike price K1 and a written put with strike price K2. 15. Bear: an investor betting on a decrease in market value of an asset 16. Bear spread: written call with lower strike price K1 and purchased call with higher strike price K2 17. Ratio spread: purchasing m calls at one strike price K1 and writing n calls at another strike price K2 18. Box spread: using options to create a synthetic long forward at one price and a synthetic short forward at a different price 19. Purchased collar: purchased put option with lower strike price K1 and written call option with higher strike price K2 20. Collar width: difference between the call and put strikes 21. Written collar: writing a put option with lower strike price K1 and purchasing a call option with higher strike price K2 22. Zero-cost collar: choosing the strike prices so that the cost is 0 23. Purchased straddle: purchased call and put with the same strike price K 24. Written straddle: written call and put with the same strike price K 25. Purchased strangle: purchased put with lower strike price K1 and purchased call with a higher strike price K2 26. Written strangle: written put with lower strike price K1 and written call with a higher strike price K2 27. Butterfly spread: a written straddle with a purchased strangle 28. Asymmetric butterfly spread: purchasing λ calls with strike price K1, purchasing 1 − λ calls K3 − K2 with strike price K3, and writing 1 call with strike price K2 where λ = and K1 < K2 < K3 K3 − K1 20

IFM Lesson 18: Put-Call Parity

1. Call option payoff: max(0,ST − K) 2. Put option payoff: max(0,K − ST ) 3. General Put Call Parity (PCP): −rT C(S, K, T ) − P (S, K, T ) = (F0,T − K)e 4. PCP for a non-dividend paying stock: −rT C(S, K, T ) − P (S, K, T ) = S0 − Ke 5. PCP for a dividend paying stock with discrete dividends: −rT C(S, K, T ) − P (S, K, T ) = S0 − PV0,T (Div) − Ke 6. PCP for a dividend paying stock with continuous dividends: −δT −rT C(S, K, T ) − P (S, K, T ) = S0e − Ke 7. Pre-paid forward at time t: P −r(T −t) Ft,T (S) = e Ft,T 8. Call option written at time t which lets the purchaser elect to receive ST in return for QT at time T : C(St,Qt,T − t) 9. Put option written at time t which lets the purchaser elect to give ST in return for QT at time T :

P (St,Qt,T − t) 10. PCP for exchange options: P P C(St,Qt,T − t) − P (St,Qt,T − t) = Ft,T (S) − Ft,T (Q) P (St,Qt,T − t) = C(Qt,St,T − t) P P C(St,Qt,T − t) − C(Qt,St,T − t) = Ft,T (S) − Ft,T (Q) 11. Call-Put relationship in ”domestic” currency:  1 1  KPd , ,T = Cd(x0,K,T ) x0 K  1 1  KCd , ,T = Pd(x0,K,T ) x0 K 12. Call-Put relationship in ”foreign” and ”domestic” currency:  1 1  Kx0Pf , ,T = Cd(x0,K,T ) x0 K  1 1  Kx0Cf , ,T = Pd(x0,K,T ) x0 K

13. PCP for currency options with x0 as a spot exchange rate:

−rf T −rdT Cd(x0,K,T ) − Pd(x0,K,T ) = x0e − Ke  1 1   1 1  1 1 −rdT −rf T Cf , ,T − Pf , ,T = e − e x0 K x0 K x0 K 21

IFM Lesson 19: Comparing Options

1. Inequalities for American and European call options:

P −rT S ≥ CAmer(S, K, T ) ≥ CEur(S, K, T ) ≥ max(0,F0,T (S) − Ke ) 2. Inequalities for American and European put options: −rT P K ≥ PAmer(S, K, T ) ≥ PEur(S, K, T ) ≥ max(0, Ke − F0,T (S))

3. Direction property: If K1 ≤ K2 then

C(K1) ≥ C(K2) and P (K1) ≤ P (K2)

4. Slope property: If K1 ≤ K2 then

C(K1) − C(K2) ≤ (K2 − K1) and P (K2) − P (K1) ≤ (K2 − K1)

5. Convexity property: If K1 ≤ K2 ≤ K3 then C(K2) − C(K3) C(K1) − C(K2) C(K1)(K3 − K2) + C(K3)(K2 − K1) ≤ ⇔ C(K2) ≤ K3 − K2 K2 − K1 K3 − K1 P (K2) − P (K1) P (K3) − P (K2) P (K1)(K3 − K2) + P (K3)(K2 − K1) ≤ ⇔ P (K2) ≤ K2 − K1 K3 − K2 K3 − K1 6. Maximum possible value of the difference between the two calls: −rT max(C(K1) − C(K2)) = e (K2 − K1) 22

IFM Lesson 20: Binomial Trees - Stock, One Period

The formulas below are true for calls and puts. We use calls in the notation. 1. Pricing an option using risk-neutral probability p∗ of an increase in stock price −rh ∗ ∗ C = e (p Cu + (1 − p )Cd) , where e(r−δ)h − d p∗ = u − d u − e(r−δ)h 1 − p∗ = u − d 2. Replicating portfolio for the option: C = S∆ + B, where  C − C  ∆ = u d e−δh S(u − d) uC − dC  B = e−rh d u u − d 3. To avoid arbitrage, u and d must satisfy: d < e(r−δ)h < u 4. If σ is the annualized standard deviation of stock price movements, then √ u = e(r−δ)h+σ h √ d = e(r−δ)h−σ h ∗ 1 p = √ 1 + eσ h ∗ 1 1 − p = √ 1 + e−σ h 5. If a call (put) does not pay off at the upper(lower) node, ∆ = 0. If a call or a put pays off at both the upper and the lower node, ∆ = 1. 6. If the option always pays off, one could use the Put Call Parity to evaluate its premium (the opposite option will be zero). 23

IFM Lesson 21: Binomial Trees - General

1. For options on futures contracts: √ √ u = eσ h, d = e−σ h 1 − d p∗ = u − d C − C ∆ = u d ,B = C F (u − d) −rh ∗ ∗ C = e (p Cu + (1 − p )Cd) 24

MFE Lesson 5: Risk-Neutral Pricing (Other valuation methods of pricing options).

1. The following notation is used when pricing with true probabilities: γ - for the option α - rate of return for the stock p - (true) probability that the stock will increase in value 2. Probability that the stock will increase in value e(α−δ)h − d p = u − d

3. Accumulated option value equals accumulated replicating portfolio value: (S∆ + B)eγh = S∆eαh + Berh ⇔ Ceγh = S∆eαh + Berh

4. Often used to find the annual return of an option γ (using call notation):

−γh −rh ∗ ∗ C = e (pCu + (1 − p)Cd) = e (p Cu + (1 − p )Cd) −γh −rh ∗ For example, if Cd = 0, e p = e p 5. In the tree based on forward prices, δ does not affect p∗ or p. 6. The following notation is used when pricing with utility:

7. If SH and SL are the stock prices in the high and low state, and QH ,QL - the prices of securities paying $1 when the state H or L occurs, then the state prices are calculated as follows: −δh −rh −rh −δh S0e − e SL e SH − S0e QH = and QL = SH − SL SH − SL

8. If CH ,CL - are high and low values of a derivative C, based on the stock prices, then C = CH QH + CLQL.

9. If p is the true probability of state H and UH ,UL are the utility values, expressed in terms of dollars today, that an investor attach to $1 received in the up and down state after 1 period, then

QH = pUH ,QL = (1 − p)UL

10. Initial value of stock using utility:

C0 = pUH CH + (1 − p)ULCL = QH CH + QLCL

11. Effective annual return of a risk free investment using utility, assuming a one-year horizon: 1 r = − 1 QH + QL

12. Effective annual return of a stock using utility, assuming a one-year horizon: pC + (1 − p)C pC + (1 − p)C α = H L − 1 = H L − 1 pUH CH + (1 − p)ULCL QH CH + QLCL 25

13. Risk-neutral probabilities in terms of true probabilities and utility: pU Q p∗ = H = H pUH + (1 − p)UL QH + QL ∗ p = QH (1 + r) = pUH (1 + r)

14. True probabilities in terms of risk-neutral probabilities and utility: ∗ p UL p = ∗ ∗ p UL + (1 − p )UH 26

IFM Lesson 22: Binomial Trees: Understanding Early Exercise of Options, Alternative Trees

1. It is optimal to exercise a call option early if the present value of future dividends on the stock is greater than the sum of the present value of interest on the strike price and the value of the put S(1 − e−δt) > K(1 − e−rt) + P 2. For an infinitely-lived call option on a stock with σ = 0, exercise is optimal if Sδ > Kr 3. It is optimal to exercise a put option early if the the present value of interest on the strike price is greater than the sum of present value of future dividends on the stock and the value of the call K(1 − e−rt) > S(1 − e−δt) + C 4. For an infinitely-lived put option on a stock with σ = 0, exercise is optimal if Sδ < Kr 5. Binomial tree based on forward prices: √ √ u = e(r−δ)h+σ h, d = e(r−δ)h−σ h

6. Cox-Ross-Rubinstein tree: √ √ u = eσ h, d = e−σ h 7. Lognormal (Jarrow-Rudd) tree: √ √ 2 2 u = e(r−δ−0.5σ )h+σ h, d = e(r−δ−0.5σ )h−σ h 8. In all three cases above p∗ is calculated as: e(r−δ)h − d p∗ = u − d 9. To estimate volatility from historical data: s √ P 2    P n xi 2 Si xi σˆ = N − x¯ , where xi = ln , x¯ = n − 1 n Si−1 n N is the number of periods per year, n is the number one less than the number of observations of stock price. 27

IFM Lesson 23: Modeling Stock Prices with the Lognormal Distribution

For a stock whose price St follows a lognormal model: 1. Assume that 2 2 2 2 ln(St/S0) ∈ N (m, v ), where m = (α − δ − 0.5σ )t, v = σ t 2. The stock price at time t can be represented as: √ √ (α−δ−0.5σ2)t+σ tZ µt+σ tZ St = S0e = S0e , where Z ∈ N (0, 1) 3. The expected value is (µ+0.5σ2)t (α−δ)t E[St|S0] = S0e = S0e 4. The pth percentile of the stock price at time t is: √ √ 2 (p) (α−δ−0.5σ )t+σ tZp µt+σ tZp St = S0e = S0e , th where Zp is the p percentile of Z ∈ N (0, 1). 5. The median price of the stock t years (0.5) (α−δ−0.5σ2)t µt S = S0e = S0e

6. If Z ∈ N(m, v2), then the 100(1 − α)% confidence interval is defined to be  m − zα/2v, m + zα/2v , where zα/2 is the number such that α P Z > z  = α/2 2 Z(t) 2 7. Since St/S0 = e ,Z ∈ N(m, v ), the confidence interval for St will be m−z v m+z v S0e 1−α/2 ,S0e 1−α/2 8. Probabilities of payoffs and partial expectations of stock prices are:

P r(St < K) = N(−dˆ2)

P r(St > K) = N(dˆ2) (α−δ)t PE[St|St < K] = S0e N(−dˆ1) (α−δ)t S0e N(−dˆ1) E[St|St < K] = N(−dˆ2) (α−δ)t PE[St|St > K] = S0e N(dˆ1) (α−δ)t S0e N(dˆ1) E[St|St > K] = N(dˆ2) (α−δ)t  (1) (α−δ)t  (2) S0e N dˆ1 − S0e N dˆ1 E[S |K < S < K ] = t 1 t 2  (1)  (2) N dˆ2 − N dˆ2

9. dˆ1 and dˆ2 are defined by

S0  2 ln K + (α − δ + 0.5σ )t dˆ1 = √ σ t S0  2 ln K + (α − δ − 0.5σ )t dˆ2 = √ σ t √ dˆ2 = dˆ1 − σ t 28

10. Expected call option payoff: (α−δ)t E[max(0,St − K)] = S0e N(dˆ1) − KN(dˆ2)

11. Expected put option payoff: (α−δ)t E[max(0,K − St)] = KN(−dˆ2) − S0e N(−dˆ1) 29

MFE Lesson 8: Fitting stock prices to a lognormal distribution

1. If xi are observed stock prices adjusted to remove the effect of dividends, the estimate for the continuously compounded annual return is: αˆ =µ ˆ + 0.5ˆσ2, where µˆ = Nx¯ s √ n P x2  σˆ = N i − x¯2 n − 1 n N is the number of periods per year, n is the number one less than the number of observations of   stock price, x = ln Si . i Si−1 2. A normal probability plot graphs each observation against its percentile of the normal distribu- tion on the vertical axis. However, the vertical axis is scaled according to the standard normal distribution rather than linearly. 30

IFM Lesson 24: The Black-Scholes Formula

1. Assumptions of the Black-Scholes formula: • Continuously compounded returns on the stock are normally distributed and independent over time. • Continuously compounded returns on the strike asset (e.g., the risk-free rate) are known and constant. • Volatility is known and constant. • Dividends are known and constant. • There are no transaction costs or taxes • It is possible to short-sell any amount of stock and to borrow any amount of money at the risk-free rate. 2. General form of the Black-Scholes Formula P P C(S, K, σ, r, t, δ) = F (S)N(d1) − F (K)N(d2), where P P  1 2 √ ln F (S)/F (K) + 2 σ t d1 = √ , d2 = d1 − σ t σ t Here σ is the volatility of the pre-paid forward price on the stock. Note that it is equal to the volatility of the stock in case of the continuous dividends, but otherwise differs from it. 3. The Black-Scholes Formula for a stock −δt −rt C = Se N(d1) − Ke N(d2) −rt −δt P = Ke N(−d2) − Se N(−d1), where 1 2 √ ln (S/K) + (r − δ + 2 σ )t d1 = √ , d2 = d1 − σ t σ t 4. The Black-Scholes Formula for a currency asset

−rf t −rdt C = xe N(d1) − Ke N(d2) −rdt −rf t P = Ke N(−d2) − xe N(−d1), where 1 2 √ ln (x/K) + (rd − rf + 2 σ )t d1 = √ , d2 = d1 − σ t σ t 5. The Black-Scholes Formula for futures −rt −rt C = F e N(d1) − Ke N(d2) −rt −rt P = Ke N(−d2) − F e N(−d1), where 1 2 √ ln (F/K) + 2 σ t d1 = √ , d2 = d1 − σ t σ t 31

IFM Lesson 25: The Black-Scholes Formula: Greeks

1. ∆ −δt ∆call = e N(d1) −δt −δt ∆put = ∆call − e = −e N(−d1) 2. Elasticity S∆ Ω = C 3. Connection between the volatility of an option and the volatility of the underlying stock

σoption = σstock|Ω| 4. Connection between the risk premium of an option and the risk premium of the underlying stock γ − r = Ω(α − r) 5. Sharpe ratio α − r σ 6. Greek for portfolio is the sum of Greeks 7. Elasticity for portfolio is weighted average of the elasticities 8. Option Greeks

Greek sym- Increase in option Formula Call/put relationship Shape of Graph bol price per . . .

∂C −δt Delta (∆) increase in stock ∂S ∆put = ∆call − e S shaped price (S) ∂2C Gamma (Γ) increase in stock ∂S2 Γput = Γcall Symmetric hump, peak to price (S) the left of strike price, fur- ther left with higher t ∂C Vega (V ) increase in volatil- 0.01 ∂σ Vput = Vcall Asymmetric hump, peak ity (σ) similar to Γ 1 ∂C Theta (θ) decrease in time to − 365 ∂t θput = θcall + Upside-down hump for expiry (t) rKe−rt−δSe−δt short lives, gradual de- 365 crease for long lives, unless δ is large. Almost always negative for calls, usually negative for puts unless far in-the-money ∂C Rho (ρ) increase in interest 0.01 ∂r ρput = ρcall − Increasing curve (positive rate (r) 0.01tKe−rt for calls, negative for puts) ∂C Psi (ψ) increase in divi- 0.01 ∂δ ψput = ψcall + Decreasing curve (nega- dend yield (δ) 0.01tSe−δt tive for calls, positive for puts) 32

MFE Lesson 11: The Black-Scholes formula: applications and volatility

1. Let Ct be the price of a call option at time t. Then the profit on a call sold at time 0 < t < T is: rt Profit = Ct − C0e 2. A bull spread is the purchase of a call (or put) together with the sale of an otherwise identical higher-strike call (or put). 3. A calendar spread consists of selling a call and buying another call with the same strike price on the same stock but a later expiry date. 4. Implied volatility is the opposite of historical volatility, which is estimated based on the stock data. The implied volatility is calculated (actually, back out) based on option prices and a pricing model. 33

IFM Lesson 26: Delta-Hedging

1. Overnight profit on a delta-hedged portfolio has three components: 1. The change in the value of the option. 2. ∆ times the change in the price of the stock. 3. Interest on the borrowed money.  r/365  Profit = −(C(S1) − C(S0)) + ∆(S1 − S0) − e − 1 (∆S0 − C(S0)) 2. Price movement with no gain or loss to delta-hedger: the money-maker would break even if the stock moves by about one standard deviation around the mean to either √ √ S + Sσ h or S − Sσ h 3. Delta-gamma-theta approximation 1 C(S ) = C(S ) + ∆ + Γ2 + hθ t+h t 2 4. Black-Scholes equation for market maker profit (an approximation of the profit formula above): 1 Profit = −( Γ2 + hθ + rh(∆S − C(S))) 2 |{z} | {z } | {z } time interest effect change decay in stock of price option effect 5. Boyle-Emanuel periodic variance of return when rehedging every h in period i: 1 2 V ar (R ) = S2σ2Γh h,i 2 6. Boyle-Emanuel annual variance of return when rehedging every h in period i: 1 2 V ar (R ) = S2σ2Γ h h,i 2 7. Formulas for Greeks of binomial trees  C − C  ∆(S, 0) = u d e−δh S(u − d) ∆(Su, h) − ∆(Sd, h) Γ(S, 0) ≈ Γ(S, h) = S(u − d) C(Sud) − C(S, 0) − ∆(S, 0) − 0.5Γ(S, 0)2 θ(S, 0) = 2h 34

IFM Lesson 27: Asian, Barrier, and Compound Options.

1. Geometric averages of stock prices The n stock prices S(1),S(2), ··· ,S(n) are not independent, but in the Black-Scholes framework there is no memory, so the variables Q(t) = S(t)/S(t − 1) are independent.

n !1/n Y If G = S(k) and U = ln G k=1 Then E[U] = ln(S(0)) +m, ˜ V ar(U) =v ˜2 m˜ + 1 v˜2 E[G(S)] = S(0)e 2  2  2  2  V ar (G(S)) = E[G(S)]2 ev˜ − 1 = S(0)2e2m ˜ +˜v ev˜ − 1 , where n + 1 (n + 1)(2n + 1) m˜ = m, v˜2 = v2 2 6n m = (α − δ − 0.5σ2)t, v2 = σ2t

2. Parity relationship for barrier options: Knock-in option + Knock-out option = Ordinary option 3. Maxima and Minima The following properties will be useful in expressing various claims payoffs: (a) max(S, K) = S + max(0,K − S) = K + max(0,S − K) (b) max(cS, cK) = c max(S, K), c > 0 (c) min(S, K) + max(S, K) = S + K ⇒ min(S, K) = S + K − max(S, K) 4. Parity relationships for compound options:

−rt1 CallOnCall(S, K, x, σ, r, t1, T, δ) − P utOnCall(S, K, x, σ, r, t1, T, δ) = C(S,K,σ,r,T,δ) − xe −rt1 CallOnP ut(S, K, x, σ, r, t1, T, δ) − P utOnP ut(S, K, x, σ, r, t1, T, δ) = P (S,K,σ,r,T,δ) − xe 5. Value of American call option with one discrete dividend:    −rt1 −r(T −t1) S0 − Ke + CallOnP ut S, K, D − K 1 − e 35

IFM Lesson 28: Gap, Exchange and Other Options.

1. All-or-nothing options.

Option Name Value Delta

−d 2/2 S|S > K asset-or-nothing call Se−δT N(d ) e−δT N(d ) + e−δT e √1 1 1 σ 2πT −d 2/2 S|S < K asset-or-nothing put Se−δT N(−d ) e−δT N(−d ) − e−δT e √1 1 1 σ 2πT −d 2/2 c|S > K cash-or-nothing call ce−rT N(d ) ce−rT e √2 2 Sσ 2πT −d 2/2 c|S < K cash-or-nothing put ce−rT N(−d ) −ce−rT e √2 2 Sσ 2πT

2. Gap options. Use K1 (the strike price) in the formula for C and P . Use K2 in formula for d1

−δt −rt C = Se N(d1(K2)) − K1e N(d2(K2)) −rt −δt P = K1e N(−d2(K2)) − Se N(−d1(K2)), where 1 2 √ ln (S/K2) + (r − δ + 2 σ )t d1 = √ , d2 = d1 − σ t σ t 2 −d2 /2 −δT −rT e ∆ = e N(d1) + (K2 − K1)e √ Sσ 2πT 3. Exchange options. Recieve option S for option Q. 2 2 2 Volatility: σ = σS + σQ − 2ρσSσQ

−δS T −δQT C(S, Q, T ) = Se N(d1) − Qe N(d2), where P P  2 2 ln F (S)/F (Q) + 0.5σ T ln (S/Q) + (δQ − δS + 0.5σ )T d1 = √ = √ σ T σ T √ d2 = d1 − σ T 4. Chooser options.     V = C(S, K, T ) + e−δ(T −t)P S, Ke−(r−δ)(T −t), t = P (S, K, T ) + e−δ(T −t)C S, Ke−(r−δ)(T −t), t 5. Forward start options. If you can purchase a call option with strike price cSt at time t expiring at time T , then the value of the forward start call is −δT −r(T −t)−δt FSC = S0e N(d1) − cS0e N(d2) −r(T −t)−δt −δT FSP = cS0e N(−d2) − S0e N(−d1), where 1 2 √ − ln(c)(r − δ + 2 σ )(T − t) d1 = √ , d2 = d1 − σ T − t σ T − t 6. Lookback options. Let 2 ¯ σ St = max Su,St = min Su, g = , r 6= δ 0≤u≤t 0≤u≤t 2(r − δ) 36

Let Ot be the option price at time t. Then the price of a European lookback call is:  1−1/g ! −δ(T −t) −r(T −t) St LBCt = Ste (N(d5) − gN(−d5)) − Ste N(d6) − g N(d8) , St where 2 ln (St/St) + r − δ + 0.5σ (T − t) d5 = √ σ T − t 2 ln (St/St) + r − δ − 0.5σ (T − t) √ d6 = √ = d5 − σ T − t σ T − t 2 ln (St/St) + r − δ + 0.5σ (T − t) d7 = √ σ T − t 2 ln (St/St) + r − δ − 0.5σ (T − t) √ d8 = √ = d7 − σ T − t σ T − t The price of a European lookback put is:

 1−1/g ! −δT 0 0  ¯ −r(T −t) 0 St 0 LBPt = −Ste N(−d5) − gN(d5) + Ste N(−d6) − g N(−d8) , S¯t where  2 ln St/S¯t + r − δ + 0.5σ (T − t) d0 = √ 5 σ T − t  2 ln St/S¯t + r − δ − 0.5σ (T − t) √ d0 = √ = d0 − σ T − t 6 σ T − t 5  2 ln S¯t/St + r − δ + 0.5σ (T − t) d0 = √ 7 σ T − t  2 ln S¯t/St + r − δ − 0.5σ (T − t) √ d0 = √ = d0 − σ T − t 8 σ T − t 7 In particular, for t = 0, the price of a European lookback call is: −δT −rT LBC0 = S0e (N(d5) − gN(−d5)) − S0e (1 − g)N(d6), where r − δ + 0.5σ2 T d5 = √ σ T r − δ − 0.5σ2 T √ d6 = √ = d5 − σ T σ T The price of a European lookback put is:

−δT 0 0  −r(T −t) 0 LBP0 = −S0e N(−d5) − gN(d5) + S0e (1 − g)N(−d6), where r − δ + 0.5σ2 T d0 = √ 5 σ T r − δ − 0.5σ2 T √ d0 = √ = d0 − σ T 6 σ T 5 37

IFM Lesson 30: Real Options 38

IFM Lesson 31: Actuarial Applications of Options 39

MFE Lesson 15: Monte Carlo Valuation

Generating a log-normal random number A standard normal random variable may be generated as 12 X −1 ui − 6, or as N (ui) i=1 General steps for simulating the stock value at time t with initial price S0: (1) Given u1, u2, ··· , un uniformly distributed on [0, 1], −1 (2) Find z1, z2, ··· , zn where zj = N (ui) normally distributed with m = 0, v = 1 2 2 2 (3) Generate n1, n2, ··· , nn where nj = m + vzj, m = (α − δ − 0.5σ )t, v = σ t (j) nj 1 Pn (j) (4) Then St = S0e and St = n j=1 St A Lognormal Model of Stock Prices An expression for the stock price: √ n 2 o St = S0 exp α − δ − 0.5σ t + σ tz , z ∈ N(0, 1) The expected stock price:  1  E[S ] = S exp α − δ − 0.5σ2 t + σ2t = S exp {(α − δ) t} t 0 2 0 The median stock price:  2 S0 exp α − δ − 0.5σ t Variance Reduction Methods For the control variate method, X∗ = X¯ + E[Y ] − Y¯  V ar(X∗) = V ar(X¯) + V ar(Y¯ ) − 2Cov(X,¯ Y¯ ) For the Boyle modification, X∗ = X¯ + β E[Y ] − Y¯  , V ar(X∗) = V ar(X¯) + β2V ar(Y¯ ) − 2βCov(X,¯ Y¯ ) β is chosen to minimize the expression for V ar(X∗): Cov(X,¯ Y¯ ) P x y − X¯Y¯ /n β = = i i ¯ P 2 ¯ 2 V ar(Y ) yi − Y /n 40

MFE Lesson 16: Brownian Motion.

Part 1. Brownian Motion. Assume that 1. The continuously compounded rate of return for a stock is α 2. The continuously compounded dividend yield is δ 3. The volatility is σ

Brownian motion: {Z(t)} - a collection of random variable, defined by the following properties: 1. Z(0) = 0 2. Let t be the latest time for which you know Z(t). Then Z(t + s)|Z(t) has a normal distribution with m = Z(t) and v2 = s 3. Increments are independent: Z(t + s1) − Z(t) is independent of Z(t) − Z(t − s2) 4. More generally, non-overlapping increments are independently distributed, i.e. for t1 < t2 ≤ t3 < t4,Z(t2) − Z(t1) and Z(t4) − Z(t3) are independent. 5. Z(t) is continuous in t. Arithmetic Brownian motion: Let Z(t) be Brownian motion. Then X(t) = X(0) + αt + σZ(t) or dX(t) = αdt + σdZ(t) is Arithmetic Brownian motion. Here α is the drift and σ is the volatility of the process. Let t be the latest time for which you know X(t). Then 1. X(t + s)|X(t) has a normal distribution with m = X(t) + αs and v2 = σ2s 2. X(t + s) − X(t) has a normal distribution with m = αs and v2 = σ2s 3. Non-overlapping increments are independently distributed, i.e. for t1 < t2 < t3 < t4,X(t2)−X(t1) and X(t4) − X(t3) are independent. 4. If X(0) = 0, then for any t, u such that 0 < t < u, Cov(X(t),X(u)) = σ2t Geometric Brownian motion: X(t) follows Geometric Brownian motion if ln X(t) follows Arithmetic Brownian motion. Let t be the latest time for which you know X(t). Then ln X(t+s)/X(t)|X(t) has a normal distribution with m = (α − δ − 0.5σ2)s and v2 = σ2s Let S(t) be the time t price of the stock. Then 1. The continuously compounded expected increase in the stock price is α − δ. This means E[S(t)] = S(0)e(α−δ)t 2. The increase in the stock price is also known as the capital gains return. 3. The total return on the stock is the sum of the capital gains return and the dividend yield 4. The geometric Brownian motion followed by the stock price is dS(t) = (α − δ)dt + σdZ(t) S(t) GBM is an example of an Itˆoprocess. 5. The associated arithmetic Brownian motion followed by ln S(t) is d (ln S(t)) = α − δ − 0.5σ2 dt + σdZ(t) 6. When evaluating probabilities for ranges of S(t), look up the normal table using the following parameters: m = α − δ − 0.5σ2 t, v2 = σ2t All of the following are equivalent: 41

X(t) follows a geometric Brownian motion with drift ξ and volatility σ dX(t) = ξdt + σdZ(t) X(t) d (ln X(t)) = (ξ − 0.5σ2)dt + σdZ(t) ln X(t)|X(0) is N ln X(0) + (ξ − 0.5σ2)t, σ2t ln X(t) − ln X(0) = (ξ − 0.5σ2)t + σZ(t) Z t Z t ln X(t) − ln X(0) = (ξ − 0.5σ2)ds + σdZ(s) 0 0 2 X(t) = X(0)e(ξ−0.5σ )t+σZ(t) Z t Z t X(t) − X(0) = ξX(s)ds + σX(s)dZ(s) 0 0 Stock price modeling: Let S(t) be the time t price of a stock. Assume: 1. The continuously compounded expected rate of return is α. 2. The continuously compounded dividend yield is δ. 3. The volatility is σ, in other words V ar (ln S(t)|S(0)) = σ2t. Then ξ = α = δ is the continuously compounded rate of increase in the stock price. All the statements above for geometric Brownian motion with drift ξ and volatility σ hold. For example, X(t) follows a geometric Brownian motion with drift ξ and volatility σ dS(t) = (α − δ)dt + σdZ(t) S(t) d (ln S(t)) = (α − δ − 0.5σ2)dt + σdZ(t) Multiplication rules:

dt × dt = dt × dZ = 0 dZ × dZ = dt dZ × dZ0 = ρdt 42

MFE Lesson 17: Itˆo’sLemma. Black-Scholes Equation.

Any process of the form dS(t) = ξ (S(t), t) dt + σ (S(t), t) dZ(t) is called an Itˆoprocess. Itˆo’sLemma

2 dC(S, t) = CSdS + 0.5CSS(dS) + Ctdt Black-Scholes Equation 1 σ2S2C + (r − δ)SC + C = rC 2 SS S t Using greeks, the Black-Scholes Equation is: ∆S(r − δ) + 0.5ΓS2σ2 + θ = rC Sharpe ratio Sharpe ratio: Express process in the following form: dX = (α (t, X(t)) − δ (t, X(t))) dt + σ (t, X(t)) dZ(t) X Then the Sharpe ratio is α (t, X(t)) − r φ (t, X(t)) = σ (t, X(t)) • The Sharpe ratio is based on the total return. Do not subtract the dividend rate from α (t, X(t)) in the numerator of the Sharpe ratio. • For geometric Brownian motion, the Sharpe ratio is the constant α − r η = σ • The Sharpe ratio is the same for all processes based on the same Z(t). Risk-neutral Processes. The true Itˆoprocess dS(t) = (α(t, S(t)) − δ(t, S(t))) dt + σ(t, S(t))dZ(t) can be translated into the risk-neutral process of the form: α(t, S(t)) − r(t) dS(t) = (r(t)−δ(t, S(t)))dt+σ(t, S(t))dZ˜(t), dZ˜(t) = dZ(t)+ηdt, η = − Sharpe ratio σ(t, S(t)) Z˜(t) follows an arithmetic Brownian motion. Formulas for Sa. Expected value 2 E [S(T )a] = S(0)ae[a(α−δ)+0.5a(a−1)σ ]T Forward price and pre-paid forward price a a [a(r−δ)+0.5a(a−1)σ2]T F0,T (S ) = S(0) e P a −rT a [a(r−δ)+0.5a(a−1)σ2]T F0,T (S ) = e S(0) e Itˆoprocess d(Sa) = a(α − δ) + 0.5a(a − 1)σ2 dt + aσdZ(t) Sa Total return rate γ = a(α − δ) + r 43

Ornstein-Uhlenbeck process: Definition dX(t) = λ (α − X(t)) dt + σdZ(t) Integral   Z t X(t) = X(0)e−λt + α 1 − e−λt + σ eλ(s−t)dZ(s) 0 44

MFE Lesson 18: Binomial tree models for interest rates.

Let Pt(T,T +s) be the price, to be paid at time T , for an agreement at time t to purchase a zero-coupon bond for 1 issued at time T maturing at time T + s, t < T . Omit subscript if t = T. Let Ft,T (P (T,T + s)) be the forward price at time t for an agreement to buy a bond at time T maturing at time T + s. Then P (t, T + s) F (P (T,T + s)) = t,T P (t, T ) This section considers: (1) Interest rate binomial trees (2) The Black-Derman-Toy model for interest rates (3) Pricing forwards and caps (via caplets) 45

MFE Lesson 19: The Black formula for bond options.

Black formula Let C (F,P (0,T ), σ, T ) be a call on a bond at time 0 that allows to buy a bond at time T which matures at time T + s. Let P (F,P (0,T ), σ, T ) be a put on a bond at time 0 that allows to sell a bond at time T which matures at time T + s. F = F0,T (P (T,T + s))

C (F,P (0,T ), σ, T ) = P (0,T )(FN(d1) − KN(d2))

P (F,P (0,T ), σ, T ) = P (0,T )(KN(−d2) − FN(−d1)) , where

ln(F/K) + 0.5σ2T d1 = √ σ T √ d2 = d1 − σ T and σ is the volatility of the T -year forward price of the bond. Black formula for caplets: (1) Caplet - protects a floating rate borrower by paying the amount by which the prevailing rate is greater than the cap 1 (2) Each caplet is 1 + KR puts with strike . 1+KR Black formula for floorlets: (1) Floorlet - protects a floating rate lender by paying the amount by which the prevailing rate is less than the floor 1 (2) Each floorlet is 1 + KR calls with strike . 1+KR 46

MFE Lesson 19: Equilibrium interest rate models: Vasiˇcekand Cox-Ingersoll-Ross.

Let P (r, t, T ) be the price of a zero-coupon bond purchased at time t and maturing at time T when the short term rate is r. Hedging formulas: (T − t)P (t, T ) Duration hedging bond 1 with bond 2: N = − 1 1 (T2 − t)P (t, T2) P (r, t, T ) Delta hedging bond 1 with bond 2: N = − r 1 Pr(r, t, T2) B(t, T )P (r, t, T ) N = − 1 1 for Vasiˇcekand Cox-Ingersoll-Ross B(t, T2)P (r, t, T2)

Yield to maturity on a zero-coupon bond: ln (1/P (r, t, T )) ln (P (r, t, T )) R(t, T ) = = , r(t) = lim R(t, T ) T − t t − T T →t Differential equation for bond prices: If the interest rate process is dr = a(r)dt + σ(r)dZ(t), then dP = α(r, t, T )dt − q(r, t, T )dZ(t) P where a(r)P + 0.5σ(r)2P + P α(r, t, T ) = r rr t P P = −a(b − r)B(t, T ) + 0.5σ(r)2B(t, T )2 + t for Vasiˇcekand Cox-Ingersoll-Ross P P σ(r) q(r, t, T ) = − r P = B(t, T )σ(r) for Vasiˇcekand Cox-Ingersoll-Ross

Risk-neutral process: dr = (a(r) + φ(r, t)σ(r)) dt + σ(r)dZ,˜ dZ˜(t) = dZ(t) − φ(r, t)dt, φ(r, t) is the Sharpe ratio. Black-Scholes Equation analog for bond prices: 1 σ(r)2P + (a(r) + φ(r, t)σ(r)) P + P − rP = 0 2 rr r t Sharpe ratio: α(r, t, T ) − r General φ(r, t) = q(r, t, T ) Vasiˇcek φ(r, t) = φ it is constant √ σ(r) Cox-Ingersoll-Ross φ(r, t) = φ¯ r/σ¯ (φ¯ andσ ¯ are constant, σ¯ = √ ) r

Definitions of interest rate models: General dr = a(r)dt + σ(r)dZ(t) Rendleman-Bartter dr = ardt + σrdZ(t) Vasiˇcek dr = a(b − r)dt + σdZ(t) √ Cox-Ingersoll-Ross dr = a(b − r)dt +σ ¯ rdZ(t) 47

Bond price in Vasiˇcekand CIR models: P (r, t, T ) = A(t, T )e−B(t,T )r(t) In Vasiˇcekmodel, if a 6= 0, then 1 − e−a(T −t) B(t, T ) =a ¯ = T −t a a 2 2 A(t, T ) = er¯[B(t,T )+t−T ]−B(t,T ) σ /4a r¯ = b + σφ/a − 0.5σ2/a2 If a = 0, then 2 2 3 B(t, T ) = T − t, A(t, T ) = e0.5σφ(T −t) +σ (T −t) /6

Yield-to-maturity on infinitely-lived bond: Vasiˇcek¯r = b + σφ/a − 0.5σ2/a2 q Cox-Ingersoll-Rossr ¯ = 2ab/(a − φ¯ + γ), where γ = (a − φ¯)2 + 2¯σ2