ACTS 4302 FORMULA SUMMARY Lessons 1-3: Introduction to Financial Derivatives, Forward and Futures Contracts.
P 1. If an asset pays no dividends, then the prepaid forward price is F0,T = S0. 2. 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 3. 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 . 4. Cost of carry = r − δ P rT 5. F0,T = F0,T e 6. Implied fair price: the implied value of S0 when it is unknown based on an equation relating S0 to F0,T 7. Implied repo rate: implied value of r based on the price of a stock and a forward 8. Annualized forward premium = 1 ln F0,T T S0 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 :
Ft,T 11. Price of a forward on a non-dividend paying stock: rT F0,T = S0e 12. Price of a forward on a dividend paying stock with discrete dividends: rT F0,T = S0e − CumV alue(Div) 13. Price of a forward on a dividend paying stock with continuous dividends: (r−δ)T F0,T = S0e 14. Price of a forward on a dividend paying stock with continuous dividends: (r−δ)T F0,T = S0e
15. Price of a forward expressed in domestic currency to deliver foreign currency at x0 exchange rate:
(rd−rf )T F0,T = x0e 2
Lesson 4: 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 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 T 10. Put-Call parity: C0 − P0 = C(K,T ) − P (K,T ) = (F0,T − K)v rT T 11. If the asset pays no dividends, then the no-arbitrage forward price is F0,T = S0e ⇒ S0 = F0,T v . T 12. C(K,T ) + Kv = PK,T ) + S0 13. 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 K1 and written call with higher strike price K2 16. A bull spread can also be created by a purchased put with strike price K1 and a written put with strike price K2. 17. Bear: an investor betting on a decrease in market value of an asset 18. Bear spread: written call with lower strike price K1 and purchased call with higher strike price K2 19. Ratio spread: purchasing m calls at one strike price K1 and writing n calls at another strike price K2 20. 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 K1 and written call option with higher strike price K2 22. Collar width: difference between the call and put strikes 23. Written collar: writing a put option with lower strike price K1 and purchasing a call option with higher strike price K2 24. Zero-cost collar: choosing the strike prices so that the cost is 0 25. 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 K1 and purchased call with a higher strike price K2 28. Written strangle: written put with lower strike price K1 and written call with a higher strike price K2 29. Butterfly spread: a written straddle with a purchased strangle 30. 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 4
Lesson 6: 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 5
Lesson 7: 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) 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 −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). 7
Lesson 9: 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) 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 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 9
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 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 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. 11
Lesson 12: 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 expected value is (µ+0.5σ2)t (α−δ)t E[St|S0] = S0e = S0e 3. The median price of the stock t years (0.5) µt (α−δ−0.5σ2)t S = S0e = S0e 4. 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 5. 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 6. 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
7. dˆ1 and dˆ2 are defined by