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Untitled

SomeThing Wrong English 196 cards

Cards in this deck (196)

  1. 1Price of a prepaid forward with no dividendsSt
  2. 2Price of a prepaid forward with discrete dividendsSt - PV(Divs)
  3. 3Price of a prepaid forward with continuous dividendsSte-δ(T-t)
  4. 4Relationship between a forward price and a prepaid forward priceFt,T(S) = FPt,T(S) * er(T-t)
  5. 5Price of a forward with no dividendsSter(T-t)
  6. 6Price of a forward with discrete dividendsSter(T-t) - AV(Divs)
  7. 7Price of a forward with continuous dividendsSte(r-δ)(T-t)
  8. 8Put Call Parity for stockC - P = FPt,T(S) - Ke-r(T-t)
  9. 9Meaning of C(A,B) for exchange optionC(A,B) means receive A, give up B
  10. 10Meaning of P(A,B) for exchange optionP(A,B) means give up A, receive B
  11. 11C(A,B) = P(?,?)P(B,A)
  12. 12Put Call Parity for exchange optionC(A,B) - P(A,B) = FPt,T(A) - FPt,T(B)
  13. 13Put Call Parity for currency exchangeCd(x0,K) = x0 × K × Pf(1/x0,1/K) where x0 is in d/f
  14. 14Put Call Parity for bondsC - P = Bt - PV(coupons) - Ke-r(T-t)
  15. 15When does the price of an American option equal the price of a European option?When the underlying stock pays no dividends
  16. 16A call with a higher strike price has a ________ price.Lower
  17. 17A put with a higher strike price has a _______ price.Higher
  18. 18Direction inequality for different strike prices (calls)C(small) > C(big)
  19. 19Direction inequality for different strike prices (puts)P(small)
  20. 20Slope inequality for different strike prices (calls)C(small) - C(big) < big - small
  21. 21Slope inequality for different strike prices (puts)P(big) - P(small) < big - small
  22. 22Convexity inequality for different strike prices (calls)(C(small)-C(big))/(big-small) > (C(big)-C(biggest))/(biggest-big)
  23. 23Δ for the binomial modelΔ = e-δh(Vu-Vd)/(Su-Sd)
  24. 24Convexity inequality for different strike prices (puts)(P(big)-P(small))/(big-small) < (P(biggest)-P(big))/(biggest-big)
  25. 25Equation for the value of a replicating portfolio and meanings of the variablesV = ΔS+B
    where you buy Δ shares of the underlying stock and lend B at the risk-free rate
  26. 26Probability of an up move using risk-neutral probability pricingp* = (e (r- δ)h -d)/(u-d)
  27. 27Value using risk-neutral probability pricingV = e-rh[Vup*+Vd(1-p*)]
  28. 28Probability of an up move using realistic probability pricingp = (e (α- δ)h -d)/(u-d)
  29. 29Value using realistic probability pricingV = e-γh[Vup+Vd(1-p)]
  30. 30u and d for standard binomial (forward) treesu = e (r- δ)h+σsqrt(h)
    u = e (r- δ)h-σsqrt(h)
  31. 31p* for standard binomial (forward) treesp* = 1/(1+eσsqrt(h))
  32. 32u and d for Cox-Ross-Rubinstein treeu = e σsqrt(h)
    d = e -σsqrt(h)
  33. 33u and d for lognormal (Jarrow-Rudd) treeu = e (r-δ-0.5σ^2)h+σsqrt(h)
    d = e (r-δ-0.5σ^2)h- σsqrt(h)
  34. 34For an option on a future contract using a binomial tree, what does δ approach?r
  35. 35Inequality for r, α, γcall, and γputγ put ≤ r ≤ α ≤ γ call
  36. 36Lognormal model for stock pricesln(S T /S t ) ~ N(m,v 2 )
  37. 37m and v for lognormal modelm = (α-δ-0.5σ 2 )(T-t)
    v = σsqrt(T-t)
  38. 38Median for lognormal modelMedian = Ste(α-δ-0.5σ^2)(T-t)
  39. 39Cov(St,ST) for the lognormal modelCov(S t ,S T ) = E[S T /S t ] × Var(S t |S 0 )
    where Var(S t |S 0 ) = (E[S T |S t ]) 2 (e v^2 -1)
  40. 40Probabilities for the lognormal model1. Pr(S t t > K) = N(d2hat)
    where d2hat is made with α instead of r
  41. 41Confidence interval for the lognormal modelS L T = S t e (α-δ-0.5σ^2)(T-t) + σsqrt(T-t)z LU
    S U T = S t e (α-δ-0.5σ^2)(T-t) + σsqrt(T-t)z U
    where z U = - z L = -N -1 (p/2)
  42. 42Generalized Black-Scholes formula for callsC = F P (S) × N(d 1 ) - F P (K) × N(d 2 )
    where d 1 = (ln(F P (S)/F P (K))+0.5σ 2 (T-t))/(σsqrt(T-t))
  43. 43Generalized Black-Scholes formula for putsP = F P (K) × N(-d 2 ) - F P (S) × N(-d 1 )
    where d 1 = (ln(F P (S)/F P (K))+0.5σ 2 (T-t))/(σsqrt(T-t))
  44. 44What changes between the formulas for d1 and d2Sign in front of the σ term is positive for d1 and negative for d2
  45. 45Black-Scholes formula for stockC = S t e -δ(T-t) N(d 1 ) - Ke -r(T-t) N(d 2 )
    P = Ke -r(T-t) N(-d 2 ) - S t e - δ (T-t) N(-d 1 )
  46. 46Black-Scholes formula for futuresThe same as the formula for stocks except δ goes to r and So goes to FP0,T(S)
  47. 47Meaning of deltaChange in option price over change in stock price
  48. 48Meaning of gammaChange in delta over change in stock price
  49. 49Meaning of thetaChange in option price as time advances
  50. 50Meaning of vegaChange in option price over change in volatility
  51. 51Meaning of rhoChange in option price over change in risk-free rate
  52. 52Meaning of psiChange in option price over change in dividend yield
  53. 53Delta ______ as the stock price increases.increases
  54. 54Δ's for Black-ScholesΔ call = e -δT N(d 1 )
    Δ put = -e -δT N(-d 1 )
  55. 55Meaning of elasticityPercent change in option price over percent change in stock price
  56. 56Equation for elasticityΩ = (ΔS)/V
  57. 57Relationship between the volatility of an option and the volatility of the underlying stockσoption = |Ω|σstock
  58. 58Equation for risk premiumα - r
  59. 59Relationship between risk premium of the option and risk premium of the stock if Black-Scholes appliesγ - r = Ω(α - r)
  60. 60Bounds on the elasticities of calls and putsΩ call ≥ 1
    Ω put ≤ 0
  61. 613 components of overnight profit1. Gain on stocks
    2. Gain on options
    3. Interest on borrowed/lent money
  62. 62The price movement with no gain or loss to a delta-hedger±Sσsqrt(h)
  63. 63Delta-Gamma-Theta approximationVt+h - Vt = Δε + 0.5Γε2 + θh
  64. 64Boyle-Emanuel periodic variance of return when rehedging every h in period iVar(Rh,i) = 0.5(S2σ2Γh)2
  65. 65Δ for binomial treesΔ(S,0) = e-δh(Vu-Vd)/(Su-Sd)
  66. 66Γ for binomial treesΓ(S,0) = Γ(S,h) = (Δ(Su,h)-Δ(Sd,h))/(Su-Sd)
  67. 67Equation for arithmetic average A(S)A(S) = sum from t=1 to t=N of stock prices over N
  68. 68Equation for geometric average G(S)G(S) = (product from t=1 to t=N of stock prices)1/N
  69. 69Payoff of an average price callmax(0, Avg(S) - K)
  70. 70Payoff of an average strike callmax(0, S-Avg(S))
  71. 71Payoff of an average price putmax(0, K-Avg(S))
  72. 72Payoff of an average strike putmax(0, Avg(S)-S)
  73. 73Relationship between the value of an Asian option and the value of an otherwise equivalent ordinary optionVAsian ≤ Vordinary
  74. 743 types of barrier options1. knock-in
    2. knock-out
    3. rebate
  75. 75What does a knock-in option do?Goes into existence if barrier is reached
  76. 76What does a knock-out option do?Goes out of existence if barrier is reached
  77. 77What does a rebate option do?Pays fixed amount if barrier is reached
  78. 78Relationship between S0 and Barrier for down- and up- optionsDown: S 0 0 > B
  79. 79Equation relating knock-in, knock-out, and ordinary optionsKnock-in + knock-out = ordinary
  80. 80Relationship between the price of a barrier option and the price of an ordinary optionBarrier option ≤ Ordinary option
  81. 81Put Call Parity for compound call optionCallonCall - PutonCall = C Eur - xe -rt1
    where t 1 is when the compound option expires and x is the compound option's strike price
  82. 82Put Call Parity for compound put optionCallonPut - PutonPut = P Eur - xe -rt1
    where t 1 is when the compound option expires and x is the compound option's strike price
  83. 83Meaning of K1 and K2 for gap optionsK 1 is the strike price, which determines the amount of the payoff
    K 2 is the trigger price, which determines whether the option will have a payoff
  84. 84Put Call Parity for gap optionsGapCall - GapPut = S0e-δT - K1e-rT
  85. 85Where are K1 and K2 used in pricing a gap option?K1 is used in the pricing equation while K2 is used in the calculations of d1 and d2
  86. 86Value of an exchange call optionC(A,B) = FP(A) × N(d1) - FP(B) × N(d2)
  87. 87Value of an exchange put optionP(A,B) = FP(B) × N(-d2) - FP(A) × N(-d1)
  88. 88σ of an exchange optionσ = sqrt(σA2 + σB2 - 2ρσAσB)
  89. 89Payoff of an asset optionThe asset or 0 at time T
  90. 90Payoff of a cash option$1 or 0 at time T
  91. 91Time-t price of an asset callSte-δ(T-t)N(d1)
  92. 92Time-t price of an asset putSte-δ(T-t)N(-d1)
  93. 93Time-t price of a cash calle-r(T-t)N(d2)
  94. 94Time-t price of a cash pute-r(T-t)N(-d2)
  95. 95Definition of a chooser optionAn option that allows the owner to choose at time t whether the option will become a European call or put with strike K expiring at time T
  96. 96Price of a chooser optionVt = e-δ(T-t) × max(0, Ke-(r-δ)(T-t) - St) + C(St, K, T-t)
  97. 97When to use the risk-neutral distribution versus the true distribution in Monte Carlo valuationRisk-neutral: when discounting is needed
    True: when discounting is not needed
  98. 98How to simulate (Monte Carlo) lognormal stock pricesST = Stem+vzi
  99. 99Control Variate Method to estimate option YVar(Y*) = Var( Y ~) + β 2 Var(X~) - 2βCov(X~,Y~)
  100. 100Var(Y*) is minimized when...β = Cov(X~,Y~)/Var(X~)
  101. 101When β is set to minimize Var(Y*), what does Var(Y*) equal?Var(Y*) = Var(Y~)(1-ρX~,Y~2)
  102. 1022 ways to do the antithetic variate method1. For every u i , also simulate using 1 - u i
    2. For every z i , also simulate using -z i
  103. 103How to stratify numbersnew number = left point + (Δ to get to right point)(original ui)
  104. 104Black-Scholes equation for Brownian motion (using derivatives instead of Greeks)(r-δ)SV S + 0.5σ 2 S 2 V SS + V t = (r-δ*)V
    where δ* is for the derivative and δ is for the stock
  105. 105Black-Scholes equation for Brownian motion (using Greeks instead of derivatives)(r-δ)SΔ + 0.5σ 2 S 2 Γ + θ = (r-δ*)V
    where δ* is for the derivative and δ is for the stock
  106. 1064 characteristics of standard Brownian motion1. Z(0) = 0
    2. Z(t+s) - Z(t) ~ N(0,s)
    3. Z(t+h) - Z(t) is independent of Z(t) - Z(t-s)
    4. Z(t) is continuous
  107. 1072 conditions that make Z(t) a martingale1. E[Z(t+h)-Z(t)] = 0
    2. E[Z(t+h)|Z(t)] = Z(t)
  108. 108Equation for arithmetic Brownian motiondX(t) = a dt + b dZ(t)
  109. 109Equation for Ornstein-Uhlenbeck processdX(t) = λ[α - X(t)]dt + σdZ(t)
  110. 110Equation for geometric Brownian motiondX(t)/X(t) = a dt + b dZ(t)
  111. 111Equation for d[lnX(t)]d[lnX(t)] = (a - 0.5b2)dt + b dZ(t)
  112. 112Equation for Brownian motion where the Black-Scholes framework appliesdS(t)/S(t) = (α-δ)dt + σdZ(t)
  113. 113Equation for d[lnS(t)] when Black-Scholes framework appliesd[lnS(t)] = (α-δ-0.5σ2)dt + σdZ(t)
  114. 114Ito's lemmadV = VSdS + 0.5VSS(dS)2 + Vtdt
  115. 1154 multiplication rules for dt and dZ1. dt × dt = 0
    2. dt × dZ = 0
    3. dZ × dt = 0
    4. dZ × dZ = dt
  116. 116When do two Ito processes have equal Sharpe ratios?When they depend on the same dZ(t)
  117. 117Sharpe ratioφ = (α-r)/σ
  118. 118To go from true measures to risk-neutral measures for dZ(t) and Z(t)dZ~(t) = dZ(t) + φdt
    Z~(t) = Z(t) + φt
  119. 119E[S(T)a] = ?E[S(T)a] = [S(t)]ae[a(α-δ)+0.5a(a-1)σ^2](T-t)
  120. 120Ft,T[S(T)a] = ?F t,T [S(T) a ] = [S(t)] a e [a(r-δ)+0.5a(a-1)σ^2](T-t)
  121. 121General Ito's Process for r(t) and P(r,t,T) [dr(t)=?]dr(t) = a(r)dt + σ(r)dZ(t)
  122. 122General Ito's Process for r(t) and P(r,t,T) (dP(r,t,T))dP(r,t,T)/P(r,t,T) = α(r,t,T)dt - q(r,t,T)dZ
  123. 123Definition of α(r,t,T) for interest rate modelsα(r,t,T) = (1/P)[a(r)×Pr + 0.5[σ(r)]2]×Prr + Pt]
  124. 124Definition of q(r,t,T) for interest rate modelsq(r,t,T) = -(Pr/P)σ(r)
  125. 125Sharpe ratio for interest rate modelsφ(r,t) = [α(r,t,T) - r] / q(r,t,T)
  126. 126Delta-gamma-theta approximationP(t+h,T) - P(t,T) = Δε + 0.5Γε 2 + θh
    where ε is the change in the risk-free rate
  127. 1274 types of interest rate models1. General
    2. Rendleman-Bartter
    3. Vasicek
    4. Cox-Ingersoll-Ross
  128. 128General interest rate modeldr = a(r)dt + σ(r)dZ(t)
  129. 129Rendleman-Bartter interest rate modeldr = ardt + σrdZ(t)
  130. 130Vasicek interest rate modeldr = a(b-r)dt + σdZ(t)
  131. 131CIR interest rate modeldr = a(b-r)dt + σsqrt(r)dZ(t)
  132. 132Bond price in Vasicek and CIR modelsP(r,t,T) = A(t,T)e-B(t,T)r(t)
  133. 133B(t,T) for Vasicek modelB(t,T) = (1-e-a(T-t))/a
  134. 134Yield-to-maturity on infinite bond for Vasicek modelyield = b + (σφ/a) - 0.5σ2/a2
  135. 135Equation to delta hedge bond 1 with bond 2 (N)N = -[PT(r,t,T1)]/[PT(r,t,T2)]
  136. 136Equation to duration hedge bond 1 with bond 2 (N)N = -[(T1-t)P(t,T1)]/[(T2-t)P(t,T2)]
  137. 1373 characteristics of the Black-Derman-Toy interest rate model1. Effective interest rates
    2. p*= 0.5
    3. The ratio between two consecutive nodes is e 2sqrt(h)σ t
  138. 138For the Black-Derman-Toy model, σh = ?σh = ln(rh1/rh2)/[2sqrt(h)]
  139. 139q(r,t,T) for Vasicek and CIR modelsq(r,t,T) = B(t,T)σ(r)
  140. 140Black's formula for callsC = P(0,T)[FN(d1) - KN(d2)]
  141. 141Black's formula for putsP = P(0,T)[KN(-d2) - FN(-d1)]
  142. 142What does F represent in Black's formula and how do you calculate it?F = F0,t(T,T+s) = [P0(0,T+s)]/[P0(0,T)]
  143. 143Payoff at time T of a T+1 year capletL × [max(0, RT(T+s) - K)]/[1 + RT(T+s)]
  144. 144How to value an interest rate cap1. Calculate the caplet's payoff at each node
    2. Discount the payoffs to time 0 and multiply them by their probabilities and take the sum to get each caplet
    3. Cap = sum of caplets
  145. 145How do you synthesize a risk-free asset?Choose amounts of the assets such that there is no dZ(t) term
  146. 146What is the price of a barrier option with barrier 0?The same as the price of the corresponding ordinary option
  147. 147Expected change in r using the Vasicek modela(b - r)
  148. 148Does Black-Scholes use implied volatility or historical volatility?Implied volatility
  149. 149When given stock prices at different months, how do you calculate the monthly and yearly α?α(monthly) = rbar + 0.5s 2
    α(yearly) = 12(rbar + 0.5s 2 )
  150. 150As the volatility of the stock increases, the price of the option _______.increases
  151. 151ln[E[ST]/Median[ST]] = ?0.5 σ 2 T
  152. 152When pricing Asian options using a binomial tree, what prices are used to calculate the average?The prices along the path, not all price from one time t
  153. 153Overnight profit for a market-maker-0.5 Γ ε 2 - θh - rh[ ΔS - V]
  154. 154Cov[S(t),S(u)] = ?Cov[S(t),S(u)] = S(0) 2 e ( α-δ)(t+u) [e tσ^2 -1]
  155. 155How do you use Black's formula to price a caplet?price of caplet = (1+cap rate)(put with strike 1/(1+cap rate))
  156. 156E[dZ(t)] = ?E[dZ(t) = 0
  157. 157If X(t) follows arithmetic Brownian motion, then what are Var[X(t)] and Cov[X(t),X(u)]?Var[X(t)] = σ 2 t
    Cov[X(t),X(u)] = σ 2 min(t,u)
  158. 158What is the distribution for lnS(t)?lnS(t) ~ N[lnS(0) + ( α- δ-0.5 σ 2 )t, σ 2 t ]
  159. 159What is the price of an exchange option using S and K, and what do S and K stand for?S is the underlying asset, which you receive.
    K is the strike asset, which you give up.
    V = Se -
    δ S T N(d 1 ) - Ke -δ K T N(d 2 )
  160. 160In order for two interest rate models to produce the same bond prices for all values of the short rates, what must be equal?The risk-neutral processes
  161. 161If two risk-neutral processes are equal, then _____ must be equal.the drifts
  162. 162Another method for calculating zi when given 12 ui'szi = [sum from 1 to 12 of ui] - 6
  163. 163What are the three steps you take to replicate a stock?1. Buy a call
    2. Sell a put
    3. Lend PV(K+divs)
  164. 164Cov(X,Y) = ? in sumsCov(X,Y) = (sum of XiYi) - nXbarYbar
  165. 165What does drift refer to?dt term
  166. 166When does Δ approach 1?When the option is deep in-the-money
  167. 167When does Δ approach 0?When the option is deep out-of-the-money
  168. 168d[integral from 0 to t of sZ(s)ds] = ?tZ(t)dt
  169. 169What does "excess over" imply?Subtraction
  170. 170PV(treasury bill) = ?PV(treasury bill) = [maturity value]e-r(T-t)
  171. 171Var[lnFPt,T(S)] = ?σ 2 (T-t)
  172. 172Time-t price of a derivative securityV = e-rtE*[payoff]
  173. 173As the number of periods in a binomial tree increases, returns become ______ distributed and stock prices become _______ distributed.normally, lognormally
  174. 174d1hat - d2hat = ?σsqrt(t)
  175. 175Straddle = ?Buy 1 call and buy 1 put
  176. 176For interest rate models, what is the relationship between dZ(t) and dZ~(t)?dZ~(t) = dZ(t) - φ(r,t)dt
  177. 177Var[Z(t+h)|Z(t)] = ?h
  178. 178How do you check if a function is standard Brownian motion?"d" the function. If "d" is equal to dZ or -dZ, then it is standard Brownian motion.
  179. 179θ call - θ put = ?θ call - θ put = δS t e -δ(T-t) - rKe -r(T-t)
  180. 180Γ Euro call ?? Γ Euro put=
  181. 181If Asset 1 and Asset 2 follow the same Z(t) but φ 1 > φ 2 , then what should you do to exploit the arbitrage opportunity?1. Buy 1/( σ 1 S 1 ) shares of Asset 1
    2. Sell 1/( σ 2 S 2 ) shares of Asset 2
    3. Lend 1/ σ 2 - 1/ σ 1
    so that the arbitrage profit is [φ 1 -φ 2 ]dt
  182. 182Expected payoff for a callE[Call payoff] = S t e ( α-δ)(T-t) N(d1hat) - KN(d2hat)
  183. 183Expected payoff for a putE[Put payoff] = KN(-d2hat) - S t e ( α-δ)(T-t) N(-d1hat)
  184. 184dN(di)/dS = ?dN(d i )/dS = e -0.5d i ^2 /(S σsqrt(2 πT))
  185. 185Time 0 price of a forward start optionV 0 = F 0,T P (S)[e - δ(T-t) N(d 1 ) - Xe -r(T-t) N(d 2 )
  186. 186[Utility] What does Qu equal?Qu = (p)(Uu) = (p*)(e-rh)
  187. 187[Utility] What does Qd equal?Qd = (1-p)(Ud) = (1-p*)(e-rh)
  188. 188[Utility] What does S equal in terms of Qu and Qd?S = Q u S u e δh + Q d S d e δh
  189. 189Var[aX + bY] = ?Var[aX + bY] = a2Var[X] + b2Var[Y] + 2abCov[X,Y]
  190. 190Bounds for delta call and delta put0 ≤ del ta call ≤ 1
    -1 ≤ del ta put ≤ 0
  191. 191Bounds for gamma call and gamma putgamma call ≥ 0
    gamma put ≥ 0
  192. 192Bounds for vega call and vega putVega call ≥ 0
    Vega put ≥ 0
  193. 193Bounds for rho call and rho putRho call ≥ 0
    Rho put ≤ 0
  194. 194Bounds for psi call and psi putPsi call ≤ 0
    Psi put ≥ 0
  195. 195PE[ST|ST < K] = ?S t e ( α - δ)(T-t) N(-d1hat)
  196. 196PE[ST|ST > K] = ?S t e ( σ-δ)(T-t) N(+d1hat)

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Flashcards on Untitled, created by SomeThing Wrong on 28/10/2018.
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