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51 Cards in this Set
- Front
- Back
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Monte Carlo Valuation - Simulating Standard Normal Variables z = |
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Monte Carlo Valuation - Simulating Standard Normal Variables zi = |
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Monte Carlo Valuation - Simulating Lognormal Stock Prices Not interested in the intermediate prices: ST = |
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Monte Carlo Valuation - Simulating Lognormal Stock Prices Interested in the intermediate prices: St+h = ... ST = |
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Monte Carlo Valuation - Risk neutral vs. True |
- Use the risk-neutral dist only when discounting is needed. Pricing options. - Use the true dist when discounting is not needed. True exp payoff & prob. |
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Monte Carlo Valuation - Control Variate Method A.K.A |
Boyle Modification |
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Monte Carlo Valuation - Control Variate Method Y* = |
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Monte Carlo Valuation - Control Variate Method Y* = Y bar = X = X bar = |
where Y* = control variate est for option Y Y bar = Monte Carlo est for Option Y X = Exact/True price of Option X X bar = Monte Carlo est for Option X |
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Monte Carlo Valuation - Control Variate Method Var[Y*] = |
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Monte Carlo Valuation - Control Variate Method Var[Y*] is minimized when: beta = |
The bottom Var[X bar] is the control variate. |
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Monte Carlo Valuation - Control Variate Method When beta is set to minimize Var[Y*]: Var[Y*] = |
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Monte Carlo Valuation - Antithetic Variate Method For every ui, also simulate using ... |
1 - ui
Uniform |
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Monte Carlo Valuation - Antithetic Variate Method For every zi, also simulate using ... |
-zi Standard normal |
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Monte Carlo Valuation - Stratified Sampling Define |
Break the sampling space into equal size spaces. Then, scale the uniform numbers into equal size spaces. |
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Brownian Motion Z(t): Pure/Standard Brownian Motion 4 Characteristics: |
1. Z(0) = 0
2. Z(t+s) - Z(t) ~ N(0,s) Normal dist. Z(t) - Z(0) ~ N(0,t) 3. Z(t+h) - Z(t) is independent of Z(t) - Z(t-s) 4. Z(t) is continuous |
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Brownian Motion Z(t) is a martingale if |
* E[ Z(t+h) - Z(t) ] = 0*E[ Z(t+h) | Z(t) ] = Z(t)
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Brownian Motion Properties: |
1. Quadratic variation = T 2. Cubic or higher order variation = 0 3. Total variation = ∞ |
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Arithmetic Brownian Motion dX(t) = |
a dt + b dZ(t) |
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Arithmetic Brownian Motion X(T) - X(0) = |
~aT + bZ(t) |
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Arithmetic Brownian Motion X(t) - X(0) ~ |
N(at,b^2t) |
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Ornstein-Uhlenbeck Process dX(t) = |
λ[α - X(t)]dt + σdZ(t) |
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Geometric Brownian Motion dX(t) = |
a X(t) dt + b X(t) dZ(t) |
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Geometric Brownian Motion dX(t) / X(t) = |
a dt + b dZ(t) |
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Geometric Brownian Motiond d[ ln X(t) ] = |
(a - 0.5b^2) dt + b dZ(t) |
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Geometric Brownian Motion X(t) = |
X(0) e ^[(a - 0.5b^2) t + b dZ(t)] |
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Geometric Brownian Motion ln[ X(t) / X(0) ] ~ N[ m = ... , v^2 = ... ] |
~ N[ m = (a-0.5b^2)t, v^2 = tb^2] |
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Geometric Brownian Motion - The followings are equivalent: * The Black-Scholes framework ... * dS(t) = ... * dS(t) / S(t) = ... * d [ln S(t) ] = ... * S(t) = ... * ln[ S(t) / S(0) ] ~ N[m=..., v^2=...] |
* The Black-Scholes framework applies. * dS(t) = (α-δ) S(t) dt + σ S(t) dZ(t) * dS(t) / S(t) = (α-δ) dt + σ dZ(t) * d [ln S(t) ] = (α-δ-0.5σ^2) d(t) +σ dZ(t) * S(t) = S(0) e^[(α-δ-0.5σ^2)t + σZ(t)] * ln[ S(t) / S(0) ] ~ N[m=(α-δ-0.5σ^2)t, v^2=tσ^2] |
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Ito's Lemma dV = |
Vs dS + 0.5Vss (dS)^2 + Vt dt |
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Multiplication Rules dt * dt = dt * dZ = dZ * dt = dZ * dZ = |
dt * dt = 0 dt * dZ = dZ * dt = 0 dZ * dZ = dt |
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Sharpe Ratio Φ = |
Φ = (α-r) / σ Two Ito's processes depending on the same dZ(t) will have equal Sharpe ratios. |
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Risk-free Portfolio - For a portfolio consisting of Asset 1 and Asset 2: Return on Asset 1 = |
dS1 + δ1 S1 dt |
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Risk-free Portfolio - For a portfolio consisting of Asset 1 and Asset 2: Return on Asset 2 = |
dS2 + δ2 S2 dt |
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Risk-free Portfolio - For a portfolio consisting of Asset 1 and Asset 2: Total return on the portfolio = |
N1 (dS1 + δ1 S1 dt ) + N2 ( dS2 + δ2 S2 dt ) + |
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Risk-free Portfolio The coefficient of dZ = Risk-free --> The coefficient of dZ = |
= N1 [ σ1 S1 ] + N2 [ σ2 S2 ]
= 0, risk-free |
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Risk-free Portfolio N1 = N2 = |
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Risk-neutral Pricing dS(t) / S(t) = α or r |
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Risk-neutral Pricing |
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Risk-neutral Pricing |
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Risk-neutral Pricing True Measure vs. Risk-neutral Measure |
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Proportional Portfolio x: 1-x: |
x: percentage invested in Asset A 1-x: percentage invested in Asset B |
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Proportional Portfolio W(t): |
value of the portfolio at time t |
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Proportional Portfolio dW(t) / W(t) + δw dt = |
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Proportional Portfolio If Asset A is a risk-free asset, then: dA(t) / A(t) + δA dt = so dW(t) / W(t) + δw dt = |
= r dt = x[r dt] + (1-x)[...] see previous slide. |
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The Black-Scholes Equation δ: δ*: |
δ: dividend yield on stock δ*: dividend yield on derivative/claim/security |
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The Black-Scholes Equation with Vs, Vss, Vt |
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The Black-Scholes Equation with Greeks |
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S^a (Ito Process) E[S(T)^a] = |
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S^a Ft,T[S(T)^a] = |
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S^a δ* = |
r - a(r-δ) - 0.5a(a-1)σ^2 |
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S^a Gamma γ = |
a(α-r) + r Total return rate |
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S^a dS^a / S^a = |
[a(α-δ) + 0.5a(a-1)σ^2] dt + aσ dZ(t) |
