Chapter 15: Application and extension of the Black-Scholes model Flashcards
Transaction costs and parity relations
For most contingent claims Φ(S_T ), the replicating portfolio h_t = (x_t, yt)
(replicating self-financing portfolio)
given by Theorem
14.3.1 it changes continuously with time. This would mean continually incurring transaction costs,
which is not desirable.
THIS IS UNIQUE, and MOST OF THE TIME NOT CONSTANT
Note that the hedging
portfolios h_t = (x_t, y_t) that we found in Theorem 14.3.1 are (typically) non-constant, since
y_t =∂F/∂s will generally not be constant. Worse, Corollary 14.3.3 showed that these hedging portfolios were unique: there are no other self-financing portfolio strategies, based only on cash
and stock, which replicate Φ(S_T ).
DEF 15.1.1 CONSTANT PORTFOLIO
A constant portfolio is a portfolio that we buy at time 0, and which we then
hold until time T
That is, we don’t trade stock for cash, or cash for stock, during (0, T).
ALLOWING PORTFOLIOS TO HAVE OPTIONS/CONTRACTS
Given a contingent claim Φ(ST ) with exercise date T we write
Π_t(Φ) for the price of this contingent claim at time t, and h^Φ_t = (x^Φ_t, y^Φ_t)
for its self-replicating
portfolio.
Define
Φ^cash(S_T ) = 1
value of 1 in cash at time T
Φ^stock(S_T ) = S_T
value of a unit of stock at time T
Φ^call(ST ) = max(S_T − K, 0)
value of a EU-call with strike K at time T
Φ^put(ST ) = max(K − S_T , 0)
EU-put
put-call parity
The put-call parity relation states that
Φ^put(S_T ) =
Φ^call(ST ) + KΦ^cash(S_T ) − Φ^stock(ST ).
ie
value of put at time T = Value of K in cash - 1 unit of stock and 1 call option at time T
If at time ) we buy
Ke^{irT} cash
-1 unit stock
1 call option
and wait until time T
then our portfolio will have value =
Φ^put(S_T )
put-call parity
Proof:
max(K-S_T,0) = max(S_T-k,0) + k-S_T
case 1: S_T ≥ K
True K-S_T = 0+K-S_T
Case 2: S_T ≤ K
True 0 = S_T -K +K-S_T
EXAMPLE 15.1.2
Consider a contract with a contingent claim
STRADDLE
Φ^straddle(S_T ) = |S_T − K| =
{K − S_T if S_T ≤ K
{S_T − K if S_T > K
we try to find a constant portfolio which replicates a straddle using put call and cash (EU options)
This type of contingent claim is known as a straddle, with strike price K and exercise time T.
Φ^straddle(S_T ) = |S_T − K|
graph of Φ^straddle against S_T
if S_T is less than K then bigger for smaller S_T
if S_T=K then is 0
increases as S_T increases
ie V shape
graph for cash is straight horizontal line at 1
graph for stock is linear crossing at origin and increasing as S_T increases
graph of Φ^call is flat then increases from S_T =K etc
It is easy to see that the parity relation
Φ^straddle(S_T ) = Φ^put(S_T ) + Φ^call(S_T )
value of straddle at time T= value of 1 call 1 put at time T
Hence, we can hedge a straddle by holding a portfolio of one call option, plus one put option, with the same strike price K and exercise date T.
SO a portfolio of
{1 call, 1 put} with strike K and expiry T replicates Φ^straddle(S_T ) with a constant portfolio
NOTES FOR we try to find a constant portfolio which replicates using put call and cash (EU options)
try and approximate a general contingent claim Φ(ST ) with a constant hedging portfolio consisting of cash, stock and a variety of call options with a variety of strike prices and exercise times. It turns out that this approximation is possible, at least in theory, with arbitrarily good precision for a large class of contingent claims. The formal argument, which we don’t include in this course, relies on results from analysis concerning piecewise linear approximation of functions.
Unfortunately, in some cases a large number/variety of call options are needed, which can greatly increase the transaction costs of just buying the portfolio at time 0.
In short, there is no easy answer here. Transaction costs have to be incurred at some point, and there is no ‘automatic’ strategy that is best used to minimize them.
SENSITIVITY OF REPLICATING PORTFOLIOS
sensitivity of replicating portfolios to changes in the *stock price
*model parameters, r (interest rate), µ (drift) and σ (volatility)
To quantify the sensitivity:
F(t,S_t) = value of contingent claim Φ(ST) at time t
= value of self-financing replicating portfolio for Φ(S_T)
Various derivatives of F are used to assess the sensitivity of the associated portfolio, and
they are known collectively as the Greeks. They are
dont memorize
∆ = ∂F/∂s (Delta)
how steeply does F change with stock price
Γ = ∂^2F/∂s^2
(Gamma)
interested in first 2 only ------ Θ = ∂F/∂t (Theta) how value changes with time ρ =∂F/∂r (rho) how value changes with interest rate V =∂F/∂σ (Vega)
(F doesnt dep on drift ie deriv is 0)
Suppose ∆ = ∂F/∂s =0
*If at time t ∆ = ∂F/∂s (t,S_t) approx 0 then small change in value of S_t to S_{t+ δ} ( δ small time interval)
wont change value of
F(t,S_t) t o F(t+ δ, S_{t+δ})
** [f(s+δ)- f(s)]/δ =f'(s) differentiation equating to 0 then approx equal values of f ie when delta neutral
DEF 15.3.1
DELTA NEUTRAL
A portfolio with price function F is said to be delta neutral at time t if ∆F(t,St) = 0.
*less exposed to changes in stock price, f(s) roughly f(s+delta)
F(t,S_t) is the price function
define ∆_F and Γ_F
∆_F =∂F/∂s
Γ_F =∂^2F/∂s^2
SUPPOSE
we have a portfolio including something extra into our portfolio- choose to make ∆=0 ie delta neutral
we will now look at a hedging strategy that tries to keep ∆ ≈ 0.
For a contingent claim Φ(ST), let F(t,St) be the value of the ‘usual’ hedging portfolio ht = (xt,yt), consisting of just cash and stocks, that is provided by Theorem 14.3.1. Such a portfolio is typically not delta neutral
portfolio with price function F(t,S_t), and include amount z_t of some derivative (ie some contract depending on the value of stock) which has price function Z(t,S_t)
our portfolio now has value
V(t,S_t)=F(t,S_t) + z_tZ(t,S_t)
we want this to be delta neutral at time t:
∆_v(t,S_t)=0
So we want
0=∆_v=∆_F+z_t∆_z diff wrt s
(∂F/ ∂s + zt ∂Z/∂s = 0)
and solving for zt we see that we should hold
z_t=-∆_F/∆_Z
including this amount z_t of the derivative (with price function Z) into our portfolio to make ∆_v=0 is known as a delta hedge.
EXAMPLE
suppose we sold an option that has price function P(t,S_t) we want to delta hedge this sale. So our portfolio (After selling) is F(t,S_t)=-P(t,S_t)
We want to delta hedge using the stock, so Z(t,S_t)=S_t Z(t,s)=s
V(t,S_t)=F(t,S_t) + z_tS_t
to delta hedge, need ∆_v=0 = ∆_F +z_t
=-∆_p + z_t
So z_t= ∆_p(t,S_t)
DISCRETE REBALANCED DELTA HEDGE notes
The value of zt changes with time. If, at time t, we delta hedge using zt, then at time t+ we will discover that zt+ is slightly different from zt and our delta hedge is no longer working. On the other hand, if we continually adapt our portfolio to precisely match zt then we will suffer high transaction costs. To handle this issue, there is a procedure known as discrete rebalanced delta hedge.
