ACTS 4302 FORMULA SUMMARY Lessons 1-3: Introduction to Financial Derivatives, Forward and Futures Contracts.

Transcription

1 ACTS 4302 FORMULA SUMMARY Lessons 1-3: Introduction to Financial Derivatives, Forward and Futures Contracts. 1. If an asset pays no dividends, then the prepaid forward price is F0,T P = S If an asset pays dividends with amounts D 1,..., D n at times t 1,..., t n, then the prepaid forward price is n F0,T P = S 0 D i e rt i 3. If an asset pays dividends as a percentage of the stock price at a continuous (lease) rate δ, then the prepaid forward price is F P 0,T = S 0e δt. 4. Cost of carry = r δ 5. F 0,T = F P 0,T ert 6. Implied fair price: the implied value of S 0 when it is unknown based on an equation relating S 0 to F 0,T 7. Implied repo rate: implied value of r based on the price of a stock and a forward 8. Annualized forward premium = 1 T ln ( F0,T S 0 ) 9. 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. 10. Price to be paid at time T in a forward agreement made at time t to buy an item at time T : F t,t 11. Price of a forward on a non-dividend paying stock: i=1 F 0,T = S 0 e rt 12. Price of a forward on a dividend paying stock with discrete dividends: F 0,T = S 0 e rt CumV alue(div) 13. Price of a forward on a dividend paying stock with continuous dividends: F 0,T = S 0 e (r δ)t 14. Price of a forward on a dividend paying stock with continuous dividends: F 0,T = S 0 e (r δ)t 15. Price of a forward expressed in domestic currency to deliver foreign currency at x 0 exchange rate: F 0,T = x 0 e (r d r f )T

2 2 Lesson 4: Options 1. Assets (a) S 0 = asset price at time 0; it is assumed the purchaser of the stock borrows S 0 and must repay the loan at the end of period T (b) Payoff(Purchased Asset)= S T (c) Profit(Purchased Asset)= S T S 0 e rt (d) Payoff(Short Asset)= S T (e) Profit(Short Asset)= S T + S 0 e rt 2. Calls (a) C 0 = premium for call option at time 0; it is assumed the purchaser of the call borrows C 0 and must repay the loan when the option expires (b) Payoff(Purchased Call)= max {0, S T K} (c) Profit(Purchased Call)= max {0, S T K} C 0 e rt (d) Payoff(Written Call)= max {0, S T K} (e) Profit(Written Call)= max {0, S T K} + C 0 e rt 3. Puts (a) P 0 = premium for put option at time 0; it is assumed the purchaser of the put borrows P 0 and must repay the loan when the option expires (b) Payoff(Purchased Put)= max {0, K S T } (c) Profit(Purchased Put)= max {0, K S T } P 0 e rt (d) Payoff(Written Put)= max {0, K S T } (e) Profit(Written Put)= max {0, K S T } + P 0 e rt

3 3 Lesson 5: 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 10. Put-Call parity: C 0 P 0 = C(K, T ) P (K, T ) = (F 0,T K)v T 11. If the asset pays no dividends, then the no-arbitrage forward price is F 0,T = S 0 e rt S 0 = F 0,T v T. 12. C(K, T ) + Kv T = P K, T ) + S Spread: position consisting of only calls or only puts, in which some options are purchased and some written 14. Bull: investor betting on increase in market value of an asset 15. Bull spread: purchased call with lower strike price K 1 and written call with higher strike price K A bull spread can also be created by a purchased put with strike price K 1 and a written put with strike price K Bear: an investor betting on a decrease in market value of an asset 18. Bear spread: written call with lower strike price K 1 and purchased call with higher strike price K Ratio spread: purchasing m calls at one strike price K 1 and writing n calls at another strike price K Box spread: using options to create a synthetic long forward at one price and a synthetic short forward at a different price 21. Purchased collar: purchased put option with lower strike price K 1 and written call option with higher strike price K Collar width: difference between the call and put strikes 23. Written collar: writing a put option with lower strike price K 1 and purchasing a call option with higher strike price K Zero-cost collar: choosing the strike prices so that the cost is Purchased straddle: purchased call and put with the same strike price K 26. Written straddle: written call and put with the same strike price K 27. Purchased strangle: purchased put with lower strike price K 1 and purchased call with a higher strike price K Written strangle: written put with lower strike price K 1 and written call with a higher strike price K Butterfly spread: a written straddle with a purchased strangle 30. Asymmetric butterfly spread: purchasing λ calls with strike price K 1, purchasing 1 λ calls with strike price K 3, and writing 1 call with strike price K 2 where λ = K 3 K 2 K 3 K 1 and K 1 < K 2 < K 3

4 4 1. Call option payoff: 2. Put option payoff: 3. General Put Call Parity (PCP): 4. PCP for a non-dividend paying stock: Lesson 6: Put-Call Parity max(0, S T K) max(0, K S T ) C(S, K, T ) P (S, K, T ) = (F 0,T K)e rt C(S, K, T ) P (S, K, T ) = S 0 Ke rt 5. PCP for a dividend paying stock with discrete dividends: C(S, K, T ) P (S, K, T ) = S 0 P V 0,T (Div) Ke rt 6. PCP for a dividend paying stock with continuous dividends: C(S, K, T ) P (S, K, T ) = S 0 e δt Ke rt 7. Pre-paid forward at time t: Ft,T P (S) = e r(t t) F t,t 8. Call option written at time t which lets the purchaser elect to receive S T in return for Q T at time T : C(S t, Q t, T t) 9. Put option written at time t which lets the purchaser elect to give S T in return for Q T at time T : 10. PCP for exchange options: P (S t, Q t, T t) C(S t, Q t, T t) P (S t, Q t, T t) = F P t,t (S) F P t,t (Q) P (S t, Q t, T t) = C(Q t, S t, T t) 11. Call-Put relationship in domestic currency: C(S t, Q t, T t) C(Q t, S t, T t) = F P t,t (S) F P t,t (Q) KP d ( 1 x 0, 1 K, T ) = C d (x 0, K, T ) KC d ( 1 x 0, 1 K, T ) = P d (x 0, K, T ) 12. Call-Put relationship in foreign and domestic currency: Kx 0 P f ( 1 x 0, 1 K, T ) = C d (x 0, K, T ) Kx 0 C f ( 1 x 0, 1 K, T ) = P d (x 0, K, T ) 13. PCP for currency options with x 0 as a spot exchange rate: C d (x 0, K, T ) P d (x 0, K, T ) = x 0 e r f T Ke r dt ( 1 C f, 1 ) ( 1 x 0 K, T P f, 1 ) x 0 K, T = 1 e rdt 1 x 0 K e r f T

5 5 Lesson 7: Comparing Options 1. Inequalities for American and European call options: S C Amer (S, K, T ) C Eur (S, K, T ) max(0, F P 0,T (S) Ke rt ) 2. Inequalities for American and European put options: K P Amer (S, K, T ) P Eur (S, K, T ) max(0, Ke rt F P 0,T (S)) 3. Direction property: If K 1 K 2 then 4. Slope property: If K 1 K 2 then C(K 1 ) C(K 2 ) and P (K 1 ) P (K 2 ) C(K 1 ) C(K 2 ) (K 2 K 1 ) and P (K 2 ) P (K 1 ) (K 2 K 1 ) 5. Convexity property: If K 1 K 2 K 3 then C(K 2 ) C(K 3 ) C(K 1) C(K 2 ) C(K 2 ) C(K 1)(K 3 K 2 ) + C(K 3 )(K 2 K 1 ) K 3 K 2 K 2 K 1 K 3 K 1 P (K 2 ) P (K 1 ) P (K 3) P (K 2 ) P (K 2 ) P (K 1)(K 3 K 2 ) + P (K 3 )(K 2 K 1 ) K 2 K 1 K 3 K 2 K 3 K 1 6. Maximum possible value of the difference between the two calls: max(c(k 1 ) C(K 2 )) = e rt (K 2 K 1 )

6 6 Lesson 8: 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 C = e rh (p C u + (1 p )C d ), where p = e(r δ)h d u d 1 p = u e(r δ)h u d 2. Replicating portfolio for the option: 3. To avoid arbitrage, u and d must satisfy: C = S + B, where ( ) Cu C d = e δh S(u d) B = e rh ( ucd dc u u d 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 p 1 = 1 + e σ h 1 p 1 = 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, = If the option always pays off, one could use the Put Call Parity to evaluate its premium (the opposite option will be zero). )

7 7 Lesson 9: Binomial Trees - General 1. For options on futures contracts: u = e σ h, d = e σ h p = 1 d u d = C u C d F (u d), B = C C = e rh (p C u + (1 p )C d )

8 8 Lesson 10: Risk-Neutral Pricing (Other valuation methods of pricing options). 1. The following notation is used when pricing with true probabilities: γ - rate of return 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 p = e(α δ)h d u d 3. Accumulated option value equals accumulated replicating portfolio value: (S + B)e γh = S e αh + Be rh Ce γh = S e αh + Be rh 4. Often used to find the annual return of an option γ (using call notation): C = e γh (pc u + (1 p)c d ) = e rh (p C u + (1 p )C d ) For example, if C d = 0, e γh p = e rh 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 S H and S L are the stock prices in the high and low state, and Q H, Q L - the prices of securities paying $1 when the state H or L occurs, then the state prices are calculated as follows: Q H = S 0e δh e rh S L S H S L and Q L = e rh S H S 0 e δh S H S L 8. If C H, C L - are high and low values of a derivative C, based on the stock prices, then C = C H Q H + C L Q L. 9. If p is the true probability of state H and U H, U L 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 Q H = pu H, Q L = (1 p)u L 10. Initial value of stock using utility: C 0 = pu H C H + (1 p)u L C L = Q H C H + Q L C L 11. Effective annual return of a risk free investment using utility, assuming a one-year horizon: r = 1 Q H + Q L Effective annual return of a stock using utility, assuming a one-year horizon: α = pc H + (1 p)c L pu H C H + (1 p)u L C L 1 = pc H + (1 p)c L Q H C H + Q L C L 1

9 9 13. Risk-neutral probabilities in terms of true probabilities and utility: p pu H Q H = = pu H + (1 p)u L Q H + Q L p = Q H (1 + r) = pu H (1 + r) 14. True probabilities in terms of risk-neutral probabilities and utility: p U L p = p U L + (1 p )U H

10 10 Lesson 11: Binomial Trees: Miscellaneous Topics 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 5. Binomial tree based on forward prices: 6. Cox-Ross-Rubinstein tree: 7. Lognormal (Jarrow-Rudd) tree: 8. In all three cases above p is calculated as: Sδ < Kr u = e (r δ)h+σ h, d = e (r δ)h σ h u = e σ h, d = e σ h u = e (r δ 0.5σ2 )h+σ h, d = e (r δ 0.5σ2 )h σ h p = e(r δ)h d u d 9. To estimate volatility from historical data: ˆσ = ( ) ( ) n x 2 N i n 1 n Si xi x2, where x i = ln, x = S i 1 n N is the number of periods per year, n is the number one less than the number of observations of stock price.

11 11 Lesson 12: Modeling stock prices with the lognormal distribution For a stock whose price S t follows a lognormal model: 1. Assume that ln(s t /S 0 ) N (m, v 2 ), where m = (α δ 0.5σ 2 )t, v 2 = σ 2 t 2. The expected value is E[S t S 0 ] = S 0 e (µ+0.5σ2)t = S 0 e (α δ)t 3. The median price of the stock t years S (0.5) = S 0 e µt = S 0 e (α δ 0.5σ2 )t 4. If Z N(m, v 2 ), then the 100(1 α)% confidence interval is defined to be ( m zα/2 v, m + z α/2 v ), where z α/2 is the number such that P [ ] α Z > z α/2 = 2 5. Since S t /S 0 = e Z(t), Z N(m, v 2 ), the confidence interval for S t will be ( S0 e m z1 α/2v, S 0 e m+z 1 α/2v ) 6. Probabilities of payoffs and partial expectations of stock prices are: 7. ˆd1 and ˆd 2 are defined by 8. Expected call option payoff: 9. Expected put option payoff: P r(s t < K) = N( ˆd 2 ) P r(s t > K) = N( ˆd 2 ) PE[S t S t < K] = S 0 e (α δ)t N( ˆd 1 ) E[S t S t < K] = S 0e (α δ)t N( ˆd 1 ) N( ˆd 2 ) PE[S t S t > K] = S 0 e (α δ)t N( ˆd 1 ) E[S t S t > K] = S 0e (α δ)t N( ˆd 1 ) N( ˆd 2 ) ( S 0 e (α δ)t (1) ) ( N ˆd1 S 0 e (α δ)t (2) ) N ˆd1 E[S t K 1 < S t < K 2 ] = ( (1) ) ( (2) ) N ˆd2 N ˆd2 ˆd 1 = ln ( ) S 0 K + (α δ + 0.5σ 2 )t σ t ˆd 2 = ln ( S 0 K ˆd 2 = ˆd 1 σ t ) + (α δ 0.5σ 2 )t σ t E[max(0, S t K)] = S 0 e (α δ)t N( ˆd 1 ) KN( ˆd 2 ) E[max(0, K S t )] = KN( ˆd 2 ) S 0 e (α δ)t N( ˆd 1 )

12 12 Lesson 13: Fitting stock prices to a lognormal distribution 1. If x i are observed stock prices adjusted to remove the effect of dividends, the estimate for the continuously compounded annual return is: ˆα = ˆµ + 0.5ˆσ 2, where ˆµ = N x ˆσ = N n n 1 ( x 2 i n N is the number of periods per year, n is the number one less than the number of observations of stock price, x i = ln ( Si S i 1 ). 2. A normal probability plot graphs each observation against its percentile of the normal distribution on the vertical axis. However, the vertical axis is scaled according to the standard normal distribution rather than linearly. x2 )

13 13 Lesson 14: 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 C(S, K, σ, r, t, δ) = F P (S)N(d 1 ) F P (K)N(d 2 ), where d 1 = ln ( F P (S)/F P (K) ) σ2 t σ, d 2 = d 1 σ t 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 C = Se δt N(d 1 ) Ke rt N(d 2 ) P = Ke rt N( d 2 ) Se δt N( d 1 ), where d 1 = ln (S/K) + (r δ σ2 )t σ, d 2 = d 1 σ t t 4. The Black-Scholes Formula for a currency asset C = xe r f t N(d 1 ) Ke r dt N(d 2 ) P = Ke r dt N( d 2 ) xe r f t N( d 1 ), where d 1 = ln (x/k) + (r d r f σ2 )t σ, d 2 = d 1 σ t t 5. The Black-Scholes Formula for futures C = F e rt N(d 1 ) Ke rt N(d 2 ) P = Ke rt N( d 2 ) F e rt N( d 1 ), where d 1 = ln (F/K) σ2 t σ, d 2 = d 1 σ t t

14 14 Lesson 15: The Black-Scholes Formula: Greeks Elasticity call = e δt N(d 1 ) put = call e δt = e δt N( d 1 ) Ω = S C 3. Connection between the volatility of an option and the volatility of the underlying stock σ option = σ stock 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 symbol Increase in option price per... Formula Call/put relationship Shape of Graph Delta ( ) increase in stock price (S) Gamma (Γ) increase in stock price (S) Vega (V ) Theta (θ) Rho (ρ) increase in volatility (σ) decrease in time to expiry (t) increase in interest rate (r) Psi (ψ) increase in dividend yield (δ) C S put = call e δt S shaped 2 C S 2 Γ put = Γ call Symmetric hump, peak to the left of strike price, further left with higher t 0.01 C σ V put = V call Asymmetric hump, peak similar to Γ 1 C 365 t θ put = θ call + rke rt δse δt C r ρ put = ρ call 0.01tKe rt Upside-down hump for short lives, gradual decrease for long lives, unless δ is large. Almost always negative for calls, usually negative for puts unless far in-the-money Increasing curve (positive for calls, negative for puts) 0.01 C δ ψ put = ψ call tSe δt Decreasing curve (negative for calls, positive for puts)

15 15 Lesson 16: The Black-Scholes formula: applications and volatility 1. Let C t be the price of a call option at time t. Then the profit on a call sold at time 0 < t < T is: Profit = C t C 0 e rt 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.

16 16 Lesson 17: 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. ( ) Profit = (C(S 1 ) C(S 0 )) + (S 1 S 0 ) e r/365 1 ( S 0 C(S 0 )) 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 3. Delta-gamma-theta approximation S + Sσ h or S Sσ h C(S t+h ) = C(S t ) + ɛ Γɛ2 + hθ 4. Black-Scholes equation for market maker profit (an approximation of the profit formula above): Profit = ( 1 2 Γɛ2 + }{{} hθ + rh( S C(S)) ) }{{}}{{} 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: V ar (R h,i ) = 1 ( S 2 σ 2 Γh ) Boyle-Emanuel annual variance of return when rehedging every h in period i: V ar (R h,i ) = 1 ( S 2 σ 2 Γ ) 2 h 2 7. Formulas for Greeks of binomial trees ( ) Cu C d (S, 0) = e δh S(u d) (Su, h) (Sd, h) Γ(S, 0) Γ(S, h) = S(u d) θ(s, 0) = C(Sud) C(S, 0) (S, 0)ɛ 0.5Γ(S, 0)ɛ2 2h

17 17 Lesson 18: 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 If G = S(k)) and U = ln G Then k=1 E[U] = ln(s(0)) + m, V ar(u) = ṽ 2 E[G(S)] = S(0)e m+ 1 2 ṽ2 ( ) V ar (G(S)) = E[G(S)] 2 eṽ2 1 m = n + 1 m, ṽ 2 (n + 1)(2n + 1) = 2 6n m = (α δ 0.5σ 2 )t, v 2 = σ 2 t ( ) = S(0) 2 e 2 m+ṽ2 eṽ2 1, where 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: CallOnCall(S, K, x, σ, r, t 1, T, δ) P utoncall(s, K, x, σ, r, t 1, T, δ) = C(S, K, σ, r, T, δ) xe rt 1 CallOnP ut(s, K, x, σ, r, t 1, T, δ) P utonp ut(s, K, x, σ, r, t 1, T, δ) = P (S, K, σ, r, T, δ) xe rt 1 5. Value of American call option with one discrete dividend: ( S 0 Ke rt 1 + CallOnP ut S, K, D K (1 )) e r(t t 1) v 2

18 18 Lesson 19: Gap, Exchange and Other Options. 1. Black-Scholes formula for all-or-nothing options: Option Name Value Delta S S > K asset-or-nothing call Se δt N(d 1 ) e δt N(d 1 ) + e δt e d 1 2 /2 σ 2πT S S < K asset-or-nothing put Se δt N( d 1 ) e δt N( d 1 ) e δt e d 1 2 /2 c S > K cash-or-nothing call ce rt N(d 2 ) ce rt e d 2 2 /2 Sσ 2πT c S < K cash-or-nothing put ce rt N( d 2 ) ce rt e d 2 2 /2 Sσ 2πT σ 2πT 2. Black-Scholes formula for gap options: Use K 1 (the strike price) in the formula for C and P. Use K 2 in formula for d 1 C = Se δt N(d 1 (K 2 )) K 1 e rt N(d 2 (K 2 )) P = K 1 e rt N( d 2 (K 2 )) Se δt N( d 1 (K 2 )), where d 1 = ln (S/K 2) + (r δ σ2 )t σ, d 2 = d 1 σ t t = e δt N(d 1 ) + (K 2 K 1 )e rt e d 2 2 /2 3. Exchange options: Recieve option S for option Q. 4. Chooser options: Volatility: σ 2 = σ 2 S + σ 2 Q 2ρσ S σ Q Sσ 2πT C(S, Q, T ) = Se δ ST N(d 1 ) Qe δ QT N(d 2 ), where d 1 = ln ( F P (S)/F P (Q) ) + 0.5σ 2 T σ T d 2 = d 1 σ T V = C(S, K, T ) + e δ(t t) P = ln (S/Q) + (δ Q δ S + 0.5σ 2 )T σ T ( ) ( ) 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 cs t at time t expiring at time T, then the value of the forward start option is V = Se δt N(d 1 ) cse r(t t) δt N(d 2 ) where d 1 and d 2 are computed using T t as time to expiry.

19 19 Lesson 20: Monte Carlo Valuation Generating a log-normal random number A standard normal random variable may be generated as 12 i=1 u i 6, or as N 1 (u i ) General steps for simulating the stock value at time t with initial price S 0 : (1) Given u 1, u 2,, u n uniformly distributed on [0, 1], (2) Find z 1, z 2,, z n where z j = N 1 (u i ) normally distributed with m = 0, v = 1 (3) Generate n 1, n 2,, n n where n j = m + vz j, m = (α δ 0.5σ 2 )t, v 2 = σ 2 t (4) Then S (j) t = S 0 e n j and S t = 1 n n j=1 S(j) t A Lognormal Model of Stock Prices An expression for the stock price: { (α S t = S 0 exp δ 0.5σ 2 ) t + σ } tz, z N(0, 1) The expected stock price: The median stock price: E[S t ] = S 0 exp { (α δ 0.5σ 2 ) t + 1 } 2 σ2 t = S 0 exp {(α δ) t} S 0 exp {( α δ 0.5σ 2) t } Variance Reduction Methods For the control variate method, X = X + ( E[Y ] Ȳ ) For the Boyle modification, V ar(x ) = V ar( X) + V ar(ȳ ) 2Cov( X, Ȳ ) X = X + β ( E[Y ] Ȳ ), V ar(x ) = V ar( X) + β 2 V ar(ȳ ) 2βCov( X, Ȳ ) β is chosen to minimize the expression for V ar(x ): Cov( X, Ȳ ) xi y i XȲ /n β = V ar(ȳ ) = y 2 i Ȳ 2 /n

20 20 Lesson 21: 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 v 2 = s 3. Increments are independent: Z(t + s 1 ) Z(t) is independent of Z(t) Z(t s 2 ) 4. More generally, non-overlapping increments are independently distributed, i.e. for t 1 < t 2 t 3 < t 4, Z(t 2 ) Z(t 1 ) and Z(t 4 ) Z(t 3 ) 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 Let t be the latest time for which you know X(t). Then of the process. 1. X(t + s) X(t) has a normal distribution with m = X(t) + αs and v 2 = σ 2 s 2. X(t + s) X(t) has a normal distribution with m = αs and v 2 = σ 2 s 3. Non-overlapping increments are independently distributed, i.e. for t 1 < t 2 < t 3 < t 4, X(t 2 ) X(t 1 ) and X(t 4 ) X(t 3 ) are independent. 4. If X(0) = 0, then for any t, u such that 0 < t < u, Cov(X(t), X(u)) = σ 2 t 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 0.5σ 2 )s and v 2 = σ 2 s 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) S(t) = (α δ)dt + σdz(t) GBM is an example of an Itô process. 5. The associated arithmetic Brownian motion followed by ln S(t) is d (ln S(t)) = ( α δ 0.5σ 2) dt + σdz(t) 6. When evaluating probabilities for ranges of S(t), look up the normal table using the following parameters: m = ( α δ 0.5σ 2) t, v 2 = σ 2 t

21 21 All of the following are equivalent: 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 0.5σ 2 )dt + σdz(t) ln X(t) X(0) is N ( ln X(0) + (ξ 0.5σ 2 )t, σ 2 t ) ln X(t) ln X(0) = (ξ 0.5σ 2 )t + σz(t) ln X(t) ln X(0) = t X(t) = X(0)e (ξ 0.5σ2 )t+σz(t) X(t) X(0) = t 0 0 (ξ 0.5σ 2 )ds + ξx(s)ds + t 0 t 0 σx(s)dz(s) σdz(s) 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)) = σ 2 t. 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, Multiplication rules: 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 0.5σ 2 )dt + σdz(t) dt dt = dt dz = 0 dz dz = dt dz dz = ρdt

22 22 Lesson 22: Itô s Lemma. Black-Scholes Equation. Any process of the form is called an Itô process. Itô s Lemma Black-Scholes Equation ds(t) = ξ (S(t), t) dt + σ (S(t), t) dz(t) dc(s, t) = C S ds + 0.5C SS (ds) 2 + C t dt 1 2 σ2 S 2 C SS + (r δ)sc S + C t = rc Using greeks, the Black-Scholes Equation is: S(r δ) + 0.5ΓS 2 σ 2 + 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 η = α r σ The Sharpe ratio is the same for all processes based on the same Z(t). Risk-neutral Processes. The true Itô process 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: ds(t) = (r(t) δ(t, S(t)))dt+σ(t, S(t))d Z(t), d Z(t) = dz(t)+ηdt, η = Z(t) follows an arithmetic Brownian motion. Formulas for S a. Expected value Forward price and pre-paid forward price Itô process Total return rate d(s a ) S a E [S(T ) a ] = S(0) a e [a(α δ)+0.5a(a 1)σ2 ]T F 0,T (S a ) = S(0) a e [a(r δ)+0.5a(a 1)σ2 ]T F P 0,T (S a ) = e rt S(0) a e [a(r δ)+0.5a(a 1)σ2 ]T = ( a(α δ) + 0.5a(a 1)σ 2) dt + aσdz(t) γ = a(α δ) + r α(t, S(t)) r(t) Sharpe ratio σ(t, S(t))

23 23 Ornstein-Uhlenbeck process: Definition Integral dx(t) = λ (α X(t)) dt + σdz(t) ( X(t) = X(0)e λt + α 1 e λt) + σ t 0 e λ(s t) dz(s)

24 24 Lesson 23: Binomial tree models for interest rates. Let P t (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 F t,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 F t,t (P (T, T + s)) = This section considers: (1) Interest rate binomial trees (2) The Black-Derman-Toy model for interest rates (3) Pricing forwards and caps (via caplets) P (t, T + s) P (t, T )

25 25 Lesson 24: 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 = F 0,T (P (T, T + s)) where C (F, P (0, T ), σ, T ) = P (0, T ) (F N(d 1 ) KN(d 2 )) P (F, P (0, T ), σ, T ) = P (0, T ) (KN( d 2 ) F N( d 1 )), d 1 = ln(f/k) + 0.5σ2 T σ T d 2 = d 1 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 (2) Each caplet is 1 + K R puts with strike 1 1+K R. 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 (2) Each floorlet is 1 + K R calls with strike 1 1+K R.

26 26 Lesson 20: Equilibrium interest rate models: Vasiček and 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: Duration hedging bond 1 with bond 2: N = (T 1 t)p (t, T 1 ) (T 2 t)p (t, T 2 ) Delta hedging bond 1 with bond 2: N = P r(r, t, T 1 ) P r (r, t, T 2 ) N = B(t, T 1)P (r, t, T 1 ) for Vasiček and Cox-Ingersoll-Ross B(t, T 2 )P (r, t, T 2 ) Yield to maturity on a zero-coupon bond: R(t, T ) = ln (1/P (r, t, T )) T t Differential equation for bond prices: If the interest rate process is dr = a(r)dt + σ(r)dz(t), then where dp P α(r, t, T ) = a(r)p r + 0.5σ(r) 2 P rr + P t P = ln (P (r, t, T )), r(t) = lim R(t, T ) t T T t = α(r, t, T )dt q(r, t, T )dz(t) = a(b r)b(t, T ) + 0.5σ(r) 2 B(t, T ) 2 + P t for Vasiček and Cox-Ingersoll-Ross P q(r, t, T ) = P rσ(r) P = B(t, T )σ(r) for Vasiček and Cox-Ingersoll-Ross Risk-neutral process: dr = (a(r) + φ(r, t)σ(r)) dt + σ(r)d Z, d Z(t) = dz(t) φ(r, t)dt, φ(r, t) is the Sharpe ratio. Black-Scholes Equation analog for bond prices: Sharpe ratio: 1 2 σ(r)2 P rr + (a(r) + φ(r, t)σ(r)) P r + P t rp = 0 General Vasiček Cox-Ingersoll-Ross α(r, t, T ) r φ(r, t) = q(r, t, T ) φ(r, t) = φ it is constant φ(r, t) = φ r/ σ ( φ and σ are constant, σ = σ(r) r ) Definitions of interest rate models: General Rendleman-Bartter Vasiček Cox-Ingersoll-Ross dr = a(r)dt + σ(r)dz(t) dr = ardt + σrdz(t) dr = a(b r)dt + σdz(t) dr = a(b r)dt + σ rdz(t)

27 27 Bond price in Vasiček and CIR models: In Vasiček model, if a 0, then If a = 0, then B(t,T )r(t) P (r, t, T ) = A(t, T )e 1 e a(t t) B(t, T ) = ā T t a = a A(t, T ) = e r[b(t,t )+t T ] B(t,T )2 σ 2 /4a r = b + σφ/a 0.5σ 2 /a 2 B(t, T ) = T t, A(t, T ) = e 0.5σφ(T t)2 +σ 2 (T t) 3 /6 Yield-to-maturity on infinitely-lived bond: Vasiček r = b + σφ/a 0.5σ 2 /a 2 Cox-Ingersoll-Ross r = 2ab/(a φ + γ), where γ = (a φ) σ 2