Imperial College
London
Financial Engineering
Part 1
PRICING/HEDGING DERIVATIVES WITH TREES
Enrico Biffis
Overview
One period
Two periods
Extensions
1 / 34
Imperial College
London
AGENDA
1
Overview
2
One period
3
Two periods
4
Extensions
Overview
One period
Two periods
Extensions
2 / 34
Imperial College
London
AGENDA
1
Overview
2
One period
3
Two periods
4
Extensions
Overview
One period
Two periods
Extensions
3 / 34
Imperial College
London
Overview
Main objectives of this part:
1
Develop a pricing machinery based on binomial (and multinomial) trees.
2
Address the concepts of complete and incomplete market. Understand how
they affect the pricing and hedging of derivatives.
3
Understand the value of early exercise in American derivatives.
4
Gain exposure to the most important Greeks (sensitivities) used to design
hedging strategies.
5
Be in a position to revisit the Black & Scholes pricing formula as the
continuous time limit of a tree-based pricing framework.
Overview
One period
Two periods
Extensions
4 / 34
Imperial College
London
AGENDA
1
Overview
2
One period
3
Two periods
4
Extensions
Overview
One period
Two periods
Extensions
5 / 34
Imperial College
London
One-period binomial model
The simplest possible model: one-period (two dates: 0 and 1), binomial tree.
Money market account yielding r > 0.
Stock with current price S0 , and terminal random price S1 .
Randomness is described by two outcomes, ‘up’ and ‘down’. Think of coin
flipping and sates of the world H (Head) and T (Tail), so that
S1 (H) = uS0 in state H, and S1 (T ) = dS0 in state T , for u > d. The coin
may well be biased: we assume the ‘up’ probability to be p ∈ (0, 1).
We require 0 < d < 1 + r < u:
d > 0 by limited liability;
d < 1 + r and u > 1 + r to prevent arbitrage opportunities (show how
you could make free money in case d ≥ 1 + r or u ≤ 1 + r!).
Overview
One period
Two periods
Extensions
6 / 34
Imperial College
London
One-period binomial tree (draw it)
Overview
One period
Two periods
Extensions
7 / 34
Imperial College
London
Derivatives pricing
Consider a derivative with terminal payoff V1 . In the case of (say) a call
option, we have:
V1 = max(S1 − K, 0) = (S1 − K)+ ,
where K ∈ (S1 (T ), S1 (H)) denotes the strike price.
Replication argument: is it possible to set up a trading strategy replicating
the terminal payoff of the option at maturity? If the answer is yes, then the
price of the option must coincide with the cost of setting up such a strategy
at inception (law of one price).
Let us denote by X0 the market value of the strategy at time 0, and by ∆0
the units of stock purchased according to the strategy, so that X0 − ∆0 S0 is
the amount invested in the money market account.
Overview
One period
Two periods
Extensions
8 / 34
Imperial College
London
Replicating Portfolio
We look for a pair (X0 , ∆0 ) such that X1 = V1 , i.e.,
(
X1 (H) = (X0 − ∆0 S0 )(1 + r) + ∆0 S1 (H) = V1 (H)
X1 (T ) = (X0 − ∆0 S0 )(1 + r) + ∆0 S1 (T ) = V1 (T )
The system can be easily solved to obtain
(
1 uV1 (T )−dV1 (H)
X0 − ∆0 S0 = 1+r
u−d
V1 (H)−V1 (T )
1 (T )
∆0 = SV11 (H)−V
=
(H)−S1 (T )
S0 (u−d)
Overview
One period
Two periods
(1)
Extensions
9 / 34
Imperial College
London
Risk-neutral pricing
By no arbitrage we must have V0 = X0 , and from the above we can write
1 uV1 (T ) − dV1 (H)
+ ∆0 S0
1+r
u−d
1 uV1 (T ) − dV1 (H) V1 (H) − V1 (T )
+
=
1+r
u−d
u−d
1
1+r−d
u−1−r
=
V1 (H) +
V1 (T )
1+r
u−d
u−d
1
=
(e
p V1 (H) + qe V1 (T ))
1+r
V0 =
Note that the parameters pe, qe we introduced are positive and satisfy
pe + qe = 1. We thus have the risk neutral valuation (RNV) formula:
1 e
e V1
V0 =
E [V1 ] = E
1+r
1+r
Overview
One period
Two periods
Extensions
(2)
10 / 34
Imperial College
London
Market-implied probabilities
Typically, you observe the market price of financial instruments and recover
risk-neutral probabilities via RNV. Such probabilities are therefore called
market-implied.
We now have the following system of equations, which is the counterpart of
the replicating portfolio considered in (1):
(
1
V0 = 1+r
[e
pV1 (H) + qeV1 (T )]
pe + qe = 1,
Note that V1 (H) and V1 (T ), as well as V0 , are all given. When V0 is not
available, we can recover pe and qe by writing the above system for S0 and
then obtain V0 via RNV.
Overview
One period
Two periods
Extensions
11 / 34
Imperial College
London
Comments
The pricing formula ignores the real world probabilities (p, 1 − p) of stock
movement: the only thing that matters is the size of the moves (derivatives
pricing depends on the volatility of the underlying, not its mean growth rate).
RNV says that under no arbitrage market prices must be martingales after
deflation by the money market account and under a risk adjusted probability
e (This clearly applies to S as well: apply (2) to S1 to verify it!)
measure P.
e are equivalent, in the
The condition 0 < d < 1 + r < u ensures that P and P
e is
sense that they assign zero probability to the same events. The measure P
therefore usually referred to as Equivalent Martingale Measure (EMM).
The amount invested in the underlying stock, ∆0 , is usually referred to as
Delta and is the most important sensitivity (or ‘Greek’) we discuss in this
part. It represents the (first order) sensitivity of the derivative’s market value
to (small) changes in the market price of the underlying stock.
Overview
One period
Two periods
Extensions
12 / 34
Imperial College
London
Example
Consider the following market parameters: S0 = 4, u = 2, d = 0.5,
r = 0.25, p = 60%.
Consider a call option on the stock with strike price K = 5.
We have V1 (H) = (8 − 5)+ = 3 and V1 (T ) = (2 − 5)+ = 0.
For the replicating portfolio we find ∆0 = 0.5 and X0 − ∆0 S0 = −0.8, and
hence the no arbitrage price of the call option is V0 = 1.2.
• Interpretation: if we sell the option for $1.2, we can hedge out our position
by going long 0.5 units of the stock, which can be implemented by
borrowing $0.8 at the risk free rate (it costs $2 to go long the stock!).
Verify the price via RNV, noting that pe = 50%. You will also realize that in
the above we never considered the real world probabilities (p, 1 − p).
Overview
One period
Two periods
Extensions
13 / 34
Imperial College
London
Market completeness vs. incompleteness
We were able to find a unique solution for the replicating strategy because
there is one source of risk (up or down move in the stock price), and one
tradeable asset spanning such source of risk. This is the case of a complete
market, in which every contingent claim can be perfectly replicated.
If there is “more randomness” than the tradeable assets can span, the
market is said to be incomplete: perfect replication is no longer possible;
there are infinitely many prices consistent with no arbitrage and infinitely
many EMMs.
Incomplete market pricing techniques use the notion of approximate
hedging. For given approximation criterion chosen, you determine the best
possible (optimal) strategy delivering approximate replication. The initial
cost of setting up the strategy then gives the price of the derivative of
interest. Let’s see more on this...
Overview
One period
Two periods
Extensions
14 / 34
Imperial College
London
Trinomial model
The simplest example of incomplete market model is obtained by extending our
one-period setting to a trinomial model.
There are now three states of the world at time 1 (states H, M , and T ):


with prob. pH
S1 (H) = uS0
S1 = S1 (M ) = mS0 with prob. pM


S1 (T ) = dS0
with prob. pT .
Suppose we want to price a derivative with terminal payoff V1 via RNV:
V0 =
1
1 e
E[V1 ] =
[e
pH V1 (H) + peM V1 (M ) + peT V1 (T )] ,
1+r
1+r
(3)
for some risk adjusted probabilities peH , peM , peT .
Overview
One period
Two periods
Extensions
15 / 34
Imperial College
London
Trinomial model (draw the tree)
Overview
One period
Two periods
Extensions
16 / 34
Imperial College
London
Market incompleteness
By using market information, we know that the relevant EMM (i.e., the
market implied RN probabilities) would be identified via the system∗
(
1
pH S1 (H) + peM S1 (M ) + peT S1 (T )] = S0
1+r [e
peH + peM + peT = 1,
which has infinitely many solutions (two equations with three unknowns).
There are two main ways to proceed here:
1. Market completion.
2. Incomplete market pricing methods.
∗
The system’s first equation is equivalent to peH u + peM m + peT d = 1 + r.
Overview
One period
Two periods
Extensions
17 / 34
Imperial College
London
1. Completing the market
We complete the market by adding information on a tradeable asset with random
payoff (say) U1 and current price U0 .
Think of considering not just a stock, but also an option on the stock. By
no-arbitrage, RNV must apply to both instruments simultaneously.
The relevant system of equations is now


peH u + peM m + peT d = 1 + r
peH U1 (H) + peM U1 (M ) + peT U1 (T ) = (1 + r)U0


peH + peM + peT = 1,
where the first equation is nothing else than the pricing equation for the
stock written in the previous slide.
If the system admits a unique solution (e
pH , peM , peT ), then we can price any
other derivative with payoff (say) V1 via the RNV formula (3).
Overview
One period
Two periods
Extensions
18 / 34
Imperial College
London
2. Incomplete market pricing methods
If we cannot complete the market, we can add some constraints to the problem,
for example by imposing conditions on the hedging quality we can achieve (recall
that perfect replication is no longer possible here). In the following examples,
think of a structurer selling a derivative with terminal payoff V1 to a counterparty
(i.e., V1 is the seller’s liability at maturity).
Quadratic risk minimization. Look for hedging strategies minimizing the
mean square hedging error, i.e., the following objective function:
h
i
2
E (X1 − V1 ) .
Shortfall risk minimization. Look for hedging strategies minimizing the
expected shortfall, i.e., the following objective function:
h
i
+
E (V1 − X1 ) .
Overview
One period
Two periods
Extensions
19 / 34
Imperial College
London
AGENDA
1
Overview
2
One period
3
Two periods
4
Extensions
Overview
One period
Two periods
Extensions
20 / 34
Imperial College
London
Two-period binomial model
We extend the previous model to a two-period setting with a recombining
tree (simpler to handle when covering multiple periods later on).
At time 1, each node of the first-period biniomial model can branch into two
possible outcomes for the stock price. We therefore have that S2 can take
three possible values:

2

w. prob. p2
S2 (HH) = uS1 (H) = u S0
S2 = S2 (HT ) = S2 (T H) = dS1 (H) = uS1 (T ) = udS0 w. prob. 2p(1 − p)


S2 (T T ) = dS1 (T ) = d2 S0
w. prob. (1 − p)2
Note that for a derivative with terminal value V2 we may not have
V2 (HT ) = V2 (T H) in general. You will see an exercise on path-dependent
derivatives as an example.
Overview
One period
Two periods
Extensions
21 / 34
Imperial College
London
Two-period binomial model (draw the tree)
Overview
One period
Two periods
Extensions
22 / 34
Imperial College
London
Two-period binomial pricing
We price any derivative by working on the tree from the terminal date backwards.
To fix ideas, consider a call option with payoff V2 = (S2 − K)+ . Here’s the algo:
1. Build up the tree of asset prices (S0 , S1 , S2 ) along times 0, 1, 2.
2. Write down the terminal payoff of the derivative at maturity, V2 .
3. Consider the one-period binomial trees leading to the final outcomes at the
terminal date. For each binomial tree determine the pair ∆1 , X1 across
states H and T .
4. Consider the first one-period binomial tree leading to the outcomes available
at the end of the first time period. Determine ∆0 , X0 based on the
quantities ∆1 (H), X1 (H) and ∆1 (T ), X1 (T ) determined at point 3.
5. Write down the price of the derivative as V0 = X0
Overview
One period
Two periods
Extensions
23 / 34
Imperial College
London
Details for step 3
At time 1, we solve for (X1 (H), ∆1 (H)) by replication as follows:
(
X2 (HH) = [X1 (H) − ∆1 (H)S1 (H)] (1 + r) + ∆1 (H)S2 (HH) = V2 (HH)
X2 (HT ) = [X1 (H) − ∆1 (H)S1 (H)] (1 + r) + ∆1 (H)S2 (HT ) = V2 (HT )
At time 1, we solve for (X1 (T ), ∆1 (T )) by replication as follows:
(
X2 (T H) = [X1 (T ) − ∆1 (T )S1 (T )] (1 + r) + ∆1 (T )S2 (T H) = V2 (T H)
X2 (T T ) = [X1 (T ) − ∆1 (T )S1 (T )] (1 + r) + ∆1 (T )S2 (T T ) = V2 (T T )
These are the same systems of equations as in page 9. The solutions are
similar and reported in the next page for convenience.
Overview
One period
Two periods
Extensions
24 / 34
Imperial College
London
Details for steps 3 and 4
Solutions for the cash and stock positions at time 1 in state H:
(
1 uV2 (HT )−dV2 (HH)
X1 (H) − ∆1 (H)S1 (H) = 1+r
u−d
V2 (HH)−V2 (HT )
2 (HT )
∆1 (H) = S2 (HH)−S2 (HT ) = V2 (HH)−V
S1 (H)(u−d)
(4)
Solutions for the cash and stock positions at time 1 in state T :
(
1 uV2 (T T )−dV2 (T H)
X1 (T ) − ∆1 (T )S1 (T ) = 1+r
u−d
H)−V2 (T T )
V2 (T H)−V2 (T T )
∆1 (T ) = SV22 (T
=
(T H)−S2 (T T )
S1 (T )(u−d)
The solution for the time 0 positions is finally given by:
(
1 uX1 (T )−dX1 (H)
X0 − ∆0 S0 = 1+r
u−d
X1 (H)−X1 (T )
1 (T )
∆0 = S1 (H)−S1 (T ) = X1 (H)−X
S0 (u−d)
Overview
One period
Two periods
Extensions
25 / 34
Imperial College
London
Comments
We saw that the replicating strategy now entails state contingent stock and
cash positions. This is referred to as dynamic hedging.
In particular, the Delta relevant at time 0 (∆0 ) needs to be adjusted based
on the realization of the stock price at time 1 (∆1 (H) or ∆1 (T )). The
sensitivity of Delta to changes in the underlying instrument is called Gamma.
One can verify that RNV applies at each time and state. For example, from
system (4) in the previous page we can write the market value of the
replicating strategy as follows
"
#
1
V
2
e
X1 (H) =
[e
pV2 (HH) + qeV2 (HT )] = E
F1 ,
1+r
1+r
where the risk adjusted probabilities (e
p, qe) are defined on p. 10. The above
defines V1 (H), the market value of the derivative at time 1 and in state H.
Overview
One period
Two periods
Extensions
26 / 34
Imperial College
London
American option pricing
In the case of European derivatives, exercise can occur at maturity only, and
hence any derivative payoff is captured by g(S2 ) for some payoff function g
(or g(S0 , S1 , S2 ) for path-dependent derivatives).
In the case of American derivatives, exercise can occur before maturity. We
distinguish between the market value of the derivative in case we decide to
exercise (payoff from early exercise) and the case in which we do not
(continuation value).
• The pricing/hedging algo outlined before can accommodate American
derivatives as follows: when we go back one time period in the backward
procedure, we simply need to check whether the continuation value of the
derivative, X1 , is greater than the payoff from early exercise.
• The market value of the option at time 1 is no longer simply given by
V1 = X1 , but is replaced by V1 = max (g(S1 ), X1 ).
Overview
One period
Two periods
Extensions
27 / 34
Imperial College
London
American option pricing example I
Consider the following market parameters: S0 = 4, u = 2, d = 0.5, r = 0.25.
Consider an American put option on the stock with strike price K = 5.
We have V2 (HH) = (5 − 16)+ = 0, V2 (HT ) = (5 − 4)+ = 1, and
V2 (T T ) = (5 − 1)+ = 4.
For the time-1 replicating portfolio in state H we find
(
∆1 (H) = −0.0833
X1 (H) − ∆1 (H)S1 (H) = 1.066,
and hence X1 (H) = 0.4.
We note that (5 − S1 (H))+ = (5 − 8)+ = 0 < X1 (H), and hence
continuation is optimal at time 1 in state H.
Overview
One period
Two periods
Extensions
28 / 34
Imperial College
London
American option pricing example II
For the time-1 replicating portfolio in state T we find
(
∆1 (T ) = −1
X1 (T ) − ∆1 (T )S1 (T ) = 4,
and hence X1 (T ) = 2.
We note that (5 − S1 (T ))+ = (5 − 2)+ = 3 > X1 (T ), and hence early
exercise is optimal at time 1 in state T .
• We therefore set V1 (H) = X1 (H) = 0.4, and V1 (T ) = 3 > X1 (T ).
Overview
One period
Two periods
Extensions
29 / 34
Imperial College
London
American option pricing example III
We can solve for the replicating portfolio at time 0 by just using (1):
(
1 uV1 (T )−dV1 (H)
1 u3−d0.4
X0 − ∆0 S0 = 1+r
= 1+r
= 3.0933
u−d
u−d
V1 (H)−V1 (T )
0.4−3
∆0 = S1 (H)−S1 (T ) = S0 (u−d) = −0.4333
The put option price is therefore V0 = 1.36, which is clearly larger than the
corresponding European option price of 0.96 (verify it!).
The difference represents the market value of the option to exercise early the
right to sell the stock at a fixed price (early exercise premium).
Overview
One period
Two periods
Extensions
30 / 34
Imperial College
London
AGENDA
1
Overview
2
One period
3
Two periods
4
Extensions
Overview
One period
Two periods
Extensions
31 / 34
Imperial College
London
Multi-period models
Trees are usually helpful in discretizing a continuous time problem.
In multi-period models, to be covered in the next section, each period refers
to a small enough time step of the discretization procedure.
Consider the time horizon [0, T ] and a discretization step of size h > 0.
Then, the tree branches at dates 0, h, 2h, . . . , nh = T . Hence, the time step
of the binomial model does not need to be unitary as we assumed so far.
If h is small, it is common to allow for continous compounding in the risk
free rate accrual, so that erh is what is earned on the money market account
with continuous reinvestment of proceeds.
Overview
One period
Two periods
Extensions
32 / 34
Imperial College
London
Dividends
In practice, stocks pay dividends and their impact on a derivative’s value can be
sizeable.
Assume a continuous dividend yield δ ≥ 0, which is reinvested continuously
in the stock.
Then the usual restrictions on the tree parameters now become
eδh dS0 < S0 erh < eδh uS0 , thus yielding d < e(r−δ)h < u.
The replication arguments developed earlier on carry over to the present
setting, the only difference being that we need to take into account
reinvestment of dividends. Using the one-period model as a reference
benchmark, for example, this means that the cash and stock positions of the
replicating strategy given in (1) are replaced by
(
)−dVh (H)
X0 − ∆0 S0 = e−rh uVh (Tu−d
(5)
V
(H)−V
(T
)
h (T )
∆0 = e−δh Shh (H)−Shh (T ) = e−δh Vh (H)−V
S0 (u−d)
Overview
One period
Two periods
Extensions
33 / 34
Imperial College
London
RNV with dividends
RNV applies to the case of dividends as follows:
The cumulative gain process from holding a security (capital appreciation
plus cumulative dividends) is a martingale under a risk adjusted probability
measure and after deflation by the money market account.
RNV formula (2) on p. 10, for example, now takes the form
e e−r V1 ,
V0 = E
with risk neutral probabilities
(
pe =
qe =
Overview
One period
e(r−δ) −d
u−d
u−e(r−δ)
u−d
Two periods
Extensions
34 / 34