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FDFE > Lecture 6: Monte Carlo simulation & Binomial Option Pricing > Flashcards

Lecture 6: Monte Carlo simulation & Binomial Option Pricing Flashcards

1
Q

Apart from Black-Scholes, the most popular methods to price exotic options (e.g. American options,..) are:

  1. ?? simulation
  2. ? asset pricing model
  3. ?? methods
A

Apart from Black-Scholes, the most popular methods to price exotic options (e.g. American options,..) are:

  1. Monte Carlo simulation
  2. Binominal asset pricing model
  3. Finite difference methods
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2
Q 
Monte Carlo Simulation:
Boyle (1977): 
Option price = discounted ?? in a risk neutral world at time T: 
c = ????
p = ????
A 
Monte Carlo Simulation (MCS):
Boyle (1977): 
Option price = discounted expected payoff in a risk neutral world at time T: 
c = (e^-rT) E^Q [Max(S_T - K)]
p = (e^-rT) E^Q [Max(K - S_T)]
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3
Q

Monte Carlo Simulation - Steps:

  1. Divide time period into N time steps with length Δt
    e. g. divide a year into 12 (=N) time steps (Δt=1 month)
  2. For each time interval M normally distributed random numbers (0,1)
  3. Create a table of possible paths
  4. Calculate each option’s payoff at maturiy
  5. Discount the average payoff to get option’s present value.
A

Monte Carlo Simulation - Steps:
1. Divide time period into N time steps with length Δt
e.g. divide a year into 12 (=N) time steps (Δt=1 month)
2. For each time interval M normally distributed random numbers (0,1)
3. Create a table of possible paths
4. Calculate each option’s payoff at maturiy
5. Discount the average payoff to get option’s present value.
(See q1 - practical 1 - class 2)

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4
Q

Monte Carlo & Exotic options:
e.g. average-strike Asian put (strike price = average asset price during option’s life):
Asian put’s payoff = Max (S_avg - S_T, 0)
= 1/i * [ Sum_i=0 ^ T (S_ti - S_T)
Calculate strike price by MCS & discount average payoff to get option price

A

Monte Carlo & Exotic options:
e.g. average-strike Asian put (strike price = average asset price during option’s life):
Asian put’s payoff = Max (S_avg - S_T, 0)
= 1/i * [ Sum_i=0 ^ T (S_ti - S_T)
Calculate strike price by MCS & discount average payoff to get option price

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5
Q 
Calculate Options Greeks by MCS:
e.g. Δ = ∂c/∂S
- Use ? to estimate call price c'
- assume S rises by ΔS
- Re-estimate call price to get c*
=> Δ = ???
A 
Calculate Options Greeks by MCS:
e.g. Δ = ∂c/∂S
- Use MCS to estimate call price c'
- assume S rises by ΔS
- Re-estimate call price to get c*
=> Δ = (c* - c') / ΔS
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6
Q

Advantages of MCS:

  • can use when option price depends on underlying asset’s ?.
  • can accommodate different ? processes & payment patterns (e.g. ?)
A

Advantages of MCS:

  • can use when option price depends on underlying asset’s path.
  • can accommodate different stochastic processes & payment patterns (e.g. dividends)
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7
Q

Disadvatages of MCS:

  • Time-consuming
  • can’t easily handle early-exercise like American options.
A

Disadvatages of MCS:

  • Time-consuming
  • can’t easily handle early-exercise like American options.
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8
Q

*** ? Asset Pricing Model (BAPM) by Cox, Ross & Rubinstein (?):
- 2 assets: S & R:
S: ? asset
R: ?? asset
- 2 states of the world: u & d
u: price of underlying goes ?
d: price of underlying goes ?.

A

*** Binomial Asset Pricing Model (BAPM) by Cox, Ross & Rubinstein (1979):
- 2 assets: S & R:
S: underlying asset
R: risk free asset
- 2 states of the world: u & d
u: price of underlying goes up
d: price of underlying goes down.

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9
Q

BAPM:
In absence of arbitrage, Cox, Ross & Rubinstein (1979) assume price of underlying assets follow this ? process:
S0 (t=0) => ?? or ?? at t = Δt

A

BAPM:
In absence of arbitrage, Cox, Ross & Rubinstein (1979) assume price of underlying assets follow this binomial process:
S0 (t=0) => uS0 or dS0 at t = Δt

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10
Q

BAPM:
Risk-free asset pays ? after 1 timestep no matter which state prevails:
??? (at t=0) => ? (t=Δt)
‘?? bond’ or ‘?? bond’ with face value = 1

A

BAPM:
Risk-free asset pays 1 after 1 timestep no matter which state prevails:
e^(-rΔt) (at t=0) => 1 (t=Δt)
‘zero coupon bond’ or ‘pure discount bond’ with face value = 1

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11
Q

BAPM:
d, u & r satisfy:
0 < ? < 1 < ?? < ?
- If e^(rΔt) > ? : no one would invest in ?
- If d < ?? : I can borrow money and invest in stocks because even in t’ worst case, ?? rises as least as fast as ? used to buy it.

A

BAPM:
d, u & r satisfy:
0 < d < 1 < e^(rΔt) < u
- If e^(rΔt) > u : no one would invest in stocks
- If d > e^(rΔt) : I can borrow money and invest in stocks because even in t’ worst case, stock price rises as least as fast as debt used to buy it.

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12
Q

BAPM:
u = ??
d = ??
d = 1 / ?

A

BAPM:
u = e^(σΔt)
d = e^(-σΔt)
d = 1 / u

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13
Q

BAPM - No arbitrage portfolio:

To price option at time t=0, construct a no-arbitrage portfolio containing 1 underlying asset (S0) and Φ calls c0.

A

BAPM - No arbitrage portfolio:

To price option at time t=0, construct a no-arbitrage portfolio containing 1 underlying asset (S0) and Φ calls c0.

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14
Q

BAPM:
Value of t’ no-arbitrage portfolio:
V0 = ??? (at t=0)
=> Vu = ??? OR Vd = ??? (at t = Δt)

A

BAPM:
Value of t’ no-arbitrage portfolio:
V0 = S0 + Φ * c0 (at t=0)
=> Vu = u*S0 + Φ * c_u OR Vd = d * S0 + Φ * c_d (at t = Δt)

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15
Q

BAPM - No-arbitrage condition:
V? = V? = V???
<=> ??? = ??? = (????

A

BAPM - No-arbitrage condition:
Vu = Vd = V0 * e^(rΔt)
<=> uS0 + Φ * c_u = d * S0 + Φ * c_d = (S0 + Φ * c0)e^(rΔt)

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16
Q

BAPM - No arbitrage condition:
Φ = ??? ? 0 (c_u ?? c_d)

Φ number of call options needed to include in an ? portfolio containing ? asset.

or Φ is number of call options written against underlying asset so that a portfolio containing these and ? asset is ?? => aka ‘??’.

Φ <= 0 implies we write a call option when we’re ? in underlying asset.

A

BAPM - No arbitrage condition:
Φ = [S0*( u - d)] / [c_d - c_u] <= 0 ( c_u >= c_d)

Φ number of call options needed to include in an arbitrage portfolio containing 1 asset.
or Φ is number of call options written against underlying asset so that a portfolio containing these and 1 asset is risk-free => aka ‘perfect hedge’.

Φ <= 0 implies we write a call option when we’re long in underlying asset.

17
Q

BAPM - Risk neutral probability:
Q =??? / (u - d)
0 <= Q <= 1
Risk-neutral => make real world probabilities ? in the model.

A

BAPM - Risk neutral probability:
Q = [e^(rΔt) - d] / (u - d)
0 <= Q <= 1
Risk-neutral => make real world probabilities irrelevant in the model.

18
Q

BAPM - Risk neutral probability:
Δ = -1 / ? = ( ????) / (????) >=0 :
Delta hedge ratio: how many ? we must long if we write 1 call.

A

BAPM - Risk neutral probability:
Δ = -1 / Φ = ( c_u - c_d) / (S_u - S_d) >=0 :
Delta hedge ratio: how many assets we must long if we write 1 call.

19
Q

Binomial Formula:

One period: c0 = ???
Multi-period:
c_n = ????

A

Binomial Formula:

One period: c0 = e^(-rΔt) * [ Qc_u + (1-Q)c_d ]

Multi-period:
c_n = e^(-rΔt) * [ Qc_n+1(u) + (1-Q)c_n+1(d) ]

FDFE (10 decks)

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