Stochastic Calculus Flashcards
Three general heading of derivatives
- Futures and Forwards
- Options
- Swaps
3 main differences between futures and forwards
- Organized exchange vs OTC
- No default risk vs default risk
- Mark to market each day vs settled at maturity
List 5 groups of underlying assets
- Stocks
- Currencies
- Interest rates (not asset, take a position on direction via Treasuries)
- Indexes
- Commodities
4 uses of arbitrage-free prices
- New financial product, no market data available
- Risk management, no data for hypothetical ‘worst case scenarios’
- Marking to market, may need current value of illiquid asset
- Mispricing, if market prices differ we want to take advantage of it
Arbitrage Theorem
w/ 3 assets, 2 states, 1 step
-
Positive ψ1 and ψ2 found satisfying
vec(1, St, Ct) = mx((1+r, S1(t+1), C1(t+1)), (1+r, S2(t+1), C2(t+1))) * vec(ψ1, ψ2) - Solution to this system is positive constants ψ1, ψ2
ψi = premium for insurance paying $1 in state i = State (i) price
Consequence 1 of no arbitrage opportunities: What measure do we get?
Existence of risk adjusted probabilities
Q1=(1+r)ψ1
By the matrix equation we have ΣQi = 1
Consequence 2 of no arbitrage opportunities: What asset value behavior do we force?
Discounted expected value of any asset is a martingale
St=[Q1xS1(t+1)+Q2xS2(t+1)]/(1+r)
Consequence 3 of no arbitrage opportunities: What returns do we expect?
The expected risk adjusted return of any asset is risk free rate
(Rearrange consequence 2 for 1+r)
Consequence 4 of no arbitrage opportunities: What bounds does the risk free rate have?
Total risk free rate return (1+r) is bounded above and below by the gross returns in the most favorable and least favorable future states
Ψ1R1+Ψ2R2=1 => R1<1+r<R2
If r was greater/less than all possible returns, we would buy/short bonds to short/buy as much S as possible
Discounting expected value of assets at r in real world measure results in what?
A submartingale
St<[P1xS1(t+1)+P2xS2(t+1)]/(1+r)
Risky assets in the real world need some risk premium
Our First SDE: RW dynamics of stock prices
dSt = μtStdt + σtStdWt
μ creates predicted movement, σ is the unpredictable shock over time dt
Can no arbitrage opportunities condition exist in a world with stochastic interest rate?
Neftci 17
Yes, but it is not analytically tractable. Transformation from RN to forward measure reduces complexity
Describe the setting for normalization under the Forward Measure and how we get to forward risk adjusted probabilities
In a 2 step model with 4 possible states
Existence of ψij established
Transforming to forward measure changes risk adjusted probabilities to
πij = ψij/Bt1
What are the steps to use forward measure for pricing?
Because ψij>0, πij>0
We multiple expected asset price by corresponding entry of Bt1
B is thought of as the new discount rate
Ct1 = Bt1E[πijCij]
What are 3 important consequences of the forward measure and how does it work better than RN?
Forward adjusted probs satisfy Σπij = 1
Under classic RN measure, forward interest rate is biased, but unbiased in π
You do not need to model a bivariate process in π like you would in RN
What is the approximating sun for Riemann Integral
∫f(s)ds over [0,T] and the common explanation of it
Partition [0,T] into n intervals 0=t0<t1…<tn=T
The approximating sum is Σf((ti+ti+1)/2)(ti-ti+1)
Riemann suggests integral converges to sum of n disjoint rectangles
Riemann Integral in stochastic world
What happens/why?
Riemann integration fails! In stochastic environments, the functions vary too much because they involve the Wiener process
Classical Total Differential
Also breaks down in stochastic world
Let f(St, t) be a function of 2 variables. The total differential is defined as
df = (∂f/∂St)dSt + (∂f/∂t)dt
Taylor Expansion of a function around a point
Also breaks down in stochastic world
f(x) = Σ(1/n!)fn(a)(x-a)n
Taylor Expansion for functions of two variables
dV = (∂V/∂S)dS + (∂V/∂t)dt + 1/2(∂2V/∂S2)dS2 + 1/2(∂2V/∂t2)dt2 + (∂2V/∂S∂t)dSdt + …
Name and describe two popular approaches for pricing derivatives
- Equivalent Martingale Measure
Notion of no arbitrage gives a measure where assets are martingagles, which are easy to take expectations of - Solving the Pricing PDE
Construct a risk free portfolio and get a PDE implied by no arbitrage opportunities
Give the formula for the Stochastic Total Differential
Let F(St, rt, t) be a fn of 2 s.r.v. Then the total differential is
dF = FsdSt + Frdrt + Ftdt + 1/2 Fss(dSt)2 + 1/2 Frr(drt)2 + FsrdStdrt
List 4 Statistical facts about distribution tails under/vs Normal
Neftci 5
- A dist. with heavier tals than normal means a higher probability of extreme events
- A normal dist has most ovservations occurring around the center
- Heavy tailed dists have a more sudden passage from ordinary to extreme events
- Middle region of a heavy tail dist has relatively less weight than the normal dist
What is a Markov Process (in words and math)
A discrete process X1, …Xt,…
with pdf F(x1, …xn) is Markov if
P(Xt+s<=xt+s|x1, …xt) = P(Xt+s<=xt+s|xt)
Are Jointly Markov processes individually Markov?
In general, no (but possible). Modeling a joint process univariately will usually cease to be Markov. The reverse is also true, we can make any univariate non-Markov process into a Markov one by increasing the dimensionality
Describe the interest rate model that is univariately Markov and the one that is jointly Markov
- Classic approach: Model yield curve using a single rate process rt. This is univariate and Markov, giving an SDE representation of drt = μ(rt,t)dt + σ(rt,t)dWt
- HJM approach: Model k forward rates which are assumed jointly Markov with SDE
dF(t,T) = σ(t,T,B(t,T))[∂σ/∂T]dt + [∂σ/∂T]dWt
Name three concepts of convergence for pricing derivatives
- Mean square convergence
- Almost surely convergence
- Weak convergence
Define and give basic examples of Martingale and Submartingale
Neftci 6
Martingale: trajectories display no trends or periodicities
Ex: PV of asset prices discounted by r in RN measure
Submartingale: the process increases on average
Ex: PV of asset prices discounted by r in RW measure
Define a Martingale and give the 3 conditions for it
A process St is a martingale wrt info sets It and measure P if for all t>0:
* St is adapted to the information set
* Unconditional forecasts are finite E|St| < infinity
* Best future forecast of unobserved values is the last observation Et[ST] = St for all T>t
Examples of Martingales
Let Wt be a Wiener Process
- Wt
- Wt2 - t
- Exp[θWt - 1/2 θ2t]
2 approaches for finding prob measure where discounted asset prices are martingales
- Transform the submartingales as follows:
From Bt to e-rtBt and St to e-rtSt - (Preferred) Transform the probability measure
If EtP[e-ruSt+u] > St we can find an equivalent probability measure where they are equal
Donsker Theorem: describe the limit distribution of a sum of IID variables
Let {Xn} be iid. Let Sn = X1+…Xn. If E(Xi) = μ and Var(Xi) = σ2
Then (Sn-nμ)/sqrt(n) ~ Normal(0,1) in the limit
Martingale Representation Theorem
Let X be a real valued rv, consider Mt = EP[X|Ft]. If Mt is a martingale, then we can find an adapted process γu such that Mt = M0 + ∫(0,t) γudWu
Doob Meyer Decomposition theorem
If Xt is a right continuous submartingale (increasing) and for all t, E[Xt] < infinity, then Xt admits the decomposition
Xt = Mt + At
Where Mt is a martingale and At is an increasing stochastic process
The First Stochastic Integral (Discrete)
Uses Doob Meyer to represent a new martingale
Let Zt be any martingale wrt It and P. Let Ht be any r.v. adapted to It and P. Then
Mtk = Mt0 + Σ(1,k)Hti-1[Zti-Zti-1]
will also be a martingale
First Stochastic Integral (Continuous Doob-Meyer)
Let dZt be an infinitesimal stochastic increment with mean 0 wrt It and P. Let Ht be any r.v. adapted to It. Then
Mt = M0 + ∫(0,t)HudZu
will be a martingale
Two approximations of second order Taylor polynomial in stochastic environment
Neftci 7
Since x is a r.v, it cannot have 0 variance. Thus E(Δx)2>0
We have 2 approximations for the Taylor expansion with stochastic variables
f(x0+Δx)-f(x0) = fxΔx+1/2 fxxE(Δx)2 or
f(x0+Δx)-f(x0) = fxΔx+1/2 fxxE(x*)
Where x* is the mean square limit of (Δx)2
What are rare events
Neftci 8
Rare events have something to do with price discontinuity
They are supposed to occur infrequently. As the time step h goes to 0, the probability of occurrence for rare events should also go to 0.
But even when the probability goes to 0, the size may not shrink. By definition they have a large size
Problems with characterizing rare events with the Wiener Process
As h goes to 0, the tails of the normal dist carry less weight, and the size of price changes given by the WP shrink. This ther WP is not suitable for situations where prices can move large distances over an extremely short interval. We need a disturbance term capable of giving large events in short intervals without shrinking the size.
What is a model for rare events?
Jump Diffusion
dSt = a(St, t)dt + σ1(St, t)dWt + σ2(St, t)dJt
What is the finite difference approximation of SDE representation?
Neftci 9
Over a short enough interval, drift (a) and volatility (σ) will be near constant, so we integrate the SDE to get a discrete approximation
St+h - St = a(St, t)h + σ(St, t)ΔWt
Ito Integral and Properties
The proper integration of the SDE involves the Ito Integral
∫(t,t+h) σ(Su, u)dWu
Wu is a martingale and has unpredictable increments, the sum of which is itself an unpredictable increment. Thus the Ito integral is a martingale
Et[∫(t,t+h) σ(Su, u)dWu] = σ(St, t)t[∫(t,t+h) dWu] = 0
Quadratic Variation fo the Ito Integral
The quadratic variation si <I,I>t = ∫(0,t) f(Ws, s)2ds
Where the Ito integral is ∫(0,t) f(Ws, s)dWs
We can write it as the sum of increment terms as follows
It = ∫(0,t) f(Ws, s)dWs = limn>infΣf(Wti, ti)(Wti+1-Wti)
Quadratic variation of an Ito process
The quadratic variation of an Ito process is <X,X>t = ∫(0,t) σ(Xs, s)2ds
Existence of Ito integral of a general random function f(Su, u)
The General Ito Integral ∫(0,t) f(Su, u)dSu exists if the function f is continuous and non-anticipating.
The general Ito integral is a random process that has various expectations, but the martingale property ensures the first moment of the ito integral is 0
Distribution of Ito integral
I(T) = ∫(0,T) f(t)dWt ~ Normal(Mean = 0, Variance = ∫(0,T) f(t)2dt)
Connection btw Ito and Statonovich integrals
I(t) = ∫(0,t) WsdWs
S(t) = ∫(0,t)Ws o dWs = limn>infΣ(Wti+Wti+1)/2 (Wti+1-Wti)
S(t) - I(t) = t/2
Ito Integral is a martingale, Stratonovich integral is not
State Ito Lemma
Nefci 10
Let F(St, t) depend on a r.v. S and t. Let dSt = atdt + σtdWt
Then the SDE for F(St, t) is
dFt = (∂F/∂St)dSt + (∂F/∂t)dt +1/2 (∂2F/∂S2)σt2dt
or
dFt = [∂F/∂St at + ∂F/∂t + 1/2 (∂2F/∂S2)σt2]dt + (∂F/∂St)σtdWt
Equality holds in a mean square sense
Integral form of Ito Lemma
∫(0,t) FsdSu = F(St, t) - F(S0, 0) - ∫(0,t) Fu + 1/2 Fssσu2du
Give the formula for the general Ito Isometry of two Stochastic Integrals
E[∫(0,T) f(t)dWt x ∫(0,T) g(t)dWt]
=∫(0,T) f(t)g(t)dt
Ito Isometry 2
A r.v. is involved
E[∫(0,T) f(Xt, t)dWt x ∫(0,T) g(Xt, t)dWt]
=∫(0,T) E[f(Xt, t)g(Xt, t)]dt
Solution to GBM SDE
Neftci 11
The strong solution to the SDE dSt = μStdt + σStdWt
is St = S0e(μ-1/2 σ2)t + σWt
The PDE for Derivative Prices
This PDE only holds under BS assumptions
Neftci 12
(∂V/∂t) + rS(∂V/∂S) + 1/2 σ2S2(∂2V/∂S2) = rV
With Greeks:
Theta + rS(Delta) +1/2 σ2S2Gamma = rV
Simple rules in Ito Calculus
dt x dt = 0
dt x dWt = 0
dS x dt = 0
dWt x dWt = dt
Major difference between BS PDE and exotic PDE
Neftci 13
- Teh expiration value of the option may depend on some event happening over the option’s life, which makes boundary conditions much more complicated than standard BS
- Exotic derivative may have random expiration dates
- Exotic derivatives can be written on more than one asset
Seven steps to derive BS PDE
- Assume S follows a GBM and the typical BS assumptions
- Build a hedged portfolio Π (short derivative V, long delta S or vice versa)
- Use Ito’s Lemma to get an SDE for change in value dΠ over small time dt
- See that this special portfolio is free of the random term dZ
- Conclude that this portfolio is riskless and must earn r over this time interval
- Replace the dΠ and Π in the equation dΠ/Π = rdt
- The resulting PDE applies to any derivative written on the underlying asset S
Ito Lemma when S follows GBM
Let G(S,t) be a function of time and stock price S where
dS/S = μdt + σdz
The process followed by G is
dG = [(∂G/∂S)μS + ∂G/∂t + 1/2 (∂2G/∂S2)(σS)2]dt + ∂G/∂S σSdz
BS PDE for Dividend paying stock, currency options, commodities options
(∂V/∂t) + (r-x)S(∂V/∂S) + 1/2 σ2S2(∂2V/∂S2) = rV
Where x is thought as some form of ‘dividend yield’. For stocks, it is the Dividend yield D. Currency options use foreign interest rate rf. Commodities use the fraction of value that goes into the carrying cost -q
Setting of Girsanov Theorem: Novikov Condition
Neftci 14
Given information sets {It} over a period [0,T] we define a random process
ξt = exp(∫(0,t) XudWu - 1/2∫(0,t) Xu2du
where Xt is It measurable and Wt is a WP on P
We impose the Nosikov Condition on Xt:
E(exp(∫(0,t) Xu2du)) < inf
Statement of Girsanov Theorem
If Nosikov Condition is satisfied, then the process ξt will be a square integrable martingale s.t Wt~ = Wt - ∫(0,t)Xudu is a WP wrt It and measure PT~ where
PT~(A) = EP[1AξT]
and the two WPs are related by dWt~ = dWt - Xtdt
Which measure (RN or RW) should we use for pricing options vs forecasting outcomes?
Pricing derivatives have gone on without needing to model true probabilities under P or the risk premium μ. Derivatives can be priced in the risk neutral measure P~.
For forecasting, a decision maker should clearly use the real world probabiliities P and apply expecations in this measure rather than the RN one.
Pricing: Martingale approach vs PDE approach
Neftci 15
PDE: Use the steps presented to reach the BS PDE and solve it (possibly numerically)
Martingale: Find the synthetic probability measure where discounted prices are martingales and take expectations
Connection btw Martingale and PDE approaches
In trying to find the risk neutral SDE for e-rtF(St, t), we use Ito Lemma and Girsanov to see it is
d[e-rtF(St, t)] = e-rt[-rF + rFsSt + Ft + 1/2 Fss(σSt)2]dt + Fse-rtσtdWt~. Since this is a RN measure, the drift term is 0, giving
rF + rFsSt + Ft + 1/2 Fss(σSt)2 = 0, the BS PDE
Jensen’s Inequality
QFIQ 113 17
If f is a convex function and x is an r.v then
E(f(x)) >= f(E(x))
Why does RN valuation work?
QFIQ 113 17
- If you hedge correctly in BS, then all risk is eliminated. With no risk, you don’t get any risk premium, so assets will return risk free rates
- Using the GBM SDE dS/S = μdt + σdz, we see μ cancels out in deriving the BS eqn
- A price is non arbitrageable in the RW iff it is non arbitrageable in the RN world. But the RN price is always nonarbitrageable bc everything is a martingale, so the RW has no arbitrage.
What is meant by Complete and Incomplete market?
- In a complete market, derivatives can be replicated by more basic instruments like cash, bonds, and S. In incomplete markets, you cannot achieve this
- A complete markes is one for which there exists the same number of linearly independent securities as there are future states of the world
- In a complete market you can hedge a derivative with the underlying
- In a complete market, the martingale measure is unique, but this is not true for incomplete markets
Can options be priced in RW measure?
2 reasons RN pricing doesn’t work in practice: Markets are incomplete and dynamic hedging is impossible
RW Valuation: You could value an option as the real expectation of the PV of option payoff +/- some multiple of std dev. However this has 2 main problems
1. You need to know real probabilities and real drift which is very hard
2. You need to decide on a utility function or a measure of risk aversion
What is a Weiner Process
Chin
A stochastic process is a SWP if
* W0 = 0 and has continuous sample paths.
* Increments are independent, stationary, normally distributed, and variance equals elapsed time
Basic properties of SWP
Chin
- Markov property: Distribution of Wt given info up to s < t, depends only on Ws
- Strong Markov Property: For any s > 0, the r.v. Wt+s - Wt is independent of the filtration up to time t
- Martingale
Covariance of 2 SWP
Chin
Cov(Wt, Ws) = E(WtWs) = min(s,t)
What is the Covariance matrix for a set of SWP
Chin
Cov(Wt1, Wt2, Wtn) =
matrix((Cov(Wt1, Wt1), …, Cov(Wt1, Wtn)), …, (Cov(Wt1, Wtn), …, Cov(Wtn, Wtn)) =
Matrix((t1, …, t1), …, (t1, …, tn))
What is the quadratic variation of a SWP
Chin
<W,W>t = limn>infΣ(Wti+1 - Wti)2 = t
Feynman Kac Theorem
Chin
Let Xt be an Ito process. If V(Xt, t) satisfies the PDE:
∂V/∂t + 1/2 σ2∂2V/∂X2 + μ∂V/∂X - rV = 0 with boundary condition V(XT, T) = Ψ(T) then the solution to the PDE is
V(Xt, t) = E[e(-∫(t,T) r(u)du x V(XT, T)|Ft]
Ito Product Rule
Let Xt, Yt be Ito processes. The SDE for the product XtYt is given by
d(XtYt) = XtdYt = YtdXt + dXtdYt
Applying Ito Lemma to a function of a WP F(Wt, t)
d(F(Wt, t) = F’(Wt)dW(t) + [F’(t) + 1/2 ∂2/∂W2 F(W(t))]dt
SDE for Logarithm of GBM process
Let dS/S = μdt + σdW(t) and F = ln(S). Then
d(ln(S)) = (μ - σ2/2)dt + σdW(t)
SDE for PV of a GBM process
Let dS/S = μdt + σdW(t) and F = e-r(t)S. Then
d(e-r(t)S)/e-r(t)S = (μ-r(t))dt + σdW(t)
SDE for product of two GBM processes
Let dP/P = μPdt + σPdWP and dM/M = μMdt + σMdWM. Assume Corr(WP, WM) = ρ. Then
d(PM)/PM = (μP + μM + σPσMρ)dt + σPdWP + σMdWM
SDE for inverse of a GBM
Let dS/S = μdt + σdW(t) and F = 1/S. Then
d(1/S)/(1/S) = dF/F = -[(μ-σ2)dt + σdW(t)]
SDE for power of a GBM
Let dS/S = μdt + σdW(t) and F = Sn. Then
dF/F = n x [(μ + 1/2 (n-1)σ2)dt + σdW(t)]
Can use this to arrive at the correct SDE for inverse GBM
Useful tricks for stochastic calculus
- d(∫(0,t) f(s)dW(s)) = f(t)dW(t)
- d(∫(0,t) f(s)ds) = f(t)dt
- E[∫(0,t) f(Wu, u)dWu∫(0,t) g(Wu, u)dWu] = ∫(0,t) E[f(Wu, u)g(Wu, u)]du
- E[∫(0,s) f(Wu, u)dWu∫(s,t) f(Wu, u)dWu] = 0
