Option Pricing

Lecture 3: Option Pricing

August 2014

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Lecture 3: Option Pricing

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OPTION PRICING

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Lecture 3: Option Pricing

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No Arbitrage Price in Complete Markets

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In complete markets the option payoff can be replicated by trading in
the underlying risky assets and in the bank account

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Price of the option = Cost of the replicating portfolio

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Otherwise, there is arbitrage

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Finding the replicating portfolio is the basis for hedging and pricing

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Pricing and Hedging in the Binomial Tree

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Consider a single-period setting in which the stock S(0) = s only can
take two values in the future, su and sd.

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There is also a bond/bank account that pays interest r

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There is no arbitrage if and only if u > (1 + r) > d.

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The market here is complete

⇒ We can price securities by no-arbitrage by finding the cost of the
replicating portfolio that invests in the stock and the bank account

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We buy ∆ shares and invest B in the bank

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Denote by Cu and Cd the price of the derivative when the price of the
stock is us and ds, respectively

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For replication, we need to have
∆us + (1 + r)B = Cu
∆ds + (1 + r)B = Cd

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The price C(0) of the derivative has to be the same as the cost of the
replicating portfolio,
C(0) = ∆s + B

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Risk-Neutral/Martingale Pricing
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After simple algebra, we can see that
(
)
1
1+r−d
u − (1 + r)
C(0) =
Cu +
Cd
1+r
u−d
u−d

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Denote
p∗ =

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Then

1+r−d
u − (1 + r)
; (1 − p∗ ) =
u−d
u−d

[
]
1
1
∗
∗
∗
C(0) =
[p Cu + (1 − p )Cd ] = E
C(1)
1+r
1+r

⇒ Risk-neutral pricing

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Risk-Neutral/Martingale Pricing

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In classical insurance the price of liability payoff C(T ) is
C(0) = E[e−rT C(T )]
However, when there is possibility of hedging, so that X(T ) = C(T ),
then, under probability P ∗ under which e−rt X(t) is a martingale,
C(0) = X(0) = E ∗ [e−rT X(T )] = E ∗ [e−rT C(T )]

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Call Option in the Binomial Tree
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In a multi-period setting with n periods, denoting
p¯ =

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pu
1+r

one can show that
C(0) = sϕ(a, n, p¯) −

K
ϕ(a, n, p)
(1 + r)n

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where ϕ(·, ·, ·) is a binomial distribution function

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a is the minimum number of “up” jumps so that the option ends up
in-the-money

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When n → ∞, this formula converges to the Black and Scholes
formula

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Merton-Black-Scholes Pricing
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Consider the following stock dynamics
dS(t) = S(t)(µdt + σdW (t))

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with solution

{
}
1 2
S(t) = S(0) exp (µ − σ )t + σW (t)
2

There is also a bank account with price
dB(t) = rB(t)dt

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or
B(t) = B(0)ert

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Merton-Black-Scholes Pricing (cont)
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1 2

Since E[ekW (t) ] = e 2 k t , we have
E[S(t)] = S(0)eµt
[
]
S(t)
1
µ = log E
t
S(0)

and thus

Similarly, since
1
log S(t) − log S(0) = (µ − σ 2 )t + σW (t)
2
we get
σ2 =

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Var[log(S(t)) − log S(0)]
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Merton-Black-Scholes Pricing: PDE approach
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We want to find the price of the European call option on this stock,
with strike price K and maturity at T
For a claim with payoff C(T ) = g(S(T )) it is reasonable to guess that
the price will be a function C(t, S(t)) of the current time and price of
underlying.
If so, from Itˆo’s lemma, the price at time t C(t, S(t)) satisfies
]
[
1 2 2
dC = Ct + σ S Css + µSCs dt + σSCs dW
2
On the other hand, with π(t) = amount invested in stock at time t, a
self-financing wealth process satisfies:
dX(t) =

π(t)
X(t) − π(t)
dS(t) +
dB(t)
S(t)
B(t)

dX(t) = [rX(t) + (µ − r)π(t)]dt + σ(t)π(t)dW (t)
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If we want replication, C(t) = X(t), we need the dt terms to be
equal, and the dW terms to be equal

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Comparing dW terms we get that the number of shares needs to
satisfy
π(t)
= Cs (t, S(t))
S(t)

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Using this and comparing the dt terms we get the Black-Scholes PDE:

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1
Ct + σ 2 s2 Css + r(sCs − C) = 0
2
subject to the boundary condition,
C(T, s) = (s − K)+

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Black and Scholes Formula
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The solution is the Black and Scholes formula for European call
options:
C(t, S(t)) = S(t)N (d1 ) − Ke−r(T −t) N (d2 )

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where

1
N (x) := P [Z ≤ x] = √
2π

∫

x

e−

y2
2

dy

−∞

1
√
[log(S(t)/K) + (r + σ 2 /2)(T − t)]
σ T −t
1
√
=
[log(S(t)/K) + (r − σ 2 /2)(T − t)]
σ T −t
√
= d1 − σ T − t

d1 =
d2

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Black and Scholes Formula Using Martingale
Approach
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Let us find the dynamics of S under martingale probability P ∗

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Denote by W ∗ the Brownian motion under P ∗

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We claim that if we replace µ by r, that is, if the stock satisfies the
dynamics
dS(t)
= rdt + σdW ∗ (t)
S(t)

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then the discounted stock price is a P ∗ -martingale.

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Indeed, this is because Ito’s rule then gives
d(e−rt S(t)) = e−rt dS(t) + S(t)d(e−rt )
∗
¯
= e−rt [rS(t)dt+σS(t)dW ∗ (t)]−S(t)re−rt dt = 0×dt+σ S(t)dW
(t)

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Girsanov theorem

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In order to have the above dynamics and also
dS(t)
= µdt + σdW (t),
S(t)

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we need to have

µ−r
t
σ
The famous Girsanov theorem tells us that this is possible: there
exists a probability P ∗ under which so-defined W ∗ is a Brownian
motion.
W ∗ (t) = W (t) +

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Computing the price as expected value

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To find expectation under P ∗ , we note that we can write
S(T ) = S(0)eσW

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∗ (T )+(r− 1 σ 2 )T
2

We have to compute
E ∗ [e−rT (S(T ) − K)+ ]
= E ∗ [e−rT S(T )1{S(T )>K} ] − Ke−rT E ∗ [1{S(T )>K} ]

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For the second term we need to compute the price of a Digital
(Binary) option:
E ∗ e−rT 1{S(T )>K} = e−rT P ∗ (S(T ) > K)
P ∗ (S(T ) > K) = P ∗ (S(0)e(r−σ /2)T +σW
W ∗ (T )
= P ∗( √
> −d2 )
T
= N (d2 )
2

∗ (T )

> K)

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where the middle equality follows by taking logs and re-arranging.

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The first term is computed in the book using the formula
[ ( ∗
)]
∫ ∞
W (T )
1
2
√
g(x)e−x /2 dx
E g
=√
2π −∞
T

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An alternative way to find the PDE
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Under risk-neutral probability P ∗ , by Ito’s rule
1
dC(t, S(t)) = [Ct + rS(t)Cs + σ 2 S 2 (t)Css ]dt
2
+σCs S(t)dW ∗ (t)
Then, discounting,

d(e−rt C(t, S(t)))

1
= e−rt [(Ct + rS(t)Cs + σ 2 S 2 (t)Css − rC)]dt
2
+e−rt σCs S(t)dW ∗
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This has to be a P ∗ martingale, which means that the dt term has to
be zero.

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This gives the Black-Scholes PDE.

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Black and Scholes Formula for Puts

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It can be obtained from put-call parity

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or solving the same PDE with different boundary condition:
P (T, s) = (K − s)+

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or by computing the expected value under P ∗ .

Lecture 3: Option Pricing

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Plot of a Call Price

Call Option
50

25

0
0

25

50

75

100

125

Figure 7.3: Black-Scholes values for a call option.

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Plot of a Put Price

Put Option
50

25

0
0

25

50

75

100

Figure 7.4: Black-Scholes values for a put option.

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Implied Volatility

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It is the value of σ that matches the theoretical Black-Scholes price of
the option with the observed market price of the option

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In the Black - Scholes model, volatility is the same for all options on
the same underlying

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However, this is not the case for implied volatilities: volatility smile

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Plot of Volatility Smile
Volatility Smile

Implied volatility

Strike price
Figure 7.5: The level of implied volatility differs for different strike levels.

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Dividend-Paying Underlying
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Assume the stock pays a dividend at a continuous rate q

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This is appropriate when the underlying is an index, for example

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Total value of holding one share of stock is
∫ t
G(t) := S(t) +
qS(u)du
0

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Therefore, the wealth process of investing in this stock and the bank
account is
dX = (X − π)dB/B + πdG/S

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or
dX(t) = [rX(t) + π(t)(µ + q − r)]dt + π(t)σdW (t)

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Dividend-Paying Underlying (cont)
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To get the discounted wealth process to be a martingale,
dX(t) = rX(t)dt + π(t)σdW ∗ (t)

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we need to define a Brownian motion under pricing probability P ∗ as
W ∗ (t) = W (t) + t(µ + q − r)/σ

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which makes the stock dynamics
dS(t) = S(t)[(r − q)dt + σdW ∗ (t)]

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and the pricing PDE is
1
Ct + σ 2 s2 Css + (r − q)sCs − rC = 0
2

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Dividend-Paying Underlying (cont)

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The solution, for the European call option is obtained by replacing the
current underlying price s with se−q(T −t) :
C(t, s) = se−q(T −t) N (d1 ) − Ke−r(T −t) N (d2 )

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where
d1 =
d2 =

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1
[log(s/K) + (r − q + σ 2 /2)(T − t)]
σ T −t
1
√
[log(s/K) + (r − q − σ 2 /2)(T − t)]
σ T −t
√

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Options on Futures
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Assume that futures satisfy
dF (t) = F (t)[µF dt + σF dW (t)]

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Since holding futures does not cost any money, the wealth is all
invested in the bank account and it satisfies
dX =

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π
X
π
dF + dB = dF + rXdt
F
B
F

Then, the discounted wealth process is
¯=
dX

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π
¯
dF = π
¯ µF dt + π
¯ σF dW
F

Lecture 3: Option Pricing

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Options on futures (cont)
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The risk-neutral Brownian Motion is
W ∗ (t) = W (t) +

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µF
t
σF

and the PDE for path independent options is
1
Ct + σ 2 f 2 Cf f − rC = 0
2

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The solution for the call option is
C(t, f ) = e−r(T −t) [f N (d1 ) − KN (d2 )]
1
√
d1 =
[log(f /K) + (σF2 /2)(T − t)]
σF T − t
1
√
d2 =
[log(f /K) − (σF2 /2)(T − t)]
σF T − t

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Currency Options
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Consider the payoff, evaluated in the domestic currency, equal to
(Q(T ) − K)+

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where Q(T ) is time T domestic value of one unit of foreign currency

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Assume that the exchange rate process is given by
dQ(t) = Q(t)[µQ dt + σQ dW (t)]

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If we hold one unit of the foreign currency in foreign bank, we get the
foreign interest rate rf

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Pricing formula is the same as in the case of dividend-paying
underlying, but with q replaced by rf

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Currency Options (cont.)
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This is because if replicating the payoff by trading in the domestic
and foreign bank accounts, denoting B(t) := ert the time t value of
one unit of domestic currency, the dollar value of one unit of the
foreign account is
Q∗ (t) := Q(t)erf ·t

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Itˆ
o’s rule for products gives
dQ∗ = Q∗ [(µQ + rf )dt + σQ ] dW

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The wealth dynamics (in domestic currency) of a portfolio of π dollars
in the foreign account and the rest in the domestic account are
dX =

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X −π
π
dB + ∗ dQ∗ = [rX + π(µQ + rf − r)]dt + πσQ dW
B
Q

This is exactly the same as for dividends with q replaced by rf
We have
W ∗ (t) = W (t) + t(µQ − (r − rf ))/σQ
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Example: Quanto Options
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- S(t): a domestic equity index
- Payoff: S(T ) − F units of foreign currency; quanto forward
We need to have
dX(t) = rX(t)dt + π(t)σQ dW ∗ (t) + d(investment in S)

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As in the previous slide, we have
W ∗ (t) = W (t) + t(µQ − (r − rf ))/σQ
and thus

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Assume

dQ(t) = Q(t)[(r − rf )dt + σQ dW ∗ (t)]
dS(t) = S(t)[rdt + σS dZ ∗ (t)]

where BMP Z ∗ has instantaneous correlation ρ with W ∗ . We have
d(S(t)Q(t)) = S(t)Q(t)[(2r −rf +ρσQ σS )dt+σQ dW ∗ (t)+σS dZ ∗ (t)]
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Example: Quantos cont.
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S(T ) − F units of foreign currency is the same as (S(T ) − F )Q(T )
units of domestic currency. The domestic value is
e−rT (E ∗ [S(T )Q(T )] − F E ∗ [Q(T )])

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To make it equal to zero
F =

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E ∗ [S(T )Q(T )]
E ∗ [Q(T )]

(1)

We have
E ∗ [S(T )Q(T )] = S(0)Q(0)e(2r−rf +ρσS σQ )T
E ∗ [Q(T )] = Q(0)e(r−rf )T

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We get
F = S(0)e(r+ρσS σQ )T

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Merton’s jump-diffusion model
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Suppose the jumps arrive at a speed governed by a Poisson process

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That is, the number N (t) of jumps between moment 0 and t is given
by Poisson distribution:
P [N (t) = k] = e−λt

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(λt)k
k!

The stock price satisfies the following dynamics:
dS(t) = S(t)[r − λm]dt + S(t)σdW ∗ (t) + dJ(t) ,

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where m is related to the mean jump size and dJ is the actual jump
size.

Lecture 3: Option Pricing

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That is, dJ(t) = 0 if there is no jump at time t, and
dJ(t) = S(t)Xi − S(t) if the i-th jump of size Xi occurs at time t

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Therefore,
S(t) = S(0) · X1 · X2 · . . . · XN (t) · e(r−σ

2 /2−λm)t+σW ∗ (t)

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Here,

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in order to make the discounted stock price a martingale under P ∗ .

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m := E ∗ [Xi ] − 1

Lecture 3: Option Pricing

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We want to price an European option with payoff g(S(T ))

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The price of the option is,
C(0) =

∞
∑

[
E e−rT g(S(T ))
∗


]

N (T ) = k P ∗ [N (T ) = k]


k=0
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which is equal to
)]
[
(
∑∞
∗
−rT
(r−σ 2 /2−λm)T +σW ∗ (T )
E
e
g
S(0)X
·
.
.
.
X
·
e
1
k
k=0
×

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e−λT

(λT )k
k!

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If Xi ’s are lognormally distributed, the price of the option can be
represented as
∞
∑
˜ )k
˜ (λT
C(0) =
e−λT
BSk
k!
k=0

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˜ = λ(1 + m) and BSk is the Black and Scholes formula where
where λ
r = rk and σ = σk depend on k (see the book)

Lecture 3: Option Pricing

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Stochastic Volatility; Incomplete Markets

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Consider two independent BMP W1 and W2
dS(t) = S(t)[µ(t, S(t), V (t))dt + σ1 (t, S(t), V (t))dW1 (t)
+σ2 (t, S(t), V (t))dW2 (t)]
dV (t) = α(t, S(t), V (t))dt + γ(t, S(t), V (t))dW2 (t)

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Under a risk-neutral probability measure
dS(t) = S(t)[r(t)dt + σ1 (t)dW1∗ (t) + σ2 (t)dW2∗ (t)]

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Denote by κ(t) any (adapted) stochastic process

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Stochastic Vol with Inc Markets (cont)
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There are many measures P ∗ . In particular, for any such process κ we
can set
1
[µ(t) − r(t) − σ2 (t)κ(t)]dt
σ1 (t)
dW2∗ (t) = dW2 (t) + κ(t)dt
dW1∗ (t) = dW1 (t) +

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It can be checked that discounted S is then a P ∗ martingale and
dV (t) = [α(t) − κ(t)γ(t)]dt + γ(t)dW2∗ (t)

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Thus, for constant κ, the PDE becomes
1
1
Ct + Css s2 (σ12 + σ22 ) + Cvv γ 2
2
2
+ Csv γσ2 + r(sCS − C) + Cv (α − κγ) = 0

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Examples
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Complete market – CEV model
dS/S = rdt + σS p dW ∗

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Incomplete market – Heston’s model:
√
dS(t) = S(t)[rdt + V (t)dW ∗ (t)]
√
dV (t) = A(B − V (t))dt + γ V (t)dZ ∗ (t)
for some other risk-neutral Brownian motion Z ∗ having correlation ρ
with W ∗ . Price is a function C(t, s, v) satisfying
1
0 = Ct + v[s2 Css + γ 2 Cvv ] + r(sCs − C) + A(B − σ 2 )Cv + ργvsCsv
2

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Pricing theory for American options
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Denote by g¯(τ ) the discounted payoff of the option when exercised at
time τ .

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The price A(t) of the American option at time t is given by
A(t) = max Et∗ [e−r(τ −t) g(τ )]
t≤τ ≤T

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where τ is a stopping time

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When early exercise is not optimal, we have
A(t) > g(t)

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At the moment it is optimal to exercise the option, we have that
A(τ ) = g(τ )

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The Black-Scholes equation holds in the continuation region where
A(t, s) > g(t, s):
1
At + σ 2 s2 Ass + r(sAs − A) = 0
2

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In the exercise region, where if A(t, s) = g(t, s), we have:
1
At + σ 2 s2 Ass + r(sAs − A) < 0
2

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This is because if the holder does not optimally exercise it, the value
of the option falls down.

Lecture 3: Option Pricing

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Pricing in Binomial Tree

er∆t −d
u−d
values C u (T ),

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Find p∗ =

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Given

C d (T ), compute

C(T − ∆t) = e−r∆t [p∗ C u (T ) + (1 − p∗ )C d (T )]
and so on, to C(0).
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For an American option with payoff g(S(t)),
A(T − ∆t) = max{g(S(T − ∆T )), e−r∆t [p∗ Au (T ) + (1 − p∗ )Ad (T )]}

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