option pricing
DESCRIPTION
Lecture slides giving an introduction to the theory of option pricingTRANSCRIPT
Lecture 3: Option Pricing
August 2014
Cvitanic Lecture 3: Option Pricing 1
OPTION PRICING
Cvitanic Lecture 3: Option Pricing 2
No Arbitrage Price in Complete Markets
I In complete markets the option payoff can be replicated by trading inthe underlying risky assets and in the bank account
I Price of the option = Cost of the replicating portfolio
I Otherwise, there is arbitrage
I Finding the replicating portfolio is the basis for hedging and pricing
Cvitanic Lecture 3: Option Pricing 3
Pricing and Hedging in the Binomial Tree
I Consider a single-period setting in which the stock S(0) = s only cantake two values in the future, su and sd.
I There is also a bond/bank account that pays interest r
Cvitanic Lecture 3: Option Pricing 4
I There is no arbitrage if and only if u > (1 + r) > d.
I The market here is complete
⇒ We can price securities by no-arbitrage by finding the cost of thereplicating portfolio that invests in the stock and the bank account
Cvitanic Lecture 3: Option Pricing 5
I We buy ∆ shares and invest B in the bank
I Denote by Cu and Cd the price of the derivative when the price of thestock is us and ds, respectively
I For replication, we need to have
∆us+ (1 + r)B = Cu
∆ds+ (1 + r)B = Cd
I The price C(0) of the derivative has to be the same as the cost of thereplicating portfolio,
C(0) = ∆s+B
Cvitanic Lecture 3: Option Pricing 6
Risk-Neutral/Martingale Pricing
I After simple algebra, we can see that
C(0) =1
1 + r
(1 + r − d
u− dCu +
u− (1 + r)
u− dCd
)I Denote
p∗ =1 + r − d
u− d; (1− p∗) =
u− (1 + r)
u− d
I Then
C(0) =1
1 + r[p∗Cu + (1− p∗)Cd] = E∗
[1
1 + rC(1)
]⇒ Risk-neutral pricing
Cvitanic Lecture 3: Option Pricing 7
Risk-Neutral/Martingale Pricing
I In classical insurance the price of liability payoff C(T ) is
C(0) = E[e−rTC(T )]
However, when there is possibility of hedging, so that X(T ) = C(T ),then, under probability P ∗ under which e−rtX(t) is a martingale,
C(0) = X(0) = E∗[e−rTX(T )] = E∗[e−rTC(T )]
Cvitanic Lecture 3: Option Pricing 8
Call Option in the Binomial Tree
I In a multi-period setting with n periods, denoting
p =pu
1 + r
I one can show that
C(0) = sϕ(a, n, p)− K
(1 + r)nϕ(a, n, p)
I where ϕ(·, ·, ·) is a binomial distribution function
I a is the minimum number of “up” jumps so that the option ends upin-the-money
I When n → ∞, this formula converges to the Black and Scholesformula
Cvitanic Lecture 3: Option Pricing 9
Merton-Black-Scholes Pricing
I Consider the following stock dynamics
dS(t) = S(t)(µdt+ σdW (t))
I with solution
S(t) = S(0) exp
{(µ− 1
2σ2)t+ σW (t)
}I There is also a bank account with price
dB(t) = rB(t)dt
I orB(t) = B(0)ert
Cvitanic Lecture 3: Option Pricing 10
Merton-Black-Scholes Pricing (cont)
I Since E[ekW (t)] = e12k2t, we have
E[S(t)] = S(0)eµt
and thus
µ =1
tlogE
[S(t)
S(0)
]Similarly, since
logS(t)− logS(0) = (µ− 1
2σ2)t+ σW (t)
we get
σ2 =Var[log(S(t))− logS(0)]
t
Cvitanic Lecture 3: Option Pricing 11
Merton-Black-Scholes Pricing: PDE approach
I We want to find the price of the European call option on this stock,with strike price K and maturity at T
I For a claim with payoff C(T ) = g(S(T )) it is reasonable to guess thatthe price will be a function C(t, S(t)) of the current time and price ofunderlying.
I If so, from Ito’s lemma, the price at time t C(t, S(t)) satisfies
dC =
[Ct +
1
2σ2S2Css + µSCs
]dt+ σSCsdW
I On the other hand, with π(t) = amount invested in stock at time t, aself-financing wealth process satisfies:
dX(t) =π(t)
S(t)dS(t) +
X(t)− π(t)
B(t)dB(t)
dX(t) = [rX(t) + (µ− r)π(t)]dt+ σ(t)π(t)dW (t)
Cvitanic Lecture 3: Option Pricing 12
I If we want replication, C(t) = X(t), we need the dt terms to beequal, and the dW terms to be equal
I Comparing dW terms we get that the number of shares needs tosatisfy
π(t)
S(t)= Cs(t, S(t))
I Using this and comparing the dt terms we get the Black-Scholes PDE:
Ct +1
2σ2s2Css + r(sCs − C) = 0
I subject to the boundary condition,
C(T, s) = (s−K)+
Cvitanic Lecture 3: Option Pricing 13
Black and Scholes Formula
I The solution is the Black and Scholes formula for European calloptions:
C(t, S(t)) = S(t)N(d1)−Ke−r(T−t)N(d2)
I where
N(x) := P [Z ≤ x] =1√2π
∫ x
−∞e−
y2
2 dy
d1 =1
σ√T − t
[log(S(t)/K) + (r + σ2/2)(T − t)]
d2 =1
σ√T − t
[log(S(t)/K) + (r − σ2/2)(T − t)]
= d1 − σ√T − t
Cvitanic Lecture 3: Option Pricing 14
Black and Scholes Formula Using MartingaleApproach
I Let us find the dynamics of S under martingale probability P ∗
I Denote by W ∗ the Brownian motion under P ∗
I We claim that if we replace µ by r, that is, if the stock satisfies thedynamics
dS(t)
S(t)= rdt+ σdW ∗(t)
I then the discounted stock price is a P ∗-martingale.
I Indeed, this is because Ito’s rule then gives
d(e−rtS(t)) = e−rtdS(t) + S(t)d(e−rt)
= e−rt[rS(t)dt+σS(t)dW ∗(t)]−S(t)re−rtdt = 0×dt+σS(t)dW ∗(t)
Cvitanic Lecture 3: Option Pricing 15
Girsanov theorem
I In order to have the above dynamics and also
dS(t)
S(t)= µdt+ σdW (t),
I we need to have
W ∗(t) = W (t) +µ− r
σt
I The famous Girsanov theorem tells us that this is possible: thereexists a probability P ∗ under which so-defined W ∗ is a Brownianmotion.
Cvitanic Lecture 3: Option Pricing 16
Computing the price as expected value
I To find expectation under P ∗, we note that we can write
S(T ) = S(0)eσW∗(T )+(r− 1
2σ2)T
I We have to compute
E∗[e−rT (S(T )−K)+]
= E∗[e−rTS(T )1{S(T )>K}]−Ke−rTE∗[1{S(T )>K}]
Cvitanic Lecture 3: Option Pricing 17
I For the second term we need to compute the price of a Digital(Binary) option:
E∗e−rT1{S(T )>K} = e−rTP ∗(S(T ) > K)
P ∗(S(T ) > K) = P ∗(S(0)e(r−σ2/2)T+σW ∗(T ) > K)
= P ∗(W ∗(T )√
T> −d2)
= N(d2)
I where the middle equality follows by taking logs and re-arranging.
I The first term is computed in the book using the formula
E
[g
(W ∗(T )√
T
)]=
1√2π
∫ ∞
−∞g(x)e−x2/2dx
Cvitanic Lecture 3: Option Pricing 18
An alternative way to find the PDE
I Under risk-neutral probability P ∗, by Ito’s rule
dC(t, S(t)) = [Ct + rS(t)Cs +1
2σ2S2(t)Css]dt
+σCsS(t)dW∗(t)
Then, discounting,d(e−rtC(t, S(t)))
= e−rt[(Ct + rS(t)Cs +1
2σ2S2(t)Css − rC)]dt
+e−rtσCsS(t)dW∗
I This has to be a P ∗ martingale, which means that the dt term has tobe zero.
I This gives the Black-Scholes PDE.
Cvitanic Lecture 3: Option Pricing 19
Black and Scholes Formula for Puts
I It can be obtained from put-call parity
I or solving the same PDE with different boundary condition:
P (T, s) = (K − s)+
I or by computing the expected value under P ∗.
Cvitanic Lecture 3: Option Pricing 20
Plot of a Call Price
Call Option
0
25
50
0 25 50 75 100 125
Figure 7.3: Black-Scholes values for a call option.
Cvitanic Lecture 3: Option Pricing 21
Plot of a Put Price
Put Option
0
25
50
0 25 50 75 100
Figure 7.4: Black-Scholes values for a put option.
Cvitanic Lecture 3: Option Pricing 22
Implied Volatility
I It is the value of σ that matches the theoretical Black-Scholes price ofthe option with the observed market price of the option
I In the Black - Scholes model, volatility is the same for all options onthe same underlying
I However, this is not the case for implied volatilities: volatility smile
Cvitanic Lecture 3: Option Pricing 23
Plot of Volatility Smile
Volatility Smile
Implied volatility
Strike price
Figure 7.5: The level of implied volatility differs for different strike levels.
Cvitanic Lecture 3: Option Pricing 24
Dividend-Paying Underlying
I Assume the stock pays a dividend at a continuous rate q
I This is appropriate when the underlying is an index, for example
I Total value of holding one share of stock is
G(t) := S(t) +
∫ t
0qS(u)du
I Therefore, the wealth process of investing in this stock and the bankaccount is
dX = (X − π)dB/B + πdG/S
I ordX(t) = [rX(t) + π(t)(µ+ q − r)]dt+ π(t)σdW (t)
Cvitanic Lecture 3: Option Pricing 25
Dividend-Paying Underlying (cont)
I To get the discounted wealth process to be a martingale,
dX(t) = rX(t)dt+ π(t)σdW ∗(t)
I we need to define a Brownian motion under pricing probability P ∗ as
W ∗(t) = W (t) + t(µ+ q − r)/σ
I which makes the stock dynamics
dS(t) = S(t)[(r − q)dt+ σdW ∗(t)]
I and the pricing PDE is
Ct +1
2σ2s2Css + (r − q)sCs − rC = 0
Cvitanic Lecture 3: Option Pricing 26
Dividend-Paying Underlying (cont)
I The solution, for the European call option is obtained by replacing thecurrent underlying price s with se−q(T−t):
C(t, s) = se−q(T−t)N(d1)−Ke−r(T−t)N(d2)
I where
d1 =1
σ√T − t
[log(s/K) + (r − q + σ2/2)(T − t)]
d2 =1
σ√T − t
[log(s/K) + (r − q − σ2/2)(T − t)]
Cvitanic Lecture 3: Option Pricing 27
Options on Futures
I Assume that futures satisfy
dF (t) = F (t)[µFdt+ σFdW (t)]
I Since holding futures does not cost any money, the wealth is allinvested in the bank account and it satisfies
dX =π
FdF +
X
BdB =
π
FdF + rXdt
I Then, the discounted wealth process is
dX =π
FdF = πµFdt+ πσFdW
Cvitanic Lecture 3: Option Pricing 28
Options on futures (cont)
I The risk-neutral Brownian Motion is
W ∗(t) = W (t) +µF
σFt
I and the PDE for path independent options is
Ct +1
2σ2f2Cff − rC = 0
I The solution for the call option is
C(t, f) = e−r(T−t)[fN(d1)−KN(d2)]
d1 =1
σF√T − t
[log(f/K) + (σ2F /2)(T − t)]
d2 =1
σF√T − t
[log(f/K)− (σ2F /2)(T − t)]
Cvitanic Lecture 3: Option Pricing 29
Currency Options
I Consider the payoff, evaluated in the domestic currency, equal to
(Q(T )−K)+
I where Q(T ) is time T domestic value of one unit of foreign currency
I Assume that the exchange rate process is given by
dQ(t) = Q(t)[µQdt+ σQdW (t)]
I If we hold one unit of the foreign currency in foreign bank, we get theforeign interest rate rf
I Pricing formula is the same as in the case of dividend-payingunderlying, but with q replaced by rf
Cvitanic Lecture 3: Option Pricing 30
Currency Options (cont.)I This is because if replicating the payoff by trading in the domestic
and foreign bank accounts, denoting B(t) := ert the time t value ofone unit of domestic currency, the dollar value of one unit of theforeign account is
Q∗(t) := Q(t)erf ·t
I Ito’s rule for products gives
dQ∗ = Q∗ [(µQ + rf )dt+ σQ] dW
I The wealth dynamics (in domestic currency) of a portfolio of π dollarsin the foreign account and the rest in the domestic account are
dX =X − π
BdB +
π
Q∗dQ∗ = [rX + π(µQ + rf − r)]dt+ πσQdW
I This is exactly the same as for dividends with q replaced by rfI We have
W ∗(t) = W (t) + t(µQ − (r − rf ))/σQ
Cvitanic Lecture 3: Option Pricing 31
Example: Quanto OptionsI - S(t): a domestic equity index
- Payoff: S(T )− F units of foreign currency; quanto forwardI We need to have
dX(t) = rX(t)dt+ π(t)σQdW∗(t) + d(investment in S)
I As in the previous slide, we have
W ∗(t) = W (t) + t(µQ − (r − rf ))/σQ
and thusdQ(t) = Q(t)[(r − rf )dt+ σQdW
∗(t)]
I AssumedS(t) = S(t)[rdt+ σSdZ
∗(t)]
where BMP Z∗ has instantaneous correlation ρ with W ∗. We have
d(S(t)Q(t)) = S(t)Q(t)[(2r−rf+ρσQσS)dt+σQdW∗(t)+σSdZ
∗(t)]
Cvitanic Lecture 3: Option Pricing 32
Example: Quantos cont.
I 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 )]− FE∗[Q(T )])
I To make it equal to zero
F =E∗[S(T )Q(T )]
E∗[Q(T )](1)
I We have
E∗[S(T )Q(T )] = S(0)Q(0)e(2r−rf+ρσSσQ)T
E∗[Q(T )] = Q(0)e(r−rf )T
I We getF = S(0)e(r+ρσSσQ)T
Cvitanic Lecture 3: Option Pricing 33
Merton’s jump-diffusion model
I Suppose the jumps arrive at a speed governed by a Poisson process
I That is, the number N(t) of jumps between moment 0 and t is givenby Poisson distribution:
P [N(t) = k] = e−λt (λt)k
k!
I The stock price satisfies the following dynamics:
dS(t) = S(t)[r − λm]dt+ S(t)σdW ∗(t) + dJ(t) ,
I where m is related to the mean jump size and dJ is the actual jumpsize.
Cvitanic Lecture 3: Option Pricing 34
I That is, dJ(t) = 0 if there is no jump at time t, anddJ(t) = S(t)Xi − S(t) if the i-th jump of size Xi occurs at time t
I Therefore,
S(t) = S(0) ·X1 ·X2 · . . . ·XN(t) · e(r−σ2/2−λm)t+σW ∗(t)
I Here,m := E∗[Xi]− 1
I in order to make the discounted stock price a martingale under P ∗.
Cvitanic Lecture 3: Option Pricing 35
I We want to price an European option with payoff g(S(T ))
I The price of the option is,
C(0) =∞∑k=0
E∗[e−rT g(S(T ))
∣∣∣∣ N(T ) = k
]P ∗[N(T ) = k]
I which is equal to∑∞k=0 E∗
[e−rT g
(S(0)X1 · . . . Xk · e(r−σ2/2−λm)T+σW ∗(T )
)]× e−λT (λT )
k
k!
Cvitanic Lecture 3: Option Pricing 36
I If Xi’s are lognormally distributed, the price of the option can berepresented as
C(0) =
∞∑k=0
e−λT (λT )k
k!BSk
I where λ = λ(1 +m) and BSk is the Black and Scholes formula wherer = rk and σ = σk depend on k (see the book)
Cvitanic Lecture 3: Option Pricing 37
Stochastic Volatility; Incomplete Markets
I 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)
I Under a risk-neutral probability measure
dS(t) = S(t)[r(t)dt+ σ1(t)dW∗1 (t) + σ2(t)dW
∗2 (t)]
I Denote by κ(t) any (adapted) stochastic process
Cvitanic Lecture 3: Option Pricing 38
Stochastic Vol with Inc Markets (cont)
I There are many measures P ∗. In particular, for any such process κ wecan set
dW ∗1 (t) = dW1(t) +
1
σ1(t)[µ(t)− r(t)− σ2(t)κ(t)]dt
dW ∗2 (t) = dW2(t) + κ(t)dt
I It can be checked that discounted S is then a P ∗ martingale and
dV (t) = [α(t)− κ(t)γ(t)]dt+ γ(t)dW ∗2 (t)
I Thus, for constant κ, the PDE becomes
Ct +1
2Csss
2(σ21 + σ2
2) +1
2Cvvγ
2
+ Csvγσ2 + r(sCS − C) + Cv(α− κγ) = 0
Cvitanic Lecture 3: Option Pricing 39
Examples
I Complete market – CEV model
dS/S = rdt+ σSpdW ∗
I 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 ρwith W ∗. Price is a function C(t, s, v) satisfying
0 = Ct +1
2v[s2Css + γ2Cvv] + r(sCs −C)+A(B− σ2)Cv + ργvsCsv
Cvitanic Lecture 3: Option Pricing 40
Pricing theory for American options
I Denote by g(τ) the discounted payoff of the option when exercised attime τ .
I The price A(t) of the American option at time t is given by
A(t) = maxt≤τ≤T
E∗t [e
−r(τ−t)g(τ)]
I where τ is a stopping time
I When early exercise is not optimal, we have
A(t) > g(t)
I At the moment it is optimal to exercise the option, we have that
A(τ) = g(τ)
Cvitanic Lecture 3: Option Pricing 41
I The Black-Scholes equation holds in the continuation region whereA(t, s) > g(t, s):
At +1
2σ2s2Ass + r(sAs −A) = 0
I In the exercise region, where if A(t, s) = g(t, s), we have:
At +1
2σ2s2Ass + r(sAs −A) < 0
I This is because if the holder does not optimally exercise it, the valueof the option falls down.
Cvitanic Lecture 3: Option Pricing 42
Pricing in Binomial Tree
I Find p∗ = er∆t−du−d
I Given values Cu(T ), Cd(T ), compute
C(T −∆t) = e−r∆t[p∗Cu(T ) + (1− p∗)Cd(T )]
and so on, to C(0).
I 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 )]}
Cvitanic Lecture 3: Option Pricing 43