Lecture 6: Monte Carlo simulation & Binomial Option Pricing

Question 1

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

Answer 1

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

Question 2

Monte Carlo Simulation:
Boyle (1977): 
Option price = discounted ?? in a risk neutral world at time T: 
c = ????
p = ????

Answer 2

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)]

Question 3

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.

Answer 3

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)

Question 4

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

Answer 4

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

Question 5

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 = ???

Answer 5

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

Question 6

Advantages of MCS:

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

Answer 6

Advantages of MCS:

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

Question 7

Disadvatages of MCS:

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

Answer 7

Disadvatages of MCS:

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

Question 8

*** ? 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 ?.

Answer 8

*** 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.

Question 9

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

Answer 9

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

Question 10

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

Answer 10

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

Question 11

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.

Answer 11

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.

Question 12

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

Answer 12

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

Question 13

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.

Answer 13

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.

Question 14

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

Answer 14

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)

Question 15

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

Answer 15

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

Question 16

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.

Answer 16

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.

Question 17

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

Answer 17

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

Question 18

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

Answer 18

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.

Question 19

Binomial Formula:

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

Answer 19

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) ]