Leonid Kogan
MIT, Sloan
15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan )
Key Points: Derivatives
15.450, Fall 2010 1 / 1

Definitions of SPD ( π ) and risk-neutral probability ( Q ).
Absence of arbitrage is equivalent to existence of the SPD or a risk-neutral probability:
P t
= E
P t
�
� u = t + 1
π u
π t
D u
�
= E
Q t
�
� u = t + 1
B t
D u
B u
�
Price of risk: under Gaussian P and Q distributions,
ε t
Q
= ε
P t
+ η t
Log-normal model (discrete version of Black-Scholes):
µ t
− r t
= σ t
η t c Leonid Kogan ( MIT, Sloan )
Key Points: Derivatives
15.450, Fall 2010 3 / 1

Brownian motion, basic properties (IID Gaussian increments, continuous trajectories, nowhere differentiable).
Quadratic variation. [ Z ]
T
= T . Heuristically,
( dZ t
)
2
= dt .
Stochastic integral.
Ito’s lemma: df ( t , X t
) =
∂ f ( t , X t
) dt
∂ t
+
∂ f ( t , X
∂ X t t
) dX t
1 ∂
2 f ( t , X t
)
+
2 ∂ X 2 t
( dX t
)
2
Multivariate Ito’s lemma. df ( t , X t
, Y t
∂ f ∂ f
) = dt + dX
∂ t ∂ X t t
∂ f
+ dY
∂ Y t t
+
1 ∂ 2 f
2 ∂ X 2 t
( dX t
)
2
+
1 ∂ 2 f
2 ∂ Y 2 t
( dY t
)
2
+
∂ 2 f
∂ X t
∂ Y t dX t dY t c Leonid Kogan ( MIT, Sloan )
Key Points: Derivatives
15.450, Fall 2010 4 / 1

Arbitrage-free pricing of options by replication.
European option with payoff H ( S
T
) .
Replicating portfolio delta is
θ t
=
∂ f ( t , S t
)
∂ S t
− r f ( t , S ) +
∂ f ( t , S )
∂ t
+ rS
∂ f ( t , S ) 1
+ σ
2
∂ S 2
S
2
∂
2 f ( t , S )
∂ S 2
= 0 with the boundary condition f ( T , S ) = H ( S ) .
c Leonid Kogan ( MIT, Sloan )
Key Points: Derivatives
15.450, Fall 2010 5 / 1

In many models cannot derive a unique price for a derivative.
Term structure models, stochastic volatility.
Price assets relative to each other. Replication argument combined with assumptions on prices of risk.
Alternatively, specify dynamics directly under Q . c Leonid Kogan ( MIT, Sloan )
Key Points: Derivatives
15.450, Fall 2010 6 / 1

General pricing formula
P t
= E
Q t
� �
�
T exp − t r s ds
� �
H
T
Need to specify dynamics of the underlying under Q .
If underlying is a stock, only one way to do this: set expected return to r .
Q dynamics is related to P through price of risk dZ
P t
= − η t dt + dZ
Q t
Risk premium
E
P t
� dS t
�
− r
S t t dt = E
P t
� dS t
S t
�
− E
Q t
� dS t
S t
� c Leonid Kogan ( MIT, Sloan )
Key Points: Derivatives
15.450, Fall 2010 7 / 1

Derive a PDE on derivative prices using Ito’s lemma.
One-factor term structure model
E t
[ df ( t , r t
)] = r t f ( t , r t
) dt
Vasicek model: dr t
= − κ ( r t
− r ) dt + σ dZ
Q t f ( t , r t
) must satisfy the PDE
∂ f ( t , r )
− κ ( r − r )
∂ f ( t , r )
∂ t with the boundary condition
∂ r
+
1
σ
2
2
∂ 2 f ( t , r )
∂ r 2
= rf ( t , r ) f ( T , r ) = 1
Expected bond returns satisfy
� dP ( t , T )
E t
P ( t , T )
�
= ( r t
+ σ
P t
t
) dt c Leonid Kogan ( MIT, Sloan )
Key Points: Derivatives
15.450, Fall 2010 8 / 1

Random number generation: inverse transform, acceptance-rejection method.
Variance reduction: antithetic variates, control variates.
Intuition behind control variates: carve out the part of the estimated moment that is known in closed form, no need to estimate that by
Monte Carlo.
Good control variates: highly correlated with the variable of interest, expectation known in closed form.
Examples of control variates: stock price, payoff of similar option, etc. c Leonid Kogan ( MIT, Sloan )
Key Points: Derivatives
15.450, Fall 2010 9 / 1

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15.450 Analytics of Finance
Fall 2010
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