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September 28, 2007, at 01:23 AM
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- Apply Itô’s lemma in the one-dimensional case.
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September 14, 2007, at 03:18 PM
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- Explain what it means to say that stock prices follow a diffusion process.
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September 13, 2007, at 10:25 AM
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September 12, 2007, at 09:37 PM
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- Calculate the value of European and American options using the Black-Scholes option-pricing model.
to:
- Calculate the value of European and American options using the Black-Scholes option-pricing model.
September 06, 2007, at 10:20 PM
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Changed lines 1-38 from:
Valuation of derivative securities
The binomial model
Basic setup
Key: replicating the purchase of a call option by buying a fractional (\Delta) shares of stock and borrowing B dollars.
Assume at time T the stock will have two values, u\cdot S and d\cdot S. Let the values of a corresponding call option be C_u and C_d. h denotes the length of one time period. Then a cash flow table shows the replicating portfolio will satisfy
\Delta \cdot dS \cdot d^{\delta h} + B\cdot e^{rh} = C_d,
\Delta \cdot uS \cdot d^{\delta h} + B\cdot e^{rh} = C_u.
Solving gives
\Delta = \left( \frac{C_u-C_d}{S(u-d)} \right) e^{-\delta h}
B = \left( \frac{uC_d-dC_u}{u-d} \right) e^{-r h}
So the call option price is
C = \Delta S + B.
The put option price is obtained entirely the same way by replacing C with P.
If the price of an option is mispriced, we can arbitrage by buying low, selling high,
using the fact that the buying a call option is equivalent to buying \Delta
shares and borrowing B dollars.
Risk-neutral pricing
If we write
p^* = \frac{ e^{(r-\delta)h} -d}{u-d},
which is called the risk neutral probability, then the formula simplifies to
C = e^{-rh} \left( p^*C_u + (1-p^*) D_d \right)
as if C is a kind of (discounted) expected value.
[In fact, a calculation also shows p^* is related to the forward price:
F_{t, t+h} = p^*uS + (1-p^*)dS.
]
Properties of \Delta and B
- The factor e^{(r-\delta)h} satisfies
d< e^{(r-\delta)h}< u
. Otherwise we can arbitrage on exchanging stock and bonds, ignoring options altogether.
- For a call option, 0<\Delta\leq 1 and B<0.
- For a put option, -1\leq \Delta <0 and B>0.
to:
September 06, 2007, at 10:16 PM
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Changed lines 2-70 from:
Put-call parity formula for stock options
Variables:
T = expiration time
K = strike price
S_0 = stock price at time 0
r = risk-free interest rate
Suppose the stock has a continuous dividend rate of \delta .
The put-call parity formula is
C(K,T)-P(K,T)=S_0e^{-\delta \, T} - Ke^{-rT},
where C is the call option price, and P is the put option price.
Justification: The key is that buying a call and selling a put amount to a
synthetic forward.
The cash flow for buying a call and selling a put
at time 0 is -C+P . There is an additional cash
flow at time T of + K .
The cash flow at time T for a buying a forward
is -forward price = - S_0e^{(r-\delta)T} .
Thus the cash flows at time 0 are
-C+P +PV(K) = PV (- S_0e^{(r-\delta)T})
and the formula follows.
General put-call parity formula
Now the underlying asset can be stock with different dividend schemes, foreign currency,
futures, or bonds.
Variables:
F_{0,T} = prepaid forward price of the underlying asset
K = strike price in dollars per share of asset
The formula is
C(K,T)-P(K,T)=F_{0,T}^P - Ke^{-rT}.
Properties
Price ranges
- A call cannot cost more than the stock price S .
- A put cannot cost more than the strike price K .
- Both C and P should be positive; this also puts lower bounds on both
C and P by the parity formula.
- American options are always going to be more expensive because they have the potential to early exercise.
We thus have the following string of inequalities:
\max[0, F_{0,T}^P - Ke^{-rT}] \leq C_{\text{Eur}} \leq C_{\text{Amer}} \leq S;
\max[0, Ke^{-rT} - F_{0,T}^P ] \leq P_{\text{Eur}} \leq P_{\text{Amer}} \leq K;
Early Exercise
Price vs. t
Price vs. K
C as a function of K is
- decreasing;
- has slope \geq -1 ;
- concave up.
P as a function of K is
- decreasing;
- has slope \leq 1 ;
- concave up.
If any of the properties are violated, we can have arbitrage.
For the first two conditions, this is done by creating a spread,
i.e., buy the option that's mispriced to be lower than it should, and sell the option
mispriced to be higher. When the third (convexity) condition is violated,
we can create an asymmetrical butterfly spread with "\lambda".
to:
September 06, 2007, at 10:16 PM
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September 06, 2007, at 10:15 PM
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Changed line 91 from:
{$$p^* = \frac{ e^{(r-\delta)h} -d}{u-d}, $$
to:
p^* = \frac{ e^{(r-\delta)h} -d}{u-d},
Changed line 100 from:
d< e^(r-\delta)h}< u
. Otherwise we can arbitrage on exchanging stock and bonds, ignoring options altogether.
to:
d< e^{(r-\delta)h}< u
. Otherwise we can arbitrage on exchanging stock and bonds, ignoring options altogether.
Changed lines 109-129 from:
- Use to determine the relationship between prices of European put and call options and to identify arbitrage opportunities.
- Calculate the value of European and American options using the binomial model.
- Calculate the value of European and American options using the Black-Scholes option-pricing model.
- Interpret the option Greeks.
- Explain the cash flow characteristics of the following exotic options: Asian, barrier, compound, gap, and exchange.
- Explain what it means to say that stock prices follow a diffusion process.
- Apply Itô’s lemma in the one-dimensional case.
- Apply option pricing concepts to actuarial problems such as equity-linked insurance.
Interest rate models
- Evaluate features of the Vasicek and Cox-Ingersoll-Ross bond price models.
- Explain why the time-zero yield curve in the Vasicek and Cox-Ingersoll-Ross bond price models cannot be exogenously prescribed.
- Construct a Black-Derman-Toy binomial model matching a given time-zero yield curve and a set of volatilities.
Risk management techniques
- Explain and demonstrate how to control risk using the method of delta-hedging.
to:
- Use put-call parity to determine the relationship between prices of European put and call options and to identify arbitrage opportunities.
- Calculate the value of European and American options using the binomial model.
- Calculate the value of European and American options using the Black-Scholes option-pricing model.
- Interpret the option Greeks.
- Explain the cash flow characteristics of the following exotic options: Asian, barrier, compound, gap, and exchange.
- Explain what it means to say that stock prices follow a diffusion process.
- Apply Itô’s lemma in the one-dimensional case.
- Apply option pricing concepts to actuarial problems such as equity-linked insurance.
Interest rate models
- Evaluate features of the Vasicek and Cox-Ingersoll-Ross bond price models.
- Explain why the time-zero yield curve in the Vasicek and Cox-Ingersoll-Ross bond price models cannot be exogenously prescribed.
- Construct a Black-Derman-Toy binomial model matching a given time-zero yield curve and a set of volatilities.
Risk management techniques
- Explain and demonstrate how to control risk using the method of delta-hedging.
September 06, 2007, at 10:13 PM
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Changed line 109 from:
- Use [put-call parity -> Actuary.MFE.parity ] to determine the relationship between prices of European put and call options and to identify arbitrage opportunities.
to:
- Use to determine the relationship between prices of European put and call options and to identify arbitrage opportunities.
September 06, 2007, at 10:12 PM
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Changed lines 107-112 from:
to:
Rational valuation of derivative securities
- Use [put-call parity -> Actuary.MFE.parity ] to determine the relationship between prices of European put and call options and to identify arbitrage opportunities.
- Calculate the value of European and American options using the binomial model.
September 06, 2007, at 10:05 PM
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September 06, 2007, at 10:04 PM
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Changed line 91 from:
{$$p^* = \frac{ e^(r-\delta)h} -d}{u-d}, $$
to:
{$$p^* = \frac{ e^{(r-\delta)h} -d}{u-d}, $$
Changed line 99 from:
- The factor e^(r-\delta)h} satisfies
to:
- The factor e^{(r-\delta)h} satisfies
September 06, 2007, at 10:03 PM
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Changed lines 71-72 from:
The binomial model
to:
The binomial model
Basic setup
Changed line 75 from:
Setup: Assume at time T the stock will have two values, u\cdot S and d\cdot S. Let the values of a corresponding call option be C_u and C_d. h denotes the length of one time period. Then a cash flow table shows the replicating portfolio will satisfy
to:
Assume at time T the stock will have two values, u\cdot S and d\cdot S. Let the values of a corresponding call option be C_u and C_d. h denotes the length of one time period. Then a cash flow table shows the replicating portfolio will satisfy
Changed lines 78-83 from:
to:
Solving gives
\Delta = \left( \frac{C_u-C_d}{S(u-d)} \right) e^{-\delta h}
B = \left( \frac{uC_d-dC_u}{u-d} \right) e^{-r h}
So the call option price is
C = \Delta S + B.
The put option price is obtained entirely the same way by replacing C with P.
If the price of an option is mispriced, we can arbitrage by buying low, selling high,
using the fact that the buying a call option is equivalent to buying \Delta
shares and borrowing B dollars.
Risk-neutral pricing
If we write
{$$p^* = \frac{ e^(r-\delta)h} -d}{u-d}, $$
which is called the risk neutral probability, then the formula simplifies to
C = e^{-rh} \left( p^*C_u + (1-p^*) D_d \right)
as if C is a kind of (discounted) expected value.
[In fact, a calculation also shows p^* is related to the forward price:
F_{t, t+h} = p^*uS + (1-p^*)dS.
]
Properties of \Delta and B
- The factor e^(r-\delta)h} satisfies
d< e^(r-\delta)h}< u
. Otherwise we can arbitrage on exchanging stock and bonds, ignoring options altogether.
- For a call option, 0<\Delta\leq 1 and B<0.
- For a put option, -1\leq \Delta <0 and B>0.
September 06, 2007, at 09:35 PM
by 24.148.11.217 -
Changed line 42 from:
C and P by the parity formula.
to:
C and P by the parity formula.
Changed lines 71-74 from:
- Calculate the value of European and American options using the binomial model.
to:
The binomial model
Key: replicating the purchase of a call option by buying a fractional (\Delta) shares of stock and borrowing B dollars.
Setup: Assume at time T the stock will have two values, u\cdot S and d\cdot S. Let the values of a corresponding call option be C_u and C_d. h denotes the length of one time period. Then a cash flow table shows the replicating portfolio will satisfy
\Delta \cdot dS \cdot d^{\delta h} + B\cdot e^{rh} = C_d,
\Delta \cdot uS \cdot d^{\delta h} + B\cdot e^{rh} = C_u.
September 06, 2007, at 09:11 PM
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Changed line 27 from:
to:
Added line 38:
Price ranges
Changed lines 46-62 from:
max[0, F_{0,T}^P - Ke^{-rT} \leq C_{Eur} \leq C_{Amer} \leq S;
Arbitrage
In general, buy something that's lower than it should be, and sell something that's higher than it should be.
Useful equivalent positions:
- borrowing money and buy (tail) a stock is the same as buying a forward.
- buying a call and selling a put is the same as buying a forward.
- short selling a stock and lending the money is the same as buying a call.
to:
\max[0, F_{0,T}^P - Ke^{-rT}] \leq C_{\text{Eur}} \leq C_{\text{Amer}} \leq S;
\max[0, Ke^{-rT} - F_{0,T}^P ] \leq P_{\text{Eur}} \leq P_{\text{Amer}} \leq K;
Early Exercise
Price vs. t
Price vs. K
C as a function of K is
- decreasing;
- has slope \geq -1 ;
- concave up.
P as a function of K is
- decreasing;
- has slope \leq 1 ;
- concave up.
If any of the properties are violated, we can have arbitrage.
For the first two conditions, this is done by creating a spread,
i.e., buy the option that's mispriced to be lower than it should, and sell the option
mispriced to be higher. When the third (convexity) condition is violated,
we can create an asymmetrical butterfly spread with "\lambda".
September 06, 2007, at 07:17 PM
by 24.148.11.217 -
Added lines 35-49:
Properties
- A call cannot cost more than the stock price S .
- A put cannot cost more than the strike price K .
- Both C and P should be positive; this also puts lower bounds on both
C and P by the parity formula.
- American options are always going to be more expensive because they have the potential to early exercise.
We thus have the following string of inequalities:
max[0, F_{0,T}^P - Ke^{-rT} \leq C_{Eur} \leq C_{Amer} \leq S;
Changed lines 54-57 from:
- buying a call and selling a put is the same as buying a synthetic forward.
-
to:
- borrowing money and buy (tail) a stock is the same as buying a forward.
- buying a call and selling a put is the same as buying a forward.
- short selling a stock and lending the money is the same as buying a call.
September 06, 2007, at 06:17 PM
by 24.148.11.217 -
Changed line 19 from:
The cash flow at time T for a buying a forward
to:
The cash flow at time T for a buying a forward
Changed line 22 from:
-C+P +PV(K) = PV (- S_0e^{(r-\delta)T)})
to:
-C+P +PV(K) = PV (- S_0e^{(r-\delta)T})
Changed lines 36-38 from:
to:
In general, buy something that's lower than it should be, and sell something that's higher than it should be.
Useful equivalent positions:
- buying a call and selling a put is the same as buying a synthetic forward.
-
September 06, 2007, at 05:18 PM
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Changed lines 1-2 from:
Valuation of derivative securities
to:
Valuation of derivative securities
September 06, 2007, at 05:17 PM
by 24.148.11.217 -
Changed lines 1-6 from:
Interest rate models
- Evaluate features of the Vasicek and Cox-Ingersoll-Ross bond price models.
- Explain why the time-zero yield curve in the Vasicek and Cox-Ingersoll-Ross bond price models cannot be exogenously prescribed.
- Construct a Black-Derman-Toy binomial model matching a given time-zero yield curve and a set of volatilities.
Rational valuation of derivative securities
to:
Valuation of derivative securities
Changed lines 47-53 from:
Risk management techniques
to:
Interest rate models
- Evaluate features of the Vasicek and Cox-Ingersoll-Ross bond price models.
- Explain why the time-zero yield curve in the Vasicek and Cox-Ingersoll-Ross bond price models cannot be exogenously prescribed.
- Construct a Black-Derman-Toy binomial model matching a given time-zero yield curve and a set of volatilities.
Risk management techniques
September 06, 2007, at 05:14 PM
by 24.148.11.217 -
Changed line 1 from:
Interest rate models
to:
Interest rate models
Changed lines 6-7 from:
Rational valuation of derivative securities
Put-call parity formula for stock options
to:
Rational valuation of derivative securities
Put-call parity formula for stock options
Changed line 27 from:
-C+P +PV(K) = PV (- S_0e^{(r-\delta)T)}
to:
-C+P +PV(K) = PV (- S_0e^{(r-\delta)T)})
Changed line 30 from:
General put-call parity formula
to:
General put-call parity formula
Changed line 40 from:
Arbitrage
to:
Arbitrage
September 06, 2007, at 05:13 PM
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Changed lines 12-13 from:
to:
r = risk-free interest rate
Changed lines 30-32 from:
to:
General put-call parity formula
Now the underlying asset can be stock with different dividend schemes, foreign currency,
futures, or bonds.
Variables:
F_{0,T} = prepaid forward price of the underlying asset
K = strike price in dollars per share of asset
The formula is
C(K,T)-P(K,T)=F_{0,T}^P - Ke^{-rT}.
Arbitrage
In general
September 06, 2007, at 03:58 PM
by 24.148.11.217 -
Changed line 19 from:
Justification: The key is that buying a call and selling a put amounts to a
to:
Justification: The key is that buying a call and selling a put amount to a
Changed line 25 from:
is -forward price = - S_0e^{(r-\delta)T .
to:
is -forward price = - S_0e^{(r-\delta)T} .
Changed line 27 from:
-C+P +PV(K) = PV (- S_0e^{(r-\delta)T)
to:
-C+P +PV(K) = PV (- S_0e^{(r-\delta)T)}
Changed lines 31-34 from:
creates a synthetic forward,
so the cash flow should be the same.
to:
September 06, 2007, at 03:56 PM
by 24.148.11.217 -
Changed lines 7-11 from:
Put-call parity
Variables:
K = strike price
to:
Put-call parity formula for stock options
Variables:
T = expiration time
K = strike price
S_0 = stock price at time 0
K = strike price
Suppose the stock has a continuous dividend rate of \delta .
The put-call parity formula is
C(K,T)-P(K,T)=S_0e^{-\delta \, T} - Ke^{-rT},
where C is the call option price, and P is the put option price.
Justification: The key is that buying a call and selling a put amounts to a
synthetic forward.
The cash flow for buying a call and selling a put
at time 0 is -C+P . There is an additional cash
flow at time T of + K .
The cash flow at time T for a buying a forward
is -forward price = - S_0e^{(r-\delta)T .
Thus the cash flows at time 0 are
-C+P +PV(K) = PV (- S_0e^{(r-\delta)T)
and the formula follows.
creates a synthetic forward,
so the cash flow should be the same.
September 06, 2007, at 02:30 PM
by 24.148.11.217 -
Changed lines 7-11 from:
- Use put-call parity to determine the relationship between prices of European put and call options and to identify arbitrage opportunities.
to:
Put-call parity
Variables:
K = strike price
June 09, 2007, at 09:47 PM
by 216.80.126.250 -
Changed lines 1-15 from:
A. Interest rate models
1. Evaluate features of the Vasicek and Cox-Ingersoll-Ross bond price models.
2. Explain why the time-zero yield curve in the Vasicek and Cox-Ingersoll-Ross bond price models cannot be exogenously prescribed.
3. Construct a Black-Derman-Toy binomial model matching a given time-zero yield curve and a set of volatilities.
B. Rational valuation of derivative securities
1. Use put-call parity to determine the relationship between prices of European put and call options and to identify arbitrage opportunities.
2. Calculate the value of European and American options using the binomial model.
3. Calculate the value of European and American options using the Black-Scholes option-pricing model.
4. Interpret the option Greeks.
5. Explain the cash flow characteristics of the following exotic options: Asian, barrier, compound, gap, and exchange.
6. Explain what it means to say that stock prices follow a diffusion process.
7. Apply Itô’s lemma in the one-dimensional case.
8. Apply option pricing concepts to actuarial problems such as equity-linked insurance.
C. Risk management techniques
1. Explain and demonstrate how to control risk using the method of delta-hedging.
to:
Interest rate models
- Evaluate features of the Vasicek and Cox-Ingersoll-Ross bond price models.
- Explain why the time-zero yield curve in the Vasicek and Cox-Ingersoll-Ross bond price models cannot be exogenously prescribed.
- Construct a Black-Derman-Toy binomial model matching a given time-zero yield curve and a set of volatilities.
Rational valuation of derivative securities
- Use put-call parity to determine the relationship between prices of European put and call options and to identify arbitrage opportunities.
- Calculate the value of European and American options using the binomial model.
- Calculate the value of European and American options using the Black-Scholes option-pricing model.
- Interpret the option Greeks.
- Explain the cash flow characteristics of the following exotic options: Asian, barrier, compound, gap, and exchange.
- Explain what it means to say that stock prices follow a diffusion process.
- Apply Itô’s lemma in the one-dimensional case.
- Apply option pricing concepts to actuarial problems such as equity-linked insurance.
Risk management techniques
- Explain and demonstrate how to control risk using the method of delta-hedging.
June 09, 2007, at 09:45 PM
by 216.80.126.250 -
Added lines 1-15:
A. Interest rate models
1. Evaluate features of the Vasicek and Cox-Ingersoll-Ross bond price models.
2. Explain why the time-zero yield curve in the Vasicek and Cox-Ingersoll-Ross bond price models cannot be exogenously prescribed.
3. Construct a Black-Derman-Toy binomial model matching a given time-zero yield curve and a set of volatilities.
B. Rational valuation of derivative securities
1. Use put-call parity to determine the relationship between prices of European put and call options and to identify arbitrage opportunities.
2. Calculate the value of European and American options using the binomial model.
3. Calculate the value of European and American options using the Black-Scholes option-pricing model.
4. Interpret the option Greeks.
5. Explain the cash flow characteristics of the following exotic options: Asian, barrier, compound, gap, and exchange.
6. Explain what it means to say that stock prices follow a diffusion process.
7. Apply Itô’s lemma in the one-dimensional case.
8. Apply option pricing concepts to actuarial problems such as equity-linked insurance.
C. Risk management techniques
1. Explain and demonstrate how to control risk using the method of delta-hedging.