Download Stochastic Calculus, Week 9 Applications of risk

Dividends
The standard Black-Scholes analysis does not allow for div-
Stochastic Calculus, Week 9
idend payments. We may introduce them in two alternative
Applications of risk-neutral valuation
ways:
• Dividend is paid continuously, with a dividend yield δ;
• Dividend is paid at discrete time points.
Outline
Continuous dividends
1. Dividends
Suppose that the dividends are paid continuously at the rate
2. Foreign exchange
δ, and that all dividends are reinvested in the stock. Then
3. Quantos
the value S̃t of a portfolio that starts with one stock evolves
4. Market price of risk
as
dS̃t = at dSt + δat St dt,
where at = S̃t /St , the number of shares at time t. Thus,
dS̃t = S̃t (µdt + σdWt ) + δ S̃t dt
= (µ + δ)S̃t dt + σ S̃t dWt .
This implies that S̃t = eδt St , and hence at = eδt .
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A strategy (φt , ψ t ) in terms of (St , Bt ) may be written as
This yields, via risk-neutral valuation:
a corresponding strategy (φ̃t , ψ t ) in terms of (S̃t , Bt ), with
φ̃t = e−δt φt . The self-financing restriction now is (with Vt =
• The forward price of St : setting e−r(T −t) EQ [(ST −
φ̃t S̃t + ψ t Bt )
Ft )|Ft ] = 0 and solving for Ft gives
Ft = e(r−δ)(T −t) St .
dVt = φ̃t dS̃t + ψ t dBt
= φt dSt + φt δSt dt + ψ t dBt .
• The price of a call option struck at K. Following the
The essential point is that, whereas the replicating portfolio
same steps as in the standard Black-Scholes model, we
is in terms of S̃t , the derivative is in terms of St . Note that
find
the measure Q which makes Z̃t = Bt−1 S̃t a martingale, does
not make
Bt−1 St
Vt = e−r(T −t) EQ [(ST − K)+ |Ft ]
a martingale.
= e−δ(T −t) St Φ(d˜1 ) − e−r(T −t) KΦ(d˜2 )
n
o
−r(T −t)
˜
˜
= e
Ft Φ(d1 ) − KΦ(d2 )
We find
dZ̃t = (µ + δ − r)Z̃t dt + σ Z̃t dWt
with
log(St /K) + [(r − δ) ± 12 σ 2 ](T − t)
˜
√
d1,2 =
σ T −t
log(Ft /K) ± 12 σ 2 (T − t)
√
.
=
σ T −t
= σ Z̃t dW̃t ,
µ+δ−r
, which is a
σ
Brownian motion under the measure Q defined by dQ/dP =
where W̃t = Wt + γt, with γ =
exp(−γWT − 12 γ 2 T ). Hence
dSt = (r − δ)St dt + σSt dW̃t ,
1 2
St = S0 exp [r − δ − 2 σ ]t + σ W̃t
2
Discrete dividends
Foreign exchange
When dividends are paid at discrete time points T1 , . . . , Tn ,
Let Ct denote the exchange rate, in US dollar per pound
then the stock goes ex-dividend, which means its price falls
sterling. We’ll assume a geometric Brownian motion for
instantaneously by the amount of the dividend. When these
Ct :
dividends are immediately reinvested, the value S̃t of that
dCt = µCt dt + σCt dWt .
strategy of course does not display these discontinuities; i.e.,
Next, consider a US cash bond Bt = ert and a UK cash bond
we simply may assume
Dt = eut ; i.e., the interest rates r and u may be different.
dS̃t = µS̃t dt + σ S̃t dWt .
From the perspective of the US investor, there are two as-
(Note: µ here should be compared with µ + δ in the contin-
sets: the domestic cash bond with price Bt , and the foreign
uous dividend model).
cash bond with price St = Ct Dt . Note that the latter is a
risky asset; its SDE is
When the dividend payments are δSt , we obtain
dSt = Ct dDt + Dt dCt
St = (1 − δ)n[t] S̃t ,
= (µ + u)St dt + σSt dWt
where n[t] is the number of dividend payments made by
= rSt dt + σSt dW̃t ,
time t.
where W̃t = Wt + γt, γ =
This implies
Q, the risk-neutral measure.
Ft = (1 − δ)n[T ]−n[t] er(T −t) St ,
and the value of a call option remains the same in terms of
Ft .
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µ+u−r
. This again defines
σ
Notice that under this measure,
Again, the value of a call option on the exchange rate struck
at K is the same as before, when expressed in terms of Ft .
EQ [CT |Ft ] = e−uT EQ [ST |Ft ]
= e−uT er(T −t) St
Change of numeraire
= e(r−u)(T −t) Ct ,
The entire analysis could be repeated from the perspective
of the UK investor, who has the choice between a sterling
which yields the uncovered interest rate parity:
cash bond Dt and the sterling value of a dollar cash bond,
EQ [CT |Ft ]
(r − u)(T − t) = log
,
Ct
S̃t = Ct−1 Bt . The discounted price then is Dt−1 Ct−1 Bt =
where the right-hand side is the conditionally expected con-
Zt−1 , where Zt = Bt−1 St . The martingale measure is not
tinuous depreciation.
the same as before: a measure which makes Zt a martingale
The forward exchange rate (for delivery at time T ) Ft should
hedge ratios are the same, regardless of the choice of the
solve
measure.
e−r(T −t) EQ [(CT − Ft )|Ft ] = 0,
1 2
and since CT = Ct exp [r − u − 2 σ ](T − t) + σ[W̃T − W̃t ] ,
Similarly, in the standard Black-Scholes model we may also
does not make Zt−1 a martingale. However, the prices and
work with a measure Q∗ which makes St−1 Bt a martingale.
this will yield
(r−u)(T −t)
Ft = e
The important thing is to make relative prices martingales –
Ct ,
the choice of the numeraire is not important.
which gives the covered interest rate parity:
(r − u)(T − t) = log
Ft
.
Ct
4
Quantos
The equivalent martingale measure now should turn both
dollar assets Bt−1 Ct St and Bt−1 Ct Dt into martingale, which
Quantos are derivatives which have a payoff in another cur-
now involves a vector γ = (γ 1 , γ 2 )0 , with
rency than the underlying asset, using a fixed, prespecified
exchange rate C̄ (e.g., one dollar per pound sterling). For
dQ
= exp −γ 0 WT − 12 γ 0 γT ,
dP
example, when St is a sterling stock price and K is a cor-
where WT = (W1 (T ), W2 (T ))0 . The actual definition of
rresponding strike price, then a quanto call has the dollar
γ follows from deriving the SDE for the discounted dollar
payoff
assets and setting the drifts to zero.
+
C̄(ST − K) .
Note that C̄St is not a tradable dollar asset; hence its dis-
In order to price such a derivative, one has to set up a joint
counted value need not be a martingale. In fact its drift is
process for (St , Ct ), which is a vector diffusion with two
(u − ρσ 1 σ 2 )C̄St dt, which in general does not equal rC̄St dt.
independent Brownian motions (W1 (t), W2 (t)):
It can be derived that the dollar forward price on C̄St will
dSt
= µdt + σ 1 dW1 (t),
St
dCt
= νdt + σ 2 dW2∗ (t)
Ct
p
= νdt + ρσ 2 dW1 (t) + 1 − ρ2 σ 2 dW2 (t),
p
∗
where W2 (t) = ρW1 (t) + 1 − ρ2 W2 (t) is a standard
be
FQt = C̄ exp(−ρσ 1 σ 2 [T − t])Ft ,
where Ft is the sterling forward price. The quanto call value
then is the usual, with Ft replaced by FQt , K replaced by
C̄K, and σ replaced by σ 1 .
Brownian motion, which has correlation ρ with W1 (t), i.e.,
EP [W1 (t)W2∗ (t)] = ρt.
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where rt is the risk-free interest rate. In models with more
Market price of risk
than one driving Brownian motion, there is a vector of mar-
The fundamental theorem of asset pricing states that the ab-
ket prices of risk, one for each source of risk (i.e., each
sence of arbitrage opportunities is equivalent to the exis-
Brownian motion).
tence of a measure Q under which all asset prices relative
to some numeraire are martingales. The equivalent martin-
Exercises
gale measure is unique if markets are complete, i.e., if any
1. Consider a bivariate geometric Brownian motion of the
claim is replicable.
form
Note that only tradable asset need to be martingales under
dS1 (t) = S1 (t) {µ1 dt + σ 11 dW1 (t) + σ 12 dW2 (t)} ,
Q; the previous examples all had a payoff in terms of a non-
dS2 (t) = S2 (t) {µ2 dt + σ 21 dW1 (t) + σ 22 dW2 (t)} ,
tradable asset, which was an explicit function of another
where W1 (t) and W2 (t) are independent Brownian mo-
tradable.
tions, and µi and σ ij are constants, i, j = 1, 2. Find
The existence of Q implies a common market price of risk
the vector γ of market prices of risks, and check that
γ t , which determines the change of measure via the CMG
e−rt S1 (t) and e−rt S2 (t) are both martingales under the
theorem. For example, if two tradable asset prices S1 (t) and
measure Q defined by this γ.
S2 (t) are driven by the same Brownian motion Wt :
2. Suppose that S1 and S2 are geometric Brownian motions,
dS1 (t) = µ1t S1 (t)dt + σ 1t S1 (t)dWt ,
driven by the same Brownian motion Wt . Show that if
dS2 (t) = µ2t S2 (t)dt + σ 2t S2 (t)dWt ,
then
γt =
both are tradable asset prices, but with a different market
price of risk, then an arbitrage opportunity exists.
µ1t − rt
µ − rt
= 2t
,
σ 1t
σ 2t
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