MFE Notes - Spring 2010 Sitting Lesson 1 - Put-Call Parity • Bull Spread : pays off if stock moves up in price – with Calls: buy C K 2 and sell C K 1 ; K 1 >K 2 – with Puts: buy P K 2 and sell P K 1 ; K 1 >K 2 • Bear Spread : pays off if the stock moves down in price – with Calls: buy C K 1 and sell C K 2 ; K 1 >K 2 – with Puts: buy P K 1 and sell P K 2 ; K 1 >K 2 • Straddle : buy a Call and a Put – same K ⇒ Payoff = |S T - S 0 | – bet on volatility – Strangle - buy P K 2 and C K 1 ; K 1 >K 2 • Synthetic Stock : Solve for S 0 : S 0 = e δt (C - P + Ke -rt ) • Synthetic Treasury : Solve for Ke -rt : Ke -rt = S 0 e -δt - C + P • Synthetic Options : Solve for C or P • Conversion – Synthetically buy a T-bill – Lend dollars • Reverse Conversion – Synthetically sell a T-bill – Borrow dollars 1
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MFE Notes - Spring 2010 Sitting
Lesson 1 - Put-Call Parity
• Bull Spread: pays off if stock moves up in price
– with Calls: buy CK2 and sell CK1 ; K1 > K2
– with Puts: buy PK2 and sell PK1 ; K1 > K2
• Bear Spread: pays off if the stock moves down in price
– with Calls: buy CK1 and sell CK2 ; K1 > K2
– with Puts: buy PK1 and sell PK2 ; K1 > K2
• Straddle: buy a Call and a Put
– same K ⇒ Payoff = |ST − S0|– bet on volatility
– Strangle - buy PK2 and CK1 ; K1 > K2
• Synthetic Stock: Solve for S0: S0 = eδt (C − P +Ke−rt)
• Synthetic Treasury: Solve for Ke−rt: Ke−rt = S0e−δt − C + P
• Synthetic Options: Solve for C or P
• Conversion
– Synthetically buy a T-bill
– Lend dollars
• Reverse Conversion
– Synthetically sell a T-bill
– Borrow dollars
1
• Converting between domestic and foreign currency options:
– A put denominated in the base currency is equivalent to somenumber of calls denominated in the foreign currency
– KPd
(1x0, 1K, T)
= Cd(x0, K, T )
– Kx0Pf
(1x0, 1K, T)
= Cd(x0, K, T )
• Bid-Ask Prices
– The verb applie to the market-maker, not the retail customer
∗ The market-maker bids the bid price when buying a share ofstock
∗ The market-maker asks the ask price when selling a share ofstock
∗ Bid Price < Ask Price
Lesson 2 - Comparing Options
• American Options:
– Calls: S ≥ CA ≥ CE ≥ max(0, F P
0,T (S)−Ke−rT , S0 −K)
– Puts: K ≥ PA ≥ PE ≥ max(0, Ke−rT − F P
0,T (S), K − S0
)• Early exercise of American Options
– Calls:
∗ lose the implicit Put
∗ if non-dividend then CA = CE
∗ not rational if PVt,T (Div) < K(1− e−r(T−t)) + P
· b/c you get stock and Divs. but pay K and lose the im-plicit Put
– Puts:
∗ lose the implicit Call
2
∗ may be rationa even if no dividends
∗ earn interest on K
• Different Strike Prices
– Direction:
∗ C1 ≤ C2 and P1 ≤ P2
∗ ∂C∂K≤ 0 and ∂P
∂K≥ 0
– Slope:
∗ C1 − C2 ≥ K2 −K1 and P1 − P2 ≤ K1 −K2
∗ ∂C∂K≥ −1 and ∂P
∂K≤ 1
– Convexity:
∗ C1−C2
K1−K2≥ C2−C3
K2−K3and P1−P2
K1−K2≥ P2−P3
K2−K3
∗ ∂2C∂K2 ≥ 0 and ∂2P
∂K2 ≤ 0
· K1 > K2 > K3
• Strike Price Increases Over Time on a Call - suppose that a stock doesnot pay dividends and the strike price increases at a rate that is less thanor equal to r:
KT ≤ Kter(T−t)
The longer the call option, the more valuable it is:C(S0, KT , T ) ≥ C(S0, Kt, t) for T > t
If the inequality above is violated, then arbitrage is available.
That is if:KT ≤ Kte
r(T−t) and C(S0, KT , T ) < C(S0, K, T )
then arbitrage can be obtained with the following steps:
1. Buy the longer option and sell the shorter one
2. At time t, the shorter option is in the money, sell stock short andlend Kt at the risk-free rate
Lesson 3 - Binomial Trees - Stock, One Period
3
• Replicating Portfolio: B stands for bond, not borrowing; amount welend
• American options on Stock with 1 discrete dividend
– CA = S0 −Ke−rt1 + CoP (S,K,D −K(1− e−r(T−t1)), t1, T )
• Asian Options - ignore initial price
Lesson 14 - Gap, Exchange and Other Options
• All-or-nothing Options
– S|S > K = S0e(r−δ)TN(d1)
– S|S < K = S0e(r−δ)TN(−d1)
– c|S > K = ce−rTN(d2)
– c|S < K = ce−rTN(−d2)
– Delta for all-or-nothing options
∗ ∂N(di)∂S
= e−d2i2
Sσ√
2πT
• Gap Options
12
– Remember that ST > trigger for Calls and ST < trigger for Puts
– Put-Call Parity applies
– If two otherwise identical gap options have different strike prices,then use linear interpolation to find the price of a third otherwiseidentical gap option with a different strike price.
• Exchange Options
– volatility measures the variance of rate of return (not the dollarreturn)i.e. 2 shares have the volatility as 1 share
• Chooser Options
– Derivation
Vt = max(C(S,K, T − t), P (S,K, T − t)) (1)
= C(S,K, T − t) +max(0, P (S,K, T − t)− C(S,K, T − t) (2)
= C(S,K, T − t) +max(0, Ke−r(T−t) − Se−δ(T−t)) (3)
@ t0 ⇒ V0 = C(S,K, T ) + e−δ(T−t) · P (S,Ke−(r−δ)(T−t), t)
• Forward Start Options
– Purchase a call @ t with K = cSt expiring @ T , then the value ofthe forward start option is:
∗ V = Se−δTN(d1)− cSe−r(T−t)−δtN(d2)
· di are computed using T − t as time to expiry
Lesson 15 - Monte Carlo Valuation
• Generating LogNormal random numbers
1. Let zj =∑12
i=1 ui − 6 where ui ∈ U [0, 1]
2. Let zj = N−1(uj)
13
• Use r to discount when pricing options
• Use α for true expected payoffs
• Control Variate Method
– Let X∗ = X + (E(Y )− Y ), Y ∼ control variate
∗ ⇒ V (X∗) = V (X) + V (Y )− 2Cov(X, Y )
∗ Always use sample variance / covariance formula
– Boyle modification:
∗ X∗ = X + β(E(Y )− Y )
· ⇒ V (X∗) = V (X) + β2V (Y )− 2βCov(X, Y )
· Optimal value for β = Cov(X,Y )
V (Y )
· Variance becomes: V (X∗) = V (X)(
1− ρ2X,Y
)• Other Variance Reduction Techniques
– Antithetic Variates: for every ui, use 1− ui
– Stratified Sampling: break sampling space into strata and thenscale uniform #s to be in these strata
∗ If you had 4 strata: [0, .25), . . . , [.75, 1) then generate sets of4 ui on [0, 1), multiply all 4 by .25, put the first in [0, .25),add .25 to 2nd number, etc.
– Latin Hypercube Sampling
– Importance Sampling
– Low Discrepancy Sequences
Lesson 16 - Brownian Motion
• Random Walk
14
1. X(0) = 0
2. For t > 0, if X(t− 1) = k, then X(t) =
k + 1, with p = 1
2
k − 1, with p = 12
3. Memoryless.
– Pr(X(t+ u) = l|X(t) = k) = Pr(X(u) = l − k)
4. X(t) is random, distance traversed is not.
– Sum of the squares of the movement is t
5. X(t) ∼ Bin(t, 1
2
)• Brownian Motion
– Move√h per h units of time and take limh→0
∗ ⇒ Cont. Random Walk and Binomial → Normal
– Properties
1. Z(0) = 0
2. Z(t+ s)|Z(t) ∼ N(Z(t), s)
3. Z(t+ s1)− Z(t) is independent of Z(t)− Z(t− s2)
4. Z(t) is cont. in t
– Expected Values Under Pure Brownian Motion
∗ E[Z(t)] = 0
∗ E[Z(t+ h)|Z(t)] = Z(t)
∗ E[Z(t+ h)− Z(t)] = 0
∗ E[dZ(t)] = 0
∗ E[dZ(t)|Z(t)] = 0
∗ E [(Z(t))2] = t
15
∗ E [(dZ(t))2] = dt
∗ E[Z(t)Z(s)] = Min(t, s)
– Variances under Pure Brownian Motion
∗ V [Z(t)] = t
∗ V [Z(t+ h)|Z(t)] = h
∗ V [Z(t+ h)− Z(t)] = h
∗ V [dZ(t)] = dt
∗ V [dZ(t)|Z(t)] = dt
– is a diffusion process - cont. process in which the absolute valueof the R.V. tends to get larger
– is a martingale - process X(t) for which E[X(t+ s)|X(t)] = X(t)
∗ ABM and GBM are martingales iff they have zero drift
• Arithmetic Brownian Motion
– X(t) = αt+ σZ(t)
– X(t+ s)−X(t) ∼ N(µs, σ2s)
– X(t+ s)|X(t) ∼ N(X(t) + µs, σ2s)
• Geometric Brownian Motion
– If ln(X(t)X(0)
)∼ N(µt, σ2t) then X(t)−X(0) ∼ LogNormal
∗ Mean = e(µ+ 12σ2)t
∗ Variance = e(2µ+σ2)t(eσ2t − 1)
• To go from GBM to ABM, you must subtract 12σ2
– When dealing with probabilities, you must convert to ABM
16
• V ar(ln(S(t))|S(0)) = V ar(ln(F0,T (S))) = V ar(ln(F P0,T (S)))
• Forms of BM
– GBM: dSS
= (α− δ)dt+ σdZ
– ABM: d(ln(S)) = (α− δ − 12σ2)dt+ σdZ
• When you add δ to total return (for Sharpe Ratio), only add to S, notC
• Portfolio Returns: Suppose that a portfolio P consists of 2 assets, Aand B. If x is the percentage of the portfolio is invested in A and(1− x) is the percentage invested in B, then the instantaneous changein the price of the portfolio is:
dP (t)P (t)
= xdA(t)A(t)
+ (1− x)dB(t)B(t)
To find the instantaneous return on the portfolio, include the dividends.
– Instantaneous Return on Portfolio
∗ dP (t)P (t)
+(xδA+(1−x)δB)dt = x[dA(t)A(t)
+ δA
]+(1−x)
[dB(t)B(t)
+ δB
]Lesson 17 - Ito’s Lemma
• dC = CSdS + 12CSS(dS)2 + Ctdt
• Multiplication rules: All → 0 except (dZ)2 = dt
• The Black-Scholes Equation
– rC = S∆(r − δ) + 12ΓS2σ2 + θh
• Sharpe Ratio(Only works for GBM)
– φ = α−rσ
∗ α is total return (includes δ)
17
– For 2 Ito processes with the same dZ, the Sharpe Ratios are equal
• Problems which give 2 processes, Prices and ask how much should beallocated to each process. Such as:
1. dS1
S1= α1dt+ σ1dZ and dS2
S2= α2dt+ σ2dZ
2. x shares of S1 and y shares of S2, r = r
(a) Solve S1 · x · α1 + S2 · y · α2 = (S1 · x+ S2 · y)r
(b) If you know x and need y, look @ σ1
σ2, that’s ratio of value of
S2 you need to buy/sell. Since S1 costs S1 · x then you need
to buy/sell S1 · x(σ1
σ2
)= y
• CAPM: αi−rσi
= ρi,M
(αM−rσM
)– φi = ρi,MφM
• Risk-Neutral Processes
– True Ito Process: dS = (α− δ)dt+ σdZ
– Risk-Neutral Ito Process: dS = (r − δ)dt+ σdZ
∗ dZ = dZ + ηdt
· where η = α−rσ
∗ E∗[Z(T )] = 0
∗ E∗[Z(T )] =(r−ασ
)T
∗ E[Z(T )] = 0
∗ E[Z(T )] =(α−rσ
)T
• Valuing a Forward on Sa
– E[S(T )a] = Sa0e[a(α−δ)+ 1
2σ2a(a−1)]T
18
– F0,T (Sa) = Sa0e[a(r−δ)+ 1
2σ2a(a−1)]T
– F P0,T (Sa) = e−rT · F0,T (Sa)
• Ito Process for Sa
– If C = Sa and dSS
= (α− δ)dt+ σdZ then
∗ dCC
= (a(α− δ) +1
2σ2a(a− 1) + δ∗)︸ ︷︷ ︸γ
dt+ σadZ
· δ∗ ∼ derivative’s dividend yield
∗ ⇒ Sharpe Ratios = γ−raσ
= α−rσ
· ⇒ γ = a(α− r) + r
• Stochastic Integration
– Regular Calculus Rules Apply (e.g. FTC)
1.∫ T
0dZ(t) = Z(T )− Z(0) ∼ N(0, T )
2.∫ T
0(dZ(t))2 =
∫ T0dt = T − 0 = T
3.∫ T
0(dZ(t))n = 0, n > 2
4. S(t) =∫ T
0sZ(s)ds⇒ dS = t · Z(t)dt
5. S(t) =∫ T
0tdZ(s)⇒ dS = dt
(∫ T0dZ(s)
)+ t(∫ T
0dZ(s)
)′=
Z(t)dt+ tdZ(s)
• Ornstein-Uhlenbeck Process
– DE: dX = λ(α−X(t))dt+ σdZ
– Integral: X(t) = X0e−λt + α(1− e−λt) + σ
∫ t0eλ(s−t)dZ(s)
• Misc. Notes
19
– volatility of Sn is n · σ (remember when working with Black-Scholes)
Lesson 18 - Binomial Tree Models for Interest Rates
• Ft,T (P (T, T + s)) ∼ forward price @ t for an agreement to buy a bond@ T maturing @ T + s
– Ft,T (P (T, T + s)) = P (t,T+s)P (t,T )
• Binomial Trees
– don’t necessarily recombine
– risk-neutral probs. are given
– list out all paths and discount by that factor
• The Black-Derman-Toy model
– Bond Price = 1(1+R)n
– Ratio between interest rates @ successive nodes is constant
∗ it is e2σt√h
– σ =12ln(RuRd
)√h
– 2 year Bond Price1 year Bond Price
= 12
(1
1+R1+ 1
1+R1e2σ
)• Pricing Forwards using BDT
– use annual not cont. compouding
• Pricing Caps using BDT
20
– Discount difference due to cap by the discount rate appropriateto the beginning of the year
– Cap pays max(
0, RT−KR1+RT
)– For multiple year trees, start @ end and calculate the value then
weigh the results and add in the additional cap values as youmove to t0
Lesson 19 - The Black Formula for Bond Options
• 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 d1 =ln( FK )+ 1
2σ2T
σ√T
and d2 = d1 − σ√T
• Pricing Caps with the Black Formula
– (1 +KR) Puts with strike price 11+KR
– Calculate @ each node and then add together and multiply thesum by (1 +KR)
Lesson 20 - Eq. Interest Rate Models: Vasicek and Cox-Ingersoll-Ross
• Eq. Models - Theory
– dr = a(r)dt+ σ(r)dZ and dPP
= α(r, t, T )dt− q(r, t, T )dZ
– by Ito’s, dP = Prdr + 12Prr(dr)
2 + Ptdt
∗ ⇒ dP = α(r, t, T )dt− q(r, t, T )dZ, where
· α(r, t, T ) = 1P
(a(r)Pr + 1
2σ2(r)Prr + Pt
)· q(r, t, T ) = 1
PPrσ(r)
21
• Black-Scholes equation for Bonds
– rP = (a(r) + σ(r)φ)Pr + 12σ2(r)Prr + Pt
• Risk Premium = σ(r)φ
• To go to Risk Neutral, add σ(r)φ
– dr = (a(r) + σ(r)φ)dt+ σ(r)dZ
– Z(t) = Z(t)− φ
• The Rendelman-Barter Model(GBM)
– dr = ardt+ σrdZ
– Interest rates cannot go negative (+)
– Volatility is proportional to interest rate (+)
– Interest rates can get arbitrarily high, no mean reversion (-)
– Determine probabilities like with any GBM problem
• The Vasicek Model
– dr = a(b− r)dt+ σdZ
– There is mean reversion (+)
– Volatility is constant (-)
– Interest rates can go negative (-)
– DE: (a(b− r) + σφ)Pr + 12σ2Prr + Pt = rP
– P (r, t, T ) = A(t, T )e−B(t,T )r
– a 6= 0
22
∗ A(t, T ) = er[B−(T−t)]−B2 σ2
4a
∗ B(t, T ) = 1−e−a(T−t)
a
∗ r = b+ σ φa− 1
2
(σa
)2
– a = 0
∗ A(t, T ) = e12σφ(T−t)2+σ2 (T−t)3
6
∗ B(t, T ) = T − t
– ∆ = Pr = −BP
– Γ = Prr = B2P
• The Cox-Ingersoll-Ross Model
– dr = a(b− r)dt+ σ√rdZ
– Interest rates cannot go negative (+)
– Volatility varies with interest rate (+)
– There is mean reversion (+)
– φσ = φr
– DE: [a(b− r) + φr]Pr + 12σ2Prr + Pt = rP
– P (r, t, T ) = A(t, T )e−B(t,T )r
– A(t, T ) =[
2γe(a−φ+γ)(T−t)/2
(a−φ+γ)(eγ(T−t)−1)+2γ
] 2abσ2
– B(t, T ) = 2(eγ(T−t)−1)
(a−φ+γ)(eγ(T−t)−1)+2γ
∗ where γ =√
(a− φ)2 + 2σ2
• Misc. Notes
– q(r, t, T ) = −PrPσ(r) = Bσ(r)
23
– Vasicek
∗ α(r, t, T ) = −a(b− r)B + 12σ2B2 + Pt
P
• Delta Hedging
– Duration Hedge: N = −T1P (r,0,T1)T2P (r,0,T2)
– Delta Hedge: N = −Pr(r,0,T1)Pr(r,0,T2)
∗ where numerator is what you are hedging
• Delta-Gamma-Theta Approximation
– P (r + ε, 0, t+ h) = P (r, 0, t) + ∆ε+ 12Γε2 + θh