Lf 2020 options

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Summary and contents.
¨ This is not a formal option class. à if any question, PLEASE interrupt.
¨ This is by no means exhaustive. à read the textbooks out there.
¨ This is meant to be an exposure to the concepts and some of the issues encountered when
dealing with options.

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Books and references.
¨ “Paul Wilmott on Quantitative Finance”, Paul Wilmott.
¨ “Options, Futures, and Other Derivatives”, John C. Hull.
¨ “Dynamic Hedging: Managing Vanilla and Exotic Options”, Nassim N. Taleb.
¨ “When Genius failed: The Rise and Fall of LTCM”, Roger Lowenstein.
¨ “Market Wizards”, Jack D. Schwager.
¨ “Reminiscence of a Stock Operator”, Edwin Lefevre.
¨ “The Education of a Speculator”, Victor Niederhoffer.
¨ Options: Perception and Deception. Position Disection, Risk Analysis and Defensive Trading
Strategies Hardcover – June 1, 1996 by Charles Cottle
¨ Fractals, Chaos, Power Laws by Manfred Schroeder
¨ Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit (Springer Finance) 2nd
Edition by Damiano Brigo

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The top 5 words.
¨ Convexity.
¨ Arbitrage.
¨ Hedging.
¨ Volatility.
¨ Correlation.
¨ Also very popular in options world:
– Skew, Smile.
– Delta, Gamma, Vega, Risk.
– Risk Neutral

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Summary
¨ Options definitions, put-call parity.
¨ Volatility and option pricing.
¨ Volatility and option trading.
¨ Convexity.
¨ Option value and Greeks.
¨ Arbitrage, Risk-neutrality and Convexity.
¨ A few products.
¨ Skew and Smile

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What are options?
¨ Options are everywhere : lottery tickets, year-end bonuses, medical plans, crop insurance,
test at the end of this course, NBA draft,…
¨ Options have been around for a while: the snake in the Garden of Eden was the first option
seller.
¨ You need three things for an option: IF………SHOULD……….THEN….

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How to define an interest-rate option?
¨ Time to expiry: T
– European
– American
– Bermuda
¨ Underlying: F
– Single index: LIBOR, CMS, CMT, FedFunds,….
– Spread: (CMS10-CMS2),…
– Basket:
– Time average
¨ Payoff

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Payoffs.
¨ Any function of the underlying.
¨ Vanilla payoffs:
– caps, floors
– any linear combination of the above: straddle, strangle, digitals, vertical spread, horizontal
spread, butterfly, condor, Christmas tree, squash,…….
¨ Exotic payoffs:
– Path dependent: Asian options, cliquet, ratchet, one-touch, two-touch, knock-out,…..

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The confirm!
¨ Legally binding, defines the option contract.
– Expiry time.
– Notification period.
– Notification procedure.
– Option settlement procedure (physical, cash,..).
– Delivery procedure.
– Fallback procedure.
– Underlying convention and resets.
– Payoff convention, daycount, rolls, holidays..
– THE CSA ! (Credit Support Annex), collateral management.

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The exchange-traded options.
¨ No confirm.
¨ Daily settlement.
¨ NO counterparty exposure.
¨ Uniform rules.

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The simplest option: European Call!
¨ Time to expiry: T
¨ Underlying: F
¨ Strike: K
¨ Payoff: Max(F-K,0)
$
F
K

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The simplest option: European call?
¨ How to price this option?
¨ Time to expiry T is known.
¨ Strike K and payoff are known.
¨ What about the underlying F, can I price a forward?
¨ Discount curve: D(T), D(T + 3 months), …
¨ à I can price bonds, swaps, FRAs, zero-coupons, ….
¨ à I know F.

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Something cute: Put-Call parity.
¨ Call Payoff: Max(F-K,0)
¨ Put Payoff: Max(K-F,0)
F
K
$

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Put-Call parity.
¨ Let’s buy a call, sell a put.
¨ Expected payoff: Max(F-K,0) – Max(K-F,0) = F-K
F
K
$

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Put-Call parity.
¨ Being long a call, short a put is equivalent to paying K and receiving Floating on a 1-period
swap (that is something we should all know how to price by now).
¨ Even though I still cannot price a Call or a Put, I can price the portfolio: (Call-Put).
¨ That is cute indeed, but I still don’t know how to price a Call.

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Pricing a Call.
¨ T is known, K is known, F is known, so… Call = Max(F-K,0) discounted back to today.

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Pricing a Call.
¨ T is known, K is known, F is known, so… Call = Max(F-K,0) discounted back to today.
¨ What’s wrong with that picture….
VOLATILITY

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Pricing a Call.
¨ At-the-money Call = Max(F-K,0) = 0.
F=K
F<K
Call=0
F>K
Call=(F-K)

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Pricing a call.
¨ The more volatile the rate, the greater the probability of a large move upwards in rates, the
more valuable the call.
¨ At zero volatility (no moves in rates), the call is equal to the terminal payoff Max(F-K,0).
¨ More exactly, from the discount curve we know the expected value of F, sometimes noted
<F> or E(F). If we assume rates F to be volatile, this is the average of F.

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The rate… it mooooves….
Typical
Option Trader

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Did you say volatility?
2.0
2.5
3.0
3.5
07/23/03
07/28/03
08/02/03
08/07/03
08/12/03
08/17/03
08/22/03
08/27/03
date O1y_S1y
7/28/2003 2.388
7/29/2003 2.539
7/30/2003 2.507
7/31/2003 2.811
8/1/2003 2.973
8/4/2003 2.818
8/5/2003 2.918
8/6/2003 2.740
8/7/2003 2.642
8/8/2003 2.615
8/11/2003 2.769
8/12/2003 2.652
8/13/2003 2.891
8/14/2003 2.989
8/15/2003 2.899
8/18/2003 2.877
8/19/2003 2.762
8/20/2003 2.840
8/21/2003 3.099
8/22/2003 3.033

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Sometimes rates move a little,..
1y1y (%)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
2/1/1998
3/1/1998
4/1/1998
5/1/1998
6/1/1998
7/1/1998
8/1/1998

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Sometimes a lot…
1y1y (%)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
12/1/1997
6/1/199812/1/1998
6/1/199912/1/1999
6/1/200012/1/2000
6/1/200112/1/2001
6/1/200212/1/2002
6/1/2003

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Different forwards move differently?
Yields (%)
2
4
6
8
10
12/1/19976/1/199812/1/19986/1/199912/1/19996/1/200012/1/20006/1/200112/1/20016/1/200212/1/20026/1/2003
2y1y
5y1y
10y1y

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Does history repeats itself?
¨ Does the future volatility depends on….
– Historical volatility.
– Particular forwards.
– Level of rates.
– Level of volatility.
– Strike.
– Time to expiry T.
– Are you a good trader?

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Does history repeats itself? (again?)
¨ Does the future volatility depends on….
– Historical volatility. Implied vs. Realised.
– Particular forwards. Term structure models.
– Level of rates. Correlation, Skew, Mean Reversion.
Normal vs. Lognormal.
– Level of volatility. Heteroskedasticity, Smile.
– Strike. Local volatility models.
– Time to expiry T. Short-dated vs. long-dated.
– Are you a good trader? Hedging strategies, sampling,..

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What volatility are we talking about anyways?
¨ Normal Volatility (basis points per day).
– You care about your margin account (absolute return).
¨ Lognormal Volatility (% per year).
– You care about your annualized returns (relative return).
Also, are you looking the volatility of the Price or the Yield (example of Eurodollar futures
and Bond)?

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Normal Volatility.
date O1y_S1y Absolute
7/28/2003 2.388 COD(bps) COD(bps)
7/29/2003 2.539 15.07 15.07
7/30/2003 2.507 -3.2 3.2
7/31/2003 2.811 30.41 30.41
8/1/2003 2.973 16.2 16.2
8/4/2003 2.818 -15.5 15.5
8/5/2003 2.918 10.05 10.05
8/6/2003 2.740 -17.86 17.86
8/7/2003 2.642 -9.73 9.73
8/8/2003 2.615 -2.73 2.73
8/11/2003 2.769 15.36 15.36
8/12/2003 2.652 -11.68 11.68
8/13/2003 2.891 23.89 23.89
8/14/2003 2.989 9.78 9.78
8/15/2003 2.899 -8.93 8.93
8/18/2003 2.877 -2.25 2.25
8/19/2003 2.762 -11.5 11.5
8/20/2003 2.840 7.8 7.8
8/21/2003 3.099 25.94 25.94
8/22/2003 3.033 -6.58 6.58
12.87 (bps/day)

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Normal Volatility.
¨ 13 ~ 13 bps/day.
¨ 13 * SQRT(7) ~ 34 bps/week.
¨ 13 * SQRT(30) ~ 71 bps/month.
¨ 13 * SQRT(90) ~ 123 bps/quarter.
¨ 13 * SQRT(360) ~ 247 bps/year.

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Normal Volatility.
¨ What about weekends and holiday?
– This is DRW, remember…not a charity. Also weekends have had lately political risk
¨ What about the Square Root? Diffusion vs. Propagation.
Normal Volatility (bps)
0
50
100
150
200
250
300
0 50 100 150 200 250 300 350 400
days

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Lognormal Volatility.
date O1y_S1y
7/28/2003 2.388 Ln(Fi/Fi-1)
7/29/2003 2.539 0.06
7/30/2003 2.507 -0.01 1) Calculate Standard Deviation.
7/31/2003 2.811 0.11 0.05
8/1/2003 2.973 0.06
8/4/2003 2.818 -0.05 2) Divide by SQRT(1/365).
8/5/2003 2.918 0.04 103 (% / year)
8/6/2003 2.740 -0.06
8/7/2003 2.642 -0.04
8/8/2003 2.615 -0.01
8/11/2003 2.769 0.06
8/12/2003 2.652 -0.04
8/13/2003 2.891 0.09
8/14/2003 2.989 0.03
8/15/2003 2.899 -0.03
8/18/2003 2.877 -0.01
8/19/2003 2.762 -0.04
8/20/2003 2.840 0.03
8/21/2003 3.099 0.09
8/22/2003 3.033 -0.02

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From Lognormal to Normal.
¨ Lognormal Volatility ~ 103 % / year.
¨ Rate Level ~ 2.4 %.
¨ 103 * 2.4 / SQRT(365) ~ 12.93 bps/day.
¨ OK, that was really 12.87 not 12.93.
¨ We will learn how to exactly equate normal volatility to lognormal volatility (i.e. so that the
price of an at-the-money option is the same in both models)

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Still trying to price the simplest option!
¨ Normal model (Louis Bachelier, 1900 Ph.D. thesis).
¨ F is normally distributed.
¨ Everyday, F goes up or down by 10bps.
F
2.2 2.72.1 2.42.3 2.5 2.82.6 3.0 3.12.9

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Normal model.
¨ Diffusion.
¨ Heat equation.
¨ Parabolic equation. (Remember the square root?)
¨ Random walk.
¨ Binomial distribution.
¨ Brownian motion. (Robert Brown, 1827).
¨ Monte-Carlo simulation.
¨ Random number generator.

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Modeling the market, one path at a time.
2.0
2.5
3.0
3.5
07/23/03
07/28/03
08/02/03
08/07/03
08/12/03
08/17/03
08/22/03
08/27/03
date O1y_S1y
7/28/2003 2.400
7/29/2003 2.300
7/30/2003 2.400
7/31/2003 2.500
8/1/2003 2.400
8/4/2003 2.300
8/5/2003 2.400
8/6/2003 2.300
8/7/2003 2.200
8/8/2003 2.300
8/11/2003 2.400
8/12/2003 2.500
8/13/2003 2.600
8/14/2003 2.700
8/15/2003 2.600
8/18/2003 2.500
8/19/2003 2.600
8/20/2003 2.700
8/21/2003 2.800
8/22/2003 2.900

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Simulating many paths, after 90 days.
¨ T=1year, K=2.4%, F=2.4%, Payoff=Max(F-K,0).
¨ Normal Volatility s=10 bps/day.
Forward F (%)
1
2
3
4
5
0
10
20
30
40
50
60
70
80
90
days

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Forward F (%)
-3
-1
1
3
5
7
9
0
20
40
60
80
100
120
140
160
180
days

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Forward F (%)
-3
-1
1
3
5
7
9
0
50
100
150
200
250
300
350
days

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Binomial distribution.
¨ At option expiry, where does the forward F end up and with which probability?
Simulation Forward Payoff
# 1 3.8 1.4
# 2 2.6 0.2
# 3 -0.8 0
# 4 4.4 2
# 5 4.6 2.2
# 6 5 2.6
# 7 1.4 0
# 8 5.6 3.2
# 9 -1.4 0
# 10 3.4 1
# 11 2.4 0
# 12 5.4 3
AVERAGE 3.03 1.30
STDEV 2.31 1.26

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Call Price (at last!)
¨ As we keep increasing the number of simulated paths, our estimate of the option price
becomes more precise.
#paths Forward Option Price
12 3.03 (+/- 2.3) 1.30 (+/- 1.3)
100 2.45 (+/- 1.8) 0.764 (+/- 0.99)
1000 2.41 (+/- 1.8) 0.749 (+/- 0.99)
10000 2.40 (+/- 1.8) 0.760 (+/- 1.12)
: : : : : :
: : : : : :
Theory 2.4 0.762

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What did we achieve?
¨ We priced the simplest option……..yeah us!
¨ I thought this was supposed to be FIXED-Income?
¨ Expiry T
¨ Strike K
¨ Forward F
¨ Volatility s }Þ C (T,K,F,s)

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Simple options rules.
¨ Option Call C(T,K,F,s).
¨ C increases when T increases.
¨ C decreases when K increases.
¨ C increases when F increases.
¨ C increases when s increases.

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Simple options rules.
¨ Option Call C(T,K,F,s)
¨ C increases when T increases. Negative time-decay (Theta).
¨ C decreases when K increases. Negative Bet.
¨ C increases when F increases. Positive Delta.
¨ C increases when s increases. Positive Vega.

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Can we improve on our model?
¨ Forward F can go negative.
¨ Constant volatility in time
¨ Constant volatility with level of forward
¨ ALSO
– We can replace our simulation with an exact analytical form (smoother risk, faster computation
time), we lose however the flexibility of adding to our model
– We have also assumed that the expected value of the forward is the current value. How
justified is that? In essence we ”fixed” the up and down probability of a jump to be 50%. Are
those the real world probabilities? The Risk Neutral probabilities?

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Avoiding negative forwards.
¨ Instead of choosing F to be normally distributed, we can choose Ln(F) to be normally
distributed.
¨ Ln(F) is Normally distributed.
¨ F is LogNormally distributed.
¨ Lognormal model.
¨ Case Sprenkle (1961).

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Avoiding Arbitrage? Fixing the probabilities.
¨ Fisher Black - Myron Scholes (1973).
¨ We’ll have to come back to that one later. That one is really tough (they don’t give away
Nobel prizes for nothing you know?).

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Why is Black-Sholes so important ?
¨ Simple version : you and I might not agree on where the market (underlying) is going, we
will still agree on the option price.
¨ Textbook version : market participants must use risk-neutral probabilities when pricing
options.

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Implementing a variable volatility.
¨ Volatility could be function of:
– Time: Absolute time, time to expiry.
– Level of rates: Correlation, skew.
– Level of vols: Smile.
– Economic release: Calendar/Business, Time Warp.

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Other possible distributions.
¨ Mixture of Lognormal.
¨ Mixture of Normal/Lognormal.
¨ Shifted Lognormal.
¨ Jump processes.
¨ Constant Elasticity models.

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Calibrating to market.
¨ Expiry T
¨ Strike K
¨ Forward F
¨ Volatility s
CMODEL (T,K,F,s)
}
CMARKET (T,K,F)
?¹
Market Implied Volatility s

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Trading options is trading volatility.
¨ Model Implied Volatility.
¨ Historical Realized Volatility.
¨ Market Implied Volatility (Future Realized Volatility).
¨ Spreading Volatility on different forwards.

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Term Structure.
FORWARD RATES (%)
20030828 1 year 2 year 3 year 4 year 5 year 7 year 10 year 15 year 20 year 30 year
1m 1.51 2.32 3.02 3.55 3.96 4.51 4.98 5.41 5.59 5.66
3m 1.70 2.56 3.24 3.74 4.12 4.63 5.08 5.48 5.65 5.71
6m 2.08 2.93 3.55 4.02 4.36 4.82 5.23 5.60 5.75 5.79
1 year 3.00 3.70 4.20 4.56 4.83 5.19 5.52 5.82 5.93 5.93
18m 3.81 4.33 4.72 5.00 5.20 5.49 5.75 6.00 6.08 6.05
2 year 4.42 4.82 5.13 5.34 5.50 5.72 5.94 6.14 6.19 6.14
3 year 5.25 5.51 5.68 5.81 5.91 6.06 6.22 6.34 6.35 6.27
4 year 5.79 5.92 6.02 6.09 6.16 6.27 6.39 6.47 6.45 6.33
5 year 6.06 6.14 6.21 6.27 6.32 6.41 6.51 6.54 6.50 6.36
7 year 6.36 6.42 6.47 6.51 6.55 6.61 6.65 6.62 6.53 6.37
10 year 6.66 6.70 6.73 6.75 6.76 6.76 6.72 6.60 6.48 6.29
20 year 6.42 6.36 6.31 6.27 6.23 6.14 6.04 5.90 5.79 5.63

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Implied Market Normal Volatility.
BP VOLATILITIES (bps/day)
1m 4.5 9.0 9.8 10.0 10.3 10.1 9.7 9.0 8.5 8.1
3m 4.7 8.7 9.6 9.8 9.9 9.7 9.3 8.7 8.2 7.4
6m 6.1 9.1 9.4 9.6 9.6 9.4 9.1 8.3 7.7 6.9
1 year 9.1 9.4 9.4 9.3 9.3 9.0 8.7 8.0 7.5 6.8
18m 9.5 9.6 9.4 9.3 9.1 8.9 8.5 7.8 7.3 6.6
2 year 9.9 9.7 9.4 9.2 9.0 8.7 8.3 7.6 7.1 6.5
3 year 9.5 9.2 9.0 8.7 8.6 8.3 7.9 7.2 6.6 6.0
4 year 8.9 8.6 8.5 8.3 8.1 7.8 7.4 6.7 6.2 5.6
5 year 8.4 8.1 8.0 7.8 7.7 7.4 7.0 6.3 5.8 5.2
7 year 7.6 7.5 7.3 7.1 7.0 6.7 6.3 5.7 5.2 4.7
10 year 6.6 6.5 6.3 6.2 6.0 5.7 5.3 4.7 4.4 4.0
20 year 4.8 4.5 4.4 4.2 4.1 4.0 3.8 3.4 3.1 2.9
• This is the canonical “Swaption Grid”. Each forward rate is
assumed to be its own independent variable, and the
implied volatility is the number that you need to plug in the
option model to recover the price of at-the-money
swaptions observed in the market

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Implied Market Lognormal Volatility.
YIELD VOLATILITIES (%/year)
1m 47.5 61.3 51.3 44.7 41.2 35.4 30.9 26.3 24.1 22.8
3m 43.4 53.8 47.0 41.5 38.0 33.2 29.2 25.1 22.9 20.5
6m 46.2 49.1 42.1 37.8 35.0 30.8 27.6 23.5 21.3 18.8
1 year 47.9 40.4 35.5 32.5 30.6 27.6 25.1 21.8 20.0 18.3
18m 39.6 35.1 31.6 29.4 27.8 25.6 23.5 20.7 19.0 17.4
2 year 35.6 31.9 29.2 27.3 25.9 24.1 22.2 19.7 18.1 16.7
3 year 28.6 26.6 25.1 23.9 23.0 21.7 20.2 18.0 16.6 15.3
4 year 24.3 23.2 22.3 21.5 20.8 19.8 18.5 16.5 15.3 14.1
5 year 21.9 21.0 20.4 19.8 19.2 18.3 17.1 15.4 14.2 13.0
7 year 19.1 18.5 18.0 17.4 16.9 16.1 15.0 13.6 12.6 11.7
10 year 15.7 15.3 15.0 14.5 14.1 13.4 12.5 11.4 10.8 10.1
20 year 11.8 11.2 11.0 10.7 10.5 10.2 10.0 9.1 8.6 8.1

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Historical Realized Lognormal Volatility (90days).
1m 57.8 63.9 55.8 48.1 43.0 36.0 30.8 25.9 23.7 22.3
3m 66.6 64.8 55.2 47.7 42.4 35.8 30.6 26.0 23.8 22.5
6m 73.9 64.5 53.9 46.8 41.8 35.5 30.6 26.0 24.0 22.7
1 year 71.1 58.6 49.6 43.8 39.7 34.2 30.0 25.8 23.9 22.9
18m 61.2 51.1 44.7 40.3 37.0 32.6 29.0 25.3 23.7 22.9
2 year 52.2 45.1 40.6 37.2 34.5 31.1 27.9 24.7 23.3 22.7
3 year 40.5 37.4 34.7 32.2 30.5 28.4 26.0 23.5 22.5 22.3
4 year 36.5 33.3 30.6 29.1 28.1 26.5 24.4 22.6 21.9 21.9
5 year 30.9 28.7 27.7 27.1 26.4 25.0 23.2 21.9 21.4 21.5
7 year 26.7 26.1 25.4 24.6 23.9 22.6 21.5 20.8 20.7 21.1
10 year 22.6 21.8 21.2 20.8 20.4 20.1 19.8 19.9 20.2 20.7

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Historical Realized Lognormal Volatility (360days).
1m 48.9 50.5 43.4 37.8 34.2 29.2 25.1 21.2 19.3 17.8
3m 55.7 50.7 43.1 37.7 33.9 29.0 25.1 21.3 19.4 17.9
6m 59.9 50.0 42.1 37.0 33.4 28.7 25.0 21.3 19.4 18.0
1 year 54.4 44.4 38.3 34.3 31.4 27.5 24.3 20.9 19.2 18.0
18m 45.5 38.6 34.4 31.5 29.2 26.1 23.4 20.4 18.9 17.9
2 year 38.6 34.2 31.5 29.2 27.3 24.9 22.5 19.8 18.5 17.7
3 year 31.2 29.4 27.5 25.8 24.6 23.0 21.0 18.8 17.8 17.3
4 year 29.2 26.8 24.9 23.8 23.0 21.7 19.8 18.1 17.3 17.0
5 year 25.2 23.7 22.8 22.2 21.6 20.4 18.8 17.4 16.8 16.7
7 year 21.8 21.3 20.8 20.1 19.4 18.2 17.1 16.3 16.0 16.3
10 year 18.8 18.1 17.3 16.8 16.4 15.9 15.5 15.2 15.4 16.0

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Trading options is trading Correlation.
CORRELATION BETWEEN FORWARDS AND NORMAL VOLATILITY (360 days).
1m 0.6 0.5 0.4 0.2 0.1 0.1 0.1 0.2 0.1 0.1
3m 0.7 0.6 0.4 0.3 0.1 0.1 0.1 0.1 0.1 0.1
6m 0.7 0.5 0.4 0.2 (0.0) 0.0 0.1 0.2 0.1 0.0
1 year 0.4 0.2 0.0 (0.1) (0.2) (0.2) (0.0) 0.0 0.1 (0.1)
18m 0.2 (0.0) (0.1) (0.2) (0.3) (0.2) (0.0) 0.0 0.1 (0.1)
2 year (0.2) (0.2) (0.2) (0.2) (0.3) (0.2) (0.0) 0.1 0.1 (0.1)
3 year (0.1) (0.1) (0.1) (0.1) (0.2) (0.1) 0.1 0.1 0.1 (0.1)
4 year (0.0) (0.0) (0.1) (0.1) (0.1) (0.0) 0.2 0.2 0.1 (0.1)
5 year 0.0 0.0 (0.0) (0.1) (0.1) 0.0 0.2 0.2 0.1 (0.1)
7 year 0.0 0.0 (0.0) (0.0) 0.0 0.1 0.2 0.2 0.1 (0.1)
10 year 0.2 0.2 0.1 0.1 0.1 0.1 0.2 0.1 0.1 (0.2)
• A non zero correlation between the level of rates and the
volatility is what people refers to as “skew”. It also shows
clearly that our assumption of constant volatility with the
level of the underlier was incorrect

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Trading options is trading Correlation.
CORRELATION BETWEEN FORWARDS AND LOGNORMAL VOLATILITY (360 days).
1m (0.5) (0.8) (0.8) (0.8) (0.8) (0.7) (0.6) (0.5) (0.4) (0.4)
3m (0.6) (0.8) (0.9) (0.9) (0.9) (0.8) (0.7) (0.7) (0.6) (0.6)
6m (0.7) (0.9) (0.9) (0.9) (0.9) (0.9) (0.8) (0.8) (0.7) (0.7)
1 year (0.9) (1.0) (1.0) (1.0) (1.0) (0.9) (0.9) (0.9) (0.8) (0.8)
18m (0.9) (1.0) (1.0) (1.0) (1.0) (0.9) (0.9) (0.9) (0.8) (0.8)
2 year (1.0) (1.0) (1.0) (1.0) (1.0) (0.9) (0.9) (0.8) (0.8) (0.8)
3 year (1.0) (0.9) (0.9) (0.9) (0.9) (0.9) (0.9) (0.8) (0.8) (0.8)
4 year (0.9) (0.9) (0.9) (0.9) (0.9) (0.9) (0.8) (0.7) (0.7) (0.8)
5 year (0.9) (0.9) (0.9) (0.9) (0.9) (0.9) (0.8) (0.7) (0.7) (0.8)
7 year (0.8) (0.8) (0.8) (0.8) (0.8) (0.8) (0.6) (0.6) (0.6) (0.8)
10 year (0.6) (0.6) (0.6) (0.6) (0.6) (0.5) (0.4) (0.5) (0.6) (0.7)
• The correlation is much stronger between rates and
lognormal volatility than between rates and normal
volatility. People tend to say that “rates behave normally”,
not “lognormally”

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Trading options is risky: Risk Parameters.
F
C
¶
¶
=D
2
2
F
C
¶
¶
=g
T
C
¶
¶
=Q
s¶
¶
=
C
Vega
Greeks Definition
Delta ($/bp)
Gamma
($/bp/bp)
Theta
($/day)
Vega ($/%)

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Why am I doing all this work?
¨ Pricing swaps and bonds was so much easier, I just needed a yield curve, et voila !
¨ Swap traders don’t care about Volatility?
¨ Why not?

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BREAK !!
¨ See you all in 10 minutes

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Why are options different from Swaps and Bonds.
¨ Arbitrage-free.
– Risk-free rate of return.
– Risk-Neutral assumption.
¨ Convexity.
¨ Let’s look at the convexity first, that’s the easy one

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What is Convexity?
¨ Change in Duration.
¨ Curvature.
¨ Second derivative.
¨ Deviation from straight line.
¨ Departure from linearity.

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What is convex?
¨ Bond price as a function of yield?
¨ Eurodollar future price as a function of forward rate?
¨ Ln(x) as a function of x?
¨ (1-2x) as a function of x?
¨ Bond price as a function of coupon?
¨ Bond price as a function of face amount?
¨ (1/x) as a function of x?

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Answers.
¨ Bond price as a function of yield? YES (+)
¨ Eurodollar future price as a function of forward rate? NO
¨ Ln(x) as a function of x? YES (-)
¨ (1-2x) as a function of x? NO
¨ Bond price as a function of coupon? NO
¨ Bond price as a function of face amount? NO
¨ (1/x) as a function of x? YES (+)

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If it’s linear, forget about Volatility.
x
f(x)

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67
x
f(x)
<x>
f(<x>)

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x
f(x)
x
f(x)
xMAXxMin <x>
f(<x>)

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x
f(x)
x
f(x)
x
f(x)
x
f(x)
xMAXxMin
f(<x>)
<x>

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¨ Average of f(x) = f(Average of x).
¨ <f(x)> = f(<x>).
¨ A linear transformation is a simple scaling (inches to centimeters, Celsius to Farenheit,…).

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If it’s convex, mind the Volatility.
x
f(x)

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xMAXxMin
x
f(x)
x
f(x)
<x>

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xMAXxMin
x
f(x)
x
f(x)
<x>

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xMAXxMin
x
f(x)
x
f(x)
<x>

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xMAXxMin
x
f(x)
x
f(x)
f(<x>)
<f(x)>
<x>

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Positively and negatively convex.
¨ If the function f is positively convex… the average of f(x) is greater than f(<x>).
¨ If the function f is negatively convex… the average of f(x) is smaller than f(<x>).

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xMAXxMin
x
f(x)
x
f(x)
Average x
f (average of x)
average of f (x)

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Positively and negatively convex, an option payoff.
¨ If the payout of an option is positively convex…
– the average of all possible option payouts is greater than the value of the payout at the
average of the underlying
¨ If the payout of an option is negatively convex…
– the average of all possible option payouts is smaller than the value of the payout at the
average of the underlying
¨ Extreme case…
– Consider an option expiring in 2 minutes that is at-the-money. The position is convex, so
the average of all possible payouts is positive, although the payout at the average of the
underlying = 0 (since the option is at-the-money)

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For the mathematically inclined.
¨ Taylor expansion (Brooke Taylor, 1715).
)().(
2
1
).()()( 32
2
2
dxdx
dx
fd
dx
dx
df
xfdxxf O+++=+

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80
Math 101, part deux.
)().(
2
1
).()()(
)().(
2
1
).()()(
32
2
2
32
2
2
dxdx
dx
fd
dx
dx
df
xfdxxf
dxdx
dx
fd
dx
dx
df
xfdxxf
O++-+=-
O+++=+

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81
Math 101, part trois.
)().(
2
1
)(
2
)()( 32
2
2
dxdx
dx
fd
xf
dxxfdxxf
O++=
þ
ý
ü
î
í
ì ++-

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82
Math 101, part quatre (sometimes called Jensen inequality)
)().(
2
1
2
)()(
2
)()( 32
2
2
dxdx
dx
fddxxdxx
f
dxxfdxxf
O++
þ
ý
ü
î
í
ì ++-
=
þ
ý
ü
î
í
ì ++-
ConvexityxofAveragefxfofAverage
dx
dx
fd
xofAveragefxfofAverage
+=
+=
)()(
).(
2
1
)()( 2
2
2
Convexity adjustments and such only work if the function is “well behaved”.
Convexity adjustments would not work on a portfolio of Digital bets for example

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Math 101, the fin.
¨ If the function f is positively convex… the average of f(x) is greater than f(<x>).
¨ If the function f is negatively convex… the average of f(x) is smaller than f(<x>).
¨ Convexity =
¨ Volatility =
2
2
dx
fd
2
)(dx

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84
A call payoff is positively convex.
F
K
$

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85
Call premium.
¨ Call = Max(F-K,0).
¨ Intrinsic Value: Payoff of the average.
¨ Call Premium : Average of the payoff.

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Call premium.
¨ Intrinsic Value: Payoff of the average.
¨ Call Premium : Average of the payoff.
¨ Intrinsic Value: Max(<F> - K, 0)
¨ Call Premium: <Max(F - K, 0)>
¨ Call premium: convexity adjusted value of the terminal payoff…. Has to depend on the
volatility…

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Time value of a call.
¨ A call premium is positively convex à always greater than the intrinsic value.
¨ (Premium – Intrinsic) = Time Value.
¨ The greater the Volatility, the greater the Time Value.
¨ The greater the convexity, the greater the Time Value.

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Call Premium.
¨ Convexity is greatest at the strike.
¨ Deep in-the-money and far out-of-the-money,
Convexity is small.
Time Value is small.
Call premium slightly greater than terminal payoff.
¨ At-the-money,
Convexity is maximum.
Time Value is maximum.
Call premium furthest away from terminal payoff.

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89
Call Premium.
F
K
$
"At the Money"
"Out of the Money" "In the Money"
Intrinsic Value.

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Call Premium.
$
Intrinsic Value.
"At the Money"
"Out of the Money" "In the Money"
F
K
Call Premium
Time Value

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Convexity smoothes out the terminal payoff.
¨ If positive convexity, <Payoff(F)> greater than Payoff(<F>).
F
K
$ Option

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¨ If negative convexity, <Payoff(F)> smaller than Payoff(<F>).
F
K
$
Option

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¨ The greater the payoff convexity,
¨ The greater the volatility
¨ The greater the time value,
¨ The further away will the option price be from the intrinsic value.

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94
¨ “Diffusion” of terminal payoff.
¨ The greater the volatility, the smoother the option price.
F
K
$

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Zero Volatility.
PVs ($)
-200,000
0
200,000
400,000
600,000
800,000
1,000,000
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100
Rate shifts (basis points)

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Low Volatility.
PVs ($)
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
900,000
1,000,000
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100

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Medium Volatility.
PVs ($)
-200,000
0
200,000
400,000
600,000
800,000
1,000,000
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100

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High Volatility.
PVs ($)
-200,000
0
200,000
400,000
600,000
800,000
1,000,000
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100

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Calls are risky.
Greeks Definition
Delta
F
C
¶
¶
=D
Gamma
2
2
F
C
¶
¶
=g
Theta
T
C
¶
¶
=Q
Vega
s¶
¶
=
C
Vega

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Profits (and Losses).
Greeks Definition Units Profits/Losses
Delta
F
C
¶
¶
=D ($/bp) )( FdD
Gamma
2
2
F
C
¶
¶
=g ($/bp/bp)
2
)(
2
1
Fdg
Theta
T
C
¶
¶
=Q ($/day) )( TdQ
Vega
s¶
¶
=
C
Vega ($/%) )(dsVega

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P&L as a Taylor expansion of the option with parameters
¨ 𝛿𝐶 =
!"
!#
. 𝛿𝑡 +
!"
!$
. 𝛿𝐹 +
!"
!%
. 𝛿𝜎 +
&
'
.
!!"
!$! . (𝛿𝐹)'
¨ 𝛿𝐶 = 𝑇ℎ𝑒𝑡𝑎 . 𝛿𝑡 + 𝐷𝑒𝑙𝑡𝑎 . 𝛿𝐹 + 𝑉𝑒𝑔𝑎 . 𝛿𝜎 +
&
'
. 𝐺𝑎𝑚𝑚𝑎 . (𝛿𝐹)'
¨ 𝛿𝐶 = Θ. 𝛿𝑡 + ∆. 𝛿𝐹 + 𝑉𝑒𝑔𝑎. 𝛿𝜎 +
&
'
. 𝛾. (𝛿𝐹)'
101

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Profits and Losses.
If DELTA is:
POSITIVE
NEGATIVE
If GAMMA is:
POSITIVE
NEGATIVE
If THETA is: Passage of time will
POSITIVE
NEGATIVE
Increase / Decrease option value
If VEGA is: You want volatility to
POSITIVE
NEGATIVE
Fall / Rise
Fall / Rise
You want the underlying to
Rise / Fall
Rise / Fall
Sit still / Make a big move

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Profits and Losses.
If DELTA is:
POSITIVE
NEGATIVE
If GAMMA is:
POSITIVE
NEGATIVE
If THETA is: Passage of time will
POSITIVE
NEGATIVE
If VEGA is: You want volatility to
POSITIVE
NEGATIVE
Fall / Rise
Fall / Rise
Rise / Fall
Rise / Fall

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Call premium, PV.
PVs ($)
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100

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Delta, Duration, DV01,..
Delta ($/bp)
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
-95 -85 -75 -65 -55 -45 -35 -25 -15 -5 5 15 25 35 45 55 65 75 85 95

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Delta, Duration, DV01,..
¨ Delta is the slope of the call PV.
¨ Far out-of-the-money Delta is close to 0.
(Call has no value, and an increase in rates still brings no value).
¨ Deep in-the-money Delta is close to 1.
(Rates are so high compared to the strike that the probability of going back under the strike and
getting a zero payoff are negligible).

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Gamma, Convexity, DV02.
Gamma ($/bp/bp)
-50
0
50
100
150
200
250
-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90

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108
Gamma, Convexity, DV02.
¨ Gamma is the slope of the Delta.
¨ Gamma is the convexity (change in Duration).
¨ Gamma is negligible on the wings, maximum at-the-money.

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Vega.
Vega ($/%)
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100

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Vega.
¨ Vega maximum at-the-money, negligible on the wings.
¨ Options far away from the money are not options anymore.. They have no Gamma, no
Vega, no time value left.

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Vega
¨ Vega
– Sensitivity of an option’s price to changes in the underlying’s volatility
– As vol moves higher, the underlying is more likely to move away from the strike in a given amount of time
K
Linear
Call
V
K
VDigital
Call
K
V
Call Price Increases as Volatility Increases
σ
(Option)
Vega
¶
¶
=
111

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112
How does the Vega change?
¨ When Volatility changes: Volga.
– Smile.
– Volatility of volatility.
¨ When Rates change: Vanna.
– Skew.
– Correlation between rates and volatility.
÷
ø
ö
ç
è
æ
¶
¶
F
Vega
÷
ø
ö
ç
è
æ
¶
¶
s
Vega

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Volga, Smile, Stochastic Volatility exposure.
Volga ($/%/%)
-200
0
200
400
600
800
1,000
1,200
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100

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Volga, Smile, Stochastic Volatility exposure.
¨ Volga is negligible for at-the-money options, the Vega of an at-the-money option does not
depend on the level of Volatility.
¨ Volga is positive for both low strike and high strike options.
¨ Low strike and high strike options will have similar exposure to the smile.

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115
Smile premium for out-of-the money options.
¨ When being long an Out-of-the-Money option, we will :
– Get longer Vega when s increases.
– Get shorter Vega when s decreases.
¨ When Vega hedging this Out-of-The-Money option, we will :
– Sell Volatility when s increases.
– Buy Volatility when s decreases.
“BUY LOW, SELL HIGH”
115

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Smile Premium
116
• If we are long Volga:
Volatility Vega Hedge P/L ?
Sell
Buy

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117
Smile premium
¨ A trader will be willing to pay up for an OTM option
¨ Vega hedging allows the trader to capture the smile premium.
¨ Note : this is almost identical to the regular option premium :
– An option exhibits positive Gamma everywhere : being long an option, we will get longer the
market as it rallies, shorter as it sells off.
– Delta hedging allows the trader to capture the option premium over time.
– Option premium is maximum for ATM options (maximum Gamma).

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Skew Exposure: How the Vega Changes with Rates
118
Vega ($/%)
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100

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Skew Exposure: Vanna
119
• The Vanna is the slope of Vega, positive below the strike,
negative above the strike
• If rates increase, we get closer to the strike for high strike
options, the Vega increases (positive Vanna)
• If rates increase, we get further away from the strike for low
strike options, the Vega decreases (negative Vanna)
• High strike options and low strike options have opposite
exposure to the skew

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120
Vanna, Skew, Correlation exposure.
Vanna ($/%/bp)
-300
-200
-100
0
100
200
300
400
-95 -85 -75 -65 -55 -45 -35 -25 -15 -5 5 15 25 35 45 55 65 75 85 95

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121
Vanna.
¨ The Vanna is the slope of Vega, positive below the strike, negative above the strike.
¨ High strike options and low strike options have opposite exposure to the skew.

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Vanna.
( )
( )Delta
F
C
F
C
Vanna
F
CC
F
Vega
F
Vanna
sss
ss
¶
¶
=÷
ø
ö
ç
è
æ
¶
¶
¶
¶
=
¶¶
¶
=
¶¶
¶
=÷
ø
ö
ç
è
æ
¶
¶
¶
¶
=
¶
¶
=
2
2
•Vanna
= Change in Vega when rates change.
= Change in Delta when Volatility changes.

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Skew premium and correlation.
1) How does the Vega change with the underlying ?
2) How does the volatility s change with the underlying ?
- Historical
- Implied

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Changes in rates and volatility.
From the model:
Vega is maximum for At-The-Money options.
– Vanna is positive below the strike.
– Vanna is negative above the strike.
From the market:
Suppose that rates and volatility are negatively correlated.
– When rates increase, yield vol decreases.
– When rates decrease, yield vol increases.

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125
1year-1year swaption : yield vol vs. ATM rates
1y1y swaption
Correlation ~ -0.9
0
10
20
30
40
50
60
70
2 3 4 5 6 7 8
ATM forward (%)
yieldvol(%)

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5year-5year swaption : yield vol vs. ATM rates
5y5y swaption
Correlation ~ -0.5
10
15
20
25
30
5 5.5 6 6.5 7 7.5 8
ATM forward (%)
yieldvol(%)

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A normal world?
Corelation between forward rates and implied normal volatility (past 360 days).
1 2 3 4 5 7 10 15 20 30
1y 2y 3y 4y 5y 7y 10y 15y 20y 30y
1m 0.64 0.49 0.35 0.23 0.07 0.09 0.15 0.16 0.13 0.11
3m 0.72 0.56 0.42 0.28 0.06 0.08 0.15 0.14 0.11 0.09
6m 0.68 0.47 0.35 0.20 0.00 0.01 0.11 0.15 0.08 0.00
1y 0.41 0.22 0.03 -0.08 -0.22 -0.16 -0.03 0.01 0.07 -0.07
18m 0.18 -0.02 -0.13 -0.20 -0.28 -0.20 -0.03 0.05 0.09 -0.08
2y -0.18 -0.19 -0.22 -0.25 -0.29 -0.21 0.00 0.11 0.12 -0.08
3y -0.13 -0.13 -0.13 -0.12 -0.18 -0.09 0.12 0.13 0.10 -0.09
4y -0.04 -0.05 -0.06 -0.09 -0.12 -0.02 0.18 0.18 0.12 -0.09
5y 0.05 0.02 -0.02 -0.05 -0.08 0.03 0.22 0.17 0.07 -0.09
7y 0.04 0.01 -0.02 -0.01 0.00 0.09 0.22 0.19 0.05 -0.11
10y 0.15 0.17 0.14 0.14 0.11 0.14 0.23 0.08 0.06 -0.21

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A normal world?
Corelation between forward rates and implied lognormal volatility (past 360 days).
1 2 3 4 5 7 10 15 20 30
1y 2y 3y 4y 5y 7y 10y 15y 20y 30y
1m -0.46 -0.77 -0.81 -0.82 -0.81 -0.73 -0.60 -0.51 -0.44 -0.45
3m -0.57 -0.83 -0.87 -0.89 -0.90 -0.84 -0.74 -0.67 -0.60 -0.61
6m -0.74 -0.90 -0.92 -0.93 -0.93 -0.90 -0.83 -0.79 -0.73 -0.74
1y -0.90 -0.96 -0.96 -0.96 -0.97 -0.95 -0.92 -0.87 -0.81 -0.84
18m -0.94 -0.97 -0.97 -0.96 -0.97 -0.95 -0.92 -0.87 -0.81 -0.84
2y -0.98 -0.97 -0.97 -0.96 -0.96 -0.95 -0.91 -0.85 -0.80 -0.83
3y -0.96 -0.95 -0.94 -0.95 -0.94 -0.92 -0.87 -0.79 -0.75 -0.81
4y -0.93 -0.92 -0.92 -0.92 -0.93 -0.90 -0.82 -0.72 -0.69 -0.79
5y -0.90 -0.90 -0.91 -0.91 -0.91 -0.87 -0.77 -0.67 -0.67 -0.77
7y -0.83 -0.83 -0.82 -0.81 -0.82 -0.76 -0.64 -0.59 -0.64 -0.75
10y -0.58 -0.58 -0.58 -0.58 -0.60 -0.53 -0.44 -0.51 -0.58 -0.75

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A lognormal world?
Corelation between forward rates and implied normal volatility (past 20 days).
1 2 3 4 5 7 10 15 20 30
1y 2y 3y 4y 5y 7y 10y 15y 20y 30y
1m -0.46 -0.77 -0.81 -0.82 -0.81 -0.73 -0.60 -0.51 -0.44 -0.45
3m -0.57 -0.83 -0.87 -0.89 -0.90 -0.84 -0.74 -0.67 -0.60 -0.61
6m -0.74 -0.90 -0.92 -0.93 -0.93 -0.90 -0.83 -0.79 -0.73 -0.74
1y -0.90 -0.96 -0.96 -0.96 -0.97 -0.95 -0.92 -0.87 -0.81 -0.84
18m -0.94 -0.97 -0.97 -0.96 -0.97 -0.95 -0.92 -0.87 -0.81 -0.84
2y -0.98 -0.97 -0.97 -0.96 -0.96 -0.95 -0.91 -0.85 -0.80 -0.83
3y -0.96 -0.95 -0.94 -0.95 -0.94 -0.92 -0.87 -0.79 -0.75 -0.81
4y -0.93 -0.92 -0.92 -0.92 -0.93 -0.90 -0.82 -0.72 -0.69 -0.79
5y -0.90 -0.90 -0.91 -0.91 -0.91 -0.87 -0.77 -0.67 -0.67 -0.77
7y -0.83 -0.83 -0.82 -0.81 -0.82 -0.76 -0.64 -0.59 -0.64 -0.75
10y -0.58 -0.58 -0.58 -0.58 -0.60 -0.53 -0.44 -0.51 -0.58 -0.75

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A lognormal world?
Corelation between forward rates and implied lognormal volatility (past 20 days).
1y 2y 3y 4y 5y 7y 10y 15y 20y 30y
1m 0.49 0.58 0.54 0.47 0.43 0.53 0.63 0.69 0.75 0.45
3m 0.40 0.54 0.56 0.53 0.48 0.64 0.72 0.75 0.75 0.18
6m 0.54 0.43 0.33 0.27 0.13 0.38 0.47 0.63 0.63 -0.01
1y 0.53 0.30 -0.13 -0.26 -0.41 -0.22 -0.07 0.17 0.37 0.49
18m 0.14 0.02 -0.25 -0.37 -0.44 -0.29 -0.05 0.27 0.42 0.51
2y -0.36 -0.28 -0.27 -0.32 -0.37 -0.23 0.03 0.41 0.54 0.58
3y -0.38 -0.24 -0.18 -0.14 -0.17 -0.06 0.03 0.25 0.37 0.42
4y -0.21 -0.13 -0.04 -0.07 -0.17 -0.16 0.06 0.07 0.27 0.28
5y -0.31 -0.18 -0.26 -0.25 -0.28 -0.17 -0.09 0.09 0.18 0.17
7y -0.27 -0.12 -0.17 -0.18 -0.17 -0.11 -0.04 -0.10 0.06 0.16
10y 0.00 0.10 0.08 0.21 0.31 0.19 0.16 -0.04 0.18 0.12

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High strike option : negative skew
¨ Suppose that we are long a high strike option:
¨ When rates increase : - Vega increases.
- Yield volatility decreases. We get
longer Vega when s decreases.
¨ When rates decrease : - Vega decreases.
- Yield volatility increases. We get
shorter Vega when s increases.
“SELL LOW, BUY HIGH”

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Low strike option : positive skew
¨ Suppose that we are long a low strike option:
¨ When rates increase : - Vega decreases.
- Yield volatility decreases. We
get shorter Vega when s decreases.
¨ When rates decrease : - Vega increases.
- Yield volatility increases. We get
longer Vega when s increases.
“BUY LOW, SELL HIGH”

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Skew premium
133
Position Rates Vega Volatility Hedge P/L ?
Sell
Long Vanna
(long high strike)
Buy
Buy
Short Vanna
(long low strike) Sell
¨ If rates and volatility are negatively correlated:

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It pays to be long low strike, short high strike.
• High strike options have positive Vanna.
• Low strike options have negative Vanna.
• When rates and yield vol are negatively correlated, it pays
to be long options with negative Vanna (buy low strikes, sell
high strikes).

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Skew and smile.
¨ Skew / Vanna / Correlation.
¨ Smile / Volga / Stochastic volatility.
÷
ø
ö
ç
è
æ
¶
¶
F
Vega
÷
ø
ö
ç
è
æ
¶
¶
s
Vega

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Higher order derivatives.
¨ It’s all Taylor expansion….
¨ Check out N. Taleb “Dynamic Hedging”.
¨ Spreading options will cause the emergence of higher order effects in the portfolio….

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Bonds and Swaps, reloaded.
¨ Wait a minute! Bond prices are convex!
¨ Discount factors are convex too!
¨ So… why can I neglect the convexity (volatility) when pricing a swap?
¨ I am still confused…..

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Bonds and Swaps, reloaded.
¨ Wait a minute! Bond prices are convex!
¨ Discount factors are convex too!
¨ So… why can I neglect the convexity (volatility) when pricing a swap?
¨ I am still confused…..
ARBITRAGE !

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Arbitrage is related to Hedging.
¨ If you can hedge a complicated structure with simpler instruments, then the price of the
structure has to be equal to the total price of the simpler market instruments.
¨ If not, there is an arbitrage.
¨ Structure = Linear sum of market instruments.

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Hedging.
¨ Static Hedge vs. Dynamic Hedge.
¨ Complete Hedge vs. Partial Hedge.
– Delta Hedge.
– Vega Hedge.
– ….

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Risk and Arbitrage.
¨ If markets willing to always arbitrage market instruments against one another….
¨ All related instruments are equally risky, there is no arbitrage possible.
¨ Risk-free world, Risk-free measure, Risk-neutral probabilities.
¨ Arbitrage-free model.

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The convexity changes the Hedge.
¨ If the structure is convex with respect to the simpler market instruments..
¨ The weights will change.
¨ There is no static hedge, there is no complete hedge.
¨ The price of S will depend on the volatility!

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Convexity is relative, only Arbitrage is absolute.
¨ (1/x) is convex with respect to (x).
¨ (x) is convex with respect to (1/x).
¨ Bond prices are convex with respect to Bond yields.
¨ Bond yields are convex with respect to Bond prices.
¨ (the two-envelopes question)….
¨ We have to choose a baseline, an absolute reference point.

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Ab$olute reference.
¨ Ca$h.
¨ $1 in the Bank.
¨ Rolling numeraire.
¨ Di$count curve.

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The secret recipe: always follow the money.
¨ $1 in the Bank, and growing….
¨ Expected Future Value of $1 deposited in the Bank.
¨ The only thing you can trust: the discount curve! (time value of money).
¨ You can only arbitrage actual cash-flows.
¨ Only the discount curve is arbitrage-free.

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A Bond is a sum of Fixed cash flows.
¨ Remember? That’s why it’s called Fixed-Income.
¨ A Bond has zero convexity as a function of the discount factors.
¨ When pricing a Bond from the discount curve, there is no convexity adjustment, so you
don’t care about the volatility of the underlying.
¨ Bonds are easy to price!

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A swap is a linear sum of discount factors.
¨ A little harder to show.
¨ A swap has zero convexity as a function of the discount factors.
¨ When pricing a swap from the discount curve, there is no convexity adjustment, so you
don’t care about the volatility of the underlying.
¨ Swaps are easy to price!

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A forward rate has convexity?
¨ A forward rate is a convex function of the discount factor.
¨ The expected value of a forward rate is convexity adjusted.
¨ By the way, a FRAs (Forward Rate Agreement) has no convexity. (see below when pricing the
floating period of a swap)

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A future has convexity?
¨ Well,….Future Price = (100 – Forward Rate).
¨ If the forward rate is convex with respect to the discount factor, so is the Future Price.
¨ Future prices are tough to price!
¨ But easy to trade

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Bond price and Bond yields.
¨ Bond prices are convex with respect to bond yields.
¨ When pricing a Bond from the yield, there IS convexity, we need to take the yield volatility
into account.
¨ Again, when pricing a bond from the discounts, there is NO convexity.

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Options have convexity?
¨ Yes they do….
¨ So will callable bonds, callable swaps, mortgages, options on mortgages, credit options,
year-end bonuses,…

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A swap is a weighted basket of forwards: AT-THE-MONEY
¨ Consider a swap with swap rate R (at-the-money swap rate)
– Nfloat periods on the Float side with forecasted forward f(i)
– indexed by i, with
– daycount fraction DCF(i),
– discount D(i)
– Notional N(i)
– Nfixed periods on the Fixed side,
– indexed by j, with
– daycount fraction DCF(j),
– discount D(j)
– Notional N(j)
𝑃𝑉 𝐹𝐿𝑂𝐴𝑇 = )
"
𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 = )
#
𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 . 𝑅 = 𝑃𝑉(𝐹𝐼𝑋𝐸𝐷)
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Standard Swap periods
¨ On the fixed side, coupon payment at the end of the period
– Period start date (psj)
– Adjusted period start date (psj_adj)
– Period end date (pej)
– Adjusted period end date (pej_adj)
– Payment date (pmj)
– PV of a period 𝑃𝑉 𝑗 = 𝐷𝐶𝐹 𝑗 . 𝐷 𝑗 . 𝑁 𝑗 . 𝑅 = 𝐷𝐶𝐹 𝑝𝑠𝑗$%#, 𝑝𝑒𝑗_𝑎𝑑𝑗 . 𝐷 𝑝𝑚# . 𝑁 𝑗 . 𝑅
¨ On the float side, floating rate sets at the beginning of the period, and pays at the end (Libor in advance or
standard Libor swap, as opposed to Libor in arrears)
– PV of a period (swaplet) 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖
– 𝐷𝐶𝐹 𝑖 = 𝐷𝐶𝐹 𝑝𝑠𝑖$%#, 𝑝𝑒𝑖$%# and 𝐷 𝑖 = 𝐷(𝑝𝑚𝑖)
– 𝐷 𝑝𝑒𝑖 = 𝐷 𝑝𝑠𝑖 ∗
&
&'()* +,",+." .0(")
or
– 𝐷𝐶𝐹 𝑝𝑒𝑖, 𝑝𝑠𝑖 . 𝑓 𝑖 = [1 −
( +."
( +,"
]
153

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Zero coupon bonds on today’s curve with zero volatility
¨ Zero-coupon bonds 𝑃 𝑡&, 𝑡3 = 𝐷 𝑡&, 𝑡3 = ⁄𝐷(𝑡3) 𝐷(𝑡&)
¨ “𝑃 𝑡&, 𝑡3 is the price at time 𝑡&of a zero-coupon bond maturing at time 𝑡3”
¨ “𝑃 𝑡&, 𝑡3 is the price at time 𝑡&of a risk-free zero-coupon bond with principal $1 maturing at time 𝑡3”
¨ IT SHOULD REALLY SAY : “Using today’s discount curve at time 𝑡4, 𝑃 𝑡&, 𝑡3 is the price of a risk-free zero-
coupon bond with principal $1 maturing at time 𝑡3, and the value of that price has been forward
discounted to time 𝑡&, again using today’s discount curve”
¨ People love the zero coupon bonds, in many cases they make those the stochastic drivers of the rates
model (HJM for example)
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Expected Values in a non deterministic world
¨ Simply compounded spot interest rate: 𝐿(𝑡&, 𝑡3)
¨ 𝐿 𝑡&, 𝑡3 =
&56(7!,7")
%80 7!,7" .6(7!,7")
or more simply 𝑃 𝑡&, 𝑡3 =
&
&'%80 7!,7" .9(7!,7")
¨ Related to how to roll the curve forward at zero volatility,
¨ Method 2
– Compute the discount factors curve
– Divide all discount factors by the overnight d(t0,t1)=d(t1)
– Use new discount factor curve starting at t1
¨ So at zero volatility, when t goes from t0 to t1, the price of a zero discount bonds 𝑃 𝑡&, 𝑡3 is unchanged:
𝑃 𝑡8, 𝑡&, 𝑡3 = 𝑃 0, 𝑡&, 𝑡3 where 𝑡8 is the “curve” time in the future.
¨ NOW, if the volatility is non zero, 𝑃 1, 𝑡&, 𝑡3 ≠ 𝑃 0, 𝑡&, 𝑡3
¨ It is only true ON AVERAGE < 𝑃 1, 𝑡&, 𝑡3 >= 𝑃 0, 𝑡&, 𝑡3 or EXP 𝑃 1, 𝑡&, 𝑡3 = 𝑃 0, 𝑡&, 𝑡3 where EXP is
the Expected value (average).
¨ This is called the rolling numeraire or “bank account” numeraire: if you deposit 𝑃 0,0, 𝑡3 today to get $1
at time t2, ON AVERAGE you should also be able to invest 𝑃 0,0, 𝑡3 until time t1, then deposit it until
time t2 and still get $1
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Fixed Leg of a swap
¨ A fixed leg of a swap is a series of fixed cash flows.
¨ Now matter how the curve moves, ON AVERAGE the price of zero coupon bonds is conserved
¨ < 𝑃 𝑡8, 𝑡&, 𝑡3 > = 𝑃 0, 𝑡&, 𝑡3 and 𝑃 𝑡8, 𝑡&, 𝑡3 =
((7#,7")
((7#,7!)
¨ In particular when t2=t1+1, 𝑃 𝑡8, 𝑡&, 𝑡& + 1 =
((7#,7!'&)
((7#,7!)
= 𝑑(𝑡8, 𝑡&)
¨ So < 𝑃 𝑡8, 𝑡&, 𝑡& + 1 > = <
( 7#,7!'&
( 7#,7!
> = < 𝑑(𝑡8, 𝑡&)> = 𝑃 0, 𝑡&, 𝑡& + 1 = 𝑑(0, 𝑡&)
¨ Also by recurrence < 𝐷 1, 𝑡& > ∗ 𝑑 0,1 = 𝐷(0, 𝑡&) at time t=0
¨ At time t=1, 𝑑 1,2 is fixed and has zero volatility (will drop when t goes from 1 to 2)
¨ So < 𝐷 2, 𝑡& > ∗ 𝑑 1,2 = 𝐷(1, 𝑡&) at time t=1
¨ So at every point in the future, if you invest then a unit of currency and ”roll” it forward (bank
numeraire), the expected gain is today’s gain if you had entered into the same contract.
¨ Still another way to say, if you invest one unit of currency for a given length of time t, it is equivalent to
investing overnight and rolling the proceeds everyday (the arbitrage free framework does not take credit
into consideration)
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Floating leg of a swap
¨ A floating swaplet pays 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 and its PV is 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖
¨ Where 𝐷𝐶𝐹 𝑝𝑒𝑖, 𝑝𝑠𝑖 . 𝑓 𝑖 = [1 −
( +."
( +,"
]
¨ We know from the fixed rate leg that < 𝐷 𝑖 >= 𝐷 𝑖 , but what about < 𝐷 𝑖 . 𝑓 𝑖 > ?
¨ Note, to be exact < 𝐷 𝑖 >= 𝐷 𝑖 should really read ∏7#:;7$
𝐸𝑋𝑃{𝑡8, 𝑑(𝑡8, 𝑡8 + 1)}, where
𝐸𝑋𝑃 𝑡8, 𝑑 𝑡8, 𝑡8 + 1 is the expected value of the overnight discount between the time (tc) and (tc+1),
observed up until time tc (because it drops off the curve after tc, and before tc, no matter where you
observe it, its expected value is equal to today’s value)
¨ 𝐸𝑋𝑃 𝑡8, 𝑑 𝑡8, 𝑡8 + 1 = 𝐸𝑋𝑃 𝑡 < 𝑡8, 𝑑 𝑡8, 𝑡8 + 1 = 𝑑(𝑡8 + 1)
¨ Back to < 𝐷 𝑖 . 𝑓 𝑖 > , there is a little trick
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Floating leg of a swap
¨ Because the forward f(i) sets at the beginning of the period, once we reach the period start, everything
is known about the payment, and it becomes a fixed cashflow.
¨ 𝑃𝑉 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝐷 𝑖 . 𝑁 𝑖 . 𝑓 𝑖 = 𝐷𝐶𝐹 𝑖 . 𝑁 𝑖 . 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖 . 𝐸𝑋𝑃{𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖, 𝑝𝑒𝑖 . 𝑓 𝑖 }
¨ 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖 = 𝐷(𝑝𝑠𝑖)
¨ 𝐷𝐶𝐹 𝑖 . 𝐸𝑋𝑃 𝑝𝑠𝑖, 𝐷 𝑝𝑠𝑖, 𝑝𝑒𝑖 . 𝑓 𝑖 = 𝐸𝑋𝑃{𝑝𝑠𝑖,
()*(")
&'()* " .0(")
. 𝑓 𝑖 }
¨ Now, magic trick,
<
&'<
=
<'&5&
&'<
=
&'<5&
&'<
= 1 −
&
&'<
¨ So, 𝐸𝑋𝑃 𝑝𝑠𝑖,
()* "
&'()* " .0 "
. 𝑓 𝑖 = 𝐸𝑋𝑃 𝑝𝑠𝑖, 1 −
&
&'()* " .0 "
= 𝐸𝑋𝑃 𝑝𝑠𝑖, 1 − 𝑃 𝑝𝑠𝑖, 𝑝𝑠𝑖, 𝑝𝑒𝑖
¨ And because the price of zero coupon bond is respected:
¨ 𝐸𝑋𝑃 𝑝𝑠𝑖,
()* "
&'()* " .0 "
. 𝑓 𝑖 = 1 − 𝑃 0, 𝑝𝑠𝑖, 𝑝𝑒𝑖 = 1 −
&
&'()* " .0 "
=
()* "
&'()* " .0 "
. 𝑓 𝑖
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Quick summary
¨ In the rolling numeraire measure,
– PV of fixed cashflows are conserved (Expected value of a fixed cashflow as the curve evolves in a stochastic manner
over time will converge to the fixed amount at the payment date)
– Price of zero coupon bonds are conserved
– Price of bonds are conserved (the price of a bond will change over time, but ON AVERAGE the price you should be
willing to pay for this bond is the price you can compute today using today’s curve, because the price of a bond
exhibits no convexity with respect to the discount curve changing, AS LONG as the discount curve changes in a manner
that respect the Arbitrage free condition, that is that a contract where you invest X today to get Y at time T, is the same
(equivalent, on average), as any contract where you invest X today, get the proceeds at some point in time in the
future, then reinvest them until T.)
– This is either painfully obvious or really deep.
– The arbitrage free assumption does NOT know about credit
– The arbitrage free assumption does NOT know about individual utility function (also called time-indifferent, it assumes
that market participants are indifferent about receiving X today versus Y at time T, where the ratio Y/X is the price of
today zero coupon bond maturing at time T, and that price will be conserved over time)
– The Floating leg of a swap ALSO exhibit zero convexity against the discount factor curve, because it can be expressed as
a linear function of discount factors, thanks to the amazing trick x=x+1-1
¨ Other markets (equity, commodities,..) do NOT have such a strong underlying constraint that needs to be
respected.. In HJM for example, we will show that respecting the arbitrage enforces zero possible choice
for the drift, once the volatility is known, everything else is.
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Outstanding questions
¨ How do you incorporate credit and utility function? Do you do it after the fact? Do you keep the
arbitrage free framework?
¨ It was almost by chance that a regular swap PV can be expressed as a linear sum of discount factor.
What happens say if the rate for each period does not set at the beginning? Or does not line up with the
period dates? Will there be convexity then? Meaning that the PV of a swap will not be a linear function
of the discount factors, and when those will evolve over time, the expected value of the PV will not the
one computed on today’s curve. Note, again at the risk of sounding obvious that in that case today yield
curve is NOT sufficient in order to be able to price such a swap, in particular you will need information
about the future dynamics of rates (volatility, correlation if include in the model, skew,…)
¨ Do you really believe in an arbitrage free world in the first place?
¨ Do you view the arbitrage free framework as the first order solution in a more complicated expansion?
How stable is that first order?
¨ Your turn
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161
Can we hedge an option?
¨ Black-Scholes (1973)… Finally!
¨ A call can be continuously dynamically Delta hedged (*) with:
- a cash position.
- a position in the underlying.
¨ The portfolio (option + hedge) on average will return the risk-free rate.
¨ See the homework tonight.
(*) within the Black-Scholes world.

Luc_Faucheux_2020
Understanding Delta Hedging
$100
$90 with 50% probability
¨ Let’s use the example of a stock...

Luc_Faucheux_2020
How Much Should you Pay for the Call ?
¨ Call struck at $100 expiring tomorrow
Call = ?

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How Much do you Want to Pay for the Call ?
¨ Call = (1/2)*($10) + (1/2)*($0) = $5

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Call PL Without Delta Hedging
$5
$0 with 50% chance à (-$5)
$10 with 50% chance à (+$5)

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Can we Delta Hedge?
¨ If we are short one unit of stock we will make $10 if the market goes down, and lose $10 if
the market rallies
¨ We need to Delta hedge with half a unit of the stock
¨ Delta = 50%

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Understanding Risk Neutral Valuation
$100
¨ Let’s change the game…

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So How Much for this Call?
¨ Still $5? Well… the market has more of a chance of a bigger move, so the option should be
more valuable…

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The Beauty of Delta Hedging
P=C–D.S
P(down) = 0 – D.(-40)
P(up) = 10 – D.10

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Delta Hedging is Removing the Risk
P(up) = P(down)
10–D.10 = 0 – D.(-40)
D = (10/50) = 0.2

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Once we Know the Delta we Know the Option Price
P(up) = 10 – (0.2) x 10 = $8
P(down) = – (0.2) x (-40) = $8
The call is worth $8!

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A Few Remarks
¨ Delta hedging eliminates the exposure to the market
¨ Delta hedging reduces the variance of the PL, not the expected PL
¨ Once we know the Delta, we can calculate the option price (this is what Black and Scholes
did in 1973)
¨ Delta hedging generates positive PL when being long Gamma

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Risk Neutral World…
¨ Delta hedging brings us into a risk-neutral world
¨ If the call is worth $8, then the probabilities have to be 80% and 20%…
$100

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Risk Neutral Probabilities…
¨ These are the probabilities that make the expected value of the stock price $100 (risk-free
rate return)
¨ In a risk-neutral world, all assets return the risk-free rate
¨ Black-Scholes showed that Delta hedging implies that we have no choice for the option
price: we have to price the option using the risk-neutral probabilities (we have to place
ourselves in a risk-neutral world)
¨ Black-Sholes simultaneously solves for the delta and for the call price
¨ In that case the portfolio of the call and the Delta hedge will also return the risk-free rate…
(ONLY on average!)

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We Understood a lot!
¨ Portfolio replication
¨ Delta hedging
¨ Risk-neutral valuation
¨ Black-Scholes

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Delta Hedging is Good…
¨ It allows us to price an option (Black-Scholes)
¨ It reduces the PL variance
¨ It allows us to capture the Time Value of the option (lock-in the value of the option)

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Delta Hedging and Delta Re-Hedging
¨ Being long an option is being long convexity
¨ Being long an option is being long Gamma
¨ We will get longer the market when it rallies
¨ We will get shorter the market when it sells off
¨ When re-hedging…
¨ We will sell the market when it rallies (SELL HIGH)
¨ We will buy the market when it sells off (BUY LOW)

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Delta Hedging
¨ By re-hedging the option over time, we accumulate positive PL on the hedge…
¨ We also lose money from time decay
¨ Delta hedging allows us to realize the Time Value of the option

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Delta Hedging
¨ If the market moves more than the implied volatility at which we bought it, we will realize
more money in the Delta hedging than we will lose in time decay
¨ If the market moves less than the implied volatility at which we bought it, we will realize less
money in the Delta hedging than we will lose in time decay
(move more or less in the regions of high convexity)

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180
Quick summary (when do you have to come talk to your
favorite option trader and not to your swap trader).
¨ A bond is a linear sum of discount factors à Swaps.
¨ A FRA is a linear function of discount factors à Swaps.
¨ A future is a convex function of discount factors à Option (?!).
¨ A swap is a linear function of discount factors à Swaps.
¨ A swap resetting in arrears is convex à Option.
¨ A swap on a CMS index is convex à Option.
¨ FINALLY… an option is a convex payoff à Option (duh!).
¨ A swap denominated in USD dollars whose CSA (Credit Support Annex) states that the collateral is Euro cash,
with a zero floor embedded in the CSA
¨ A swap that is not at-the-money facing a risky counterparty
¨ A swap that is at-the-money facing a risky counterparty

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Black-Sholes revisited
¨ Ito lemma and Ito calculus
¨ Black Sholes derivation
¨ Issues around Black-Sholes
181

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Why this whole thing about Ito calculus?
¨ Hull – White chapters 13, 14 and 15
¨ People got excited about stock prices trading as a percentage (people expect a “return”),
p.306, and so what mattered was the return of the stock 𝑆, or ⁄∆𝑆 𝑆
¨ So then they started writing things like : 𝑑𝑆 = 𝜇. 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧, (p.307)
¨ And then they got stuck, because 𝑑𝑥 = 𝑎 𝑥, 𝑡 . 𝑑𝑡 + 𝑏 𝑥, 𝑡 . 𝑑𝑧, where 𝑏 𝑥, 𝑡 is a function
of the stochastic variable 𝑥, is not something we know how to deal with (p.306, and no, it is
NOT a “small approximation” as they claim)
¨ So you need to use a ”guess” on how to deal with 𝑏 𝑥, 𝑡 , which is why it is called a “lemma”
¨ Ito (1951) guessed that you can write 𝑑𝑥 = 𝑎 𝑥, 𝑡 . 𝑑𝑡 + 𝑏 𝑥, 𝑡 . 𝑑𝑧
as ∆𝑥 = 𝑎 𝑥, 𝑡 . ∆𝑡 + 𝑏 𝑥, 𝑡 . ∆𝑧
or δ𝑥 = 𝑎 𝑥, 𝑡 . 𝛿𝑡 + 𝑏 𝑥, 𝑡 . 𝛿𝑧
¨ That seems like a good guess but then the rules of calculus are no longer applicable, you can
barely derive without making a mistake, and forget about trying to integrate (p. 311)
¨ Now you get this weird thing where 𝑑 𝑙𝑛𝑆 = ( ⁄𝑑𝑆 𝑆) − ( ⁄𝜎' 2). 𝑑𝑡, (p.312)
18
2

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Hull p.306, 10th edition
¨ Ito process
¨ A further type of stochastic process, known as an Ito process, can be defined. This is a
generalized Wiener process in which the parameters 𝑎 and 𝑏 are functions of the value of
the underlying variable 𝑥 and time 𝑡. An Ito process can therefore be written as:
𝑑𝑥 = 𝑎 𝑥, 𝑡 . 𝑑𝑡 + 𝑏 𝑥, 𝑡 . 𝑑𝑧
¨ Both the expected drift rate and variance rate of an Ito process are liable to change over
time. They are function of the current value of 𝑥 and the current time 𝑡. In a small time
interval between 𝑡 and 𝑡 + ∆𝑡 , the variable change from 𝑥 to (𝑥 + ∆𝑥), where:
∆𝑥 = 𝑎 𝑥, 𝑡 . ∆𝑡 + 𝑏 𝑥, 𝑡 . ∆𝑧
¨ This equation involves a small approximation. It assumes that the drift and variance rate of
x remains constant, equal to their values a time 𝑡, in the time interval between 𝑡 and
𝑡 + ∆𝑡 , even though during that time interval the variable 𝑥 has ”jumped” by ∆𝑥
¨ There is somewhat of a recursion in the above argument (Zeno’s arrow paradox)
¨ There is also the fact that the discrete version is NOT and approximation, it is a choice (Ito)
as opposed to an equally valid other choice (Stratonovitch for example)
183

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Hull p.329
¨ 𝑑𝑆 = 𝜇. 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧
¨ We note 𝑓 the price of a derivative contingent on 𝑆, like the price of a call option.
¨ Applying Ito lemma within Ito calculus,
¨ 𝑑𝑓 𝑆, 𝑡 =
!(
!)
. 𝑑𝑆 +
!(
!#
𝑑𝑡 +
&
'
!!(
!)! . (𝑑𝑆)'+
&
'
!!(
!#! . (𝑑𝑡)'+
!!(
!#!)
. (𝑑𝑡. 𝑑𝑆) + 𝒪 …
¨ Expressing this in terms of the variables 𝑑𝑡 and 𝑑𝑧
¨ 𝑑𝑓 𝑆, 𝑡 =
!(
!)
. 𝜇. 𝑆. 𝑑𝑡 +
!(
!#
𝑑𝑡 +
&
'
!!(
!)! . 𝜎'. 𝑆'. 𝑑𝑡 +
!(
!)
. 𝜎. 𝑆. 𝑑𝑧
¨ 𝑑𝑓 𝑆, 𝑡 =
!(
!)
. 𝜇. 𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' . 𝑑𝑡 +
!(
!)
184

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Hull p.330
¨ 𝑑𝑓 𝑆, 𝑡 =
!(
!)
. 𝜇. 𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' . 𝑑𝑡 +
!(
!)
¨ This is the SDE that 𝑓 𝑆, 𝑡 follows
¨ The term in front of the stochastic driver is quite complicated:
!(
!)
. 𝜎. 𝑆
¨ What if we were to create a portfolio composed of this contingent claim 𝑓 𝑆, 𝑡 and some
units of the underlying stock 𝑆?
¨ More precisely let’s construct a portfolio Π = 𝑓 − (𝐷𝑒𝑙𝑡𝑎). 𝑆
¨ Π = 𝑓 − (𝐷𝑒𝑙𝑡𝑎). 𝑆
¨ 𝑑Π = 𝑑𝑓 − (𝐷𝑒𝑙𝑡𝑎). 𝑑𝑆 and 𝑑𝑆 = 𝜇. 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧
¨ 𝑑Π =
!(
!)
. 𝜇. 𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' − 𝐷𝑒𝑙𝑡𝑎 . 𝜇. 𝑆 . 𝑑𝑡 + (
!(
!)
. 𝜎. 𝑆 − 𝐷𝑒𝑙𝑡𝑎 . 𝜎. 𝑆). 𝑑𝑧
185

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The beauty of Delta hedging: from SDE to PDE
¨ 𝑑Π =
!(
!)
. 𝜇. 𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' − 𝐷𝑒𝑙𝑡𝑎 . 𝜇. 𝑆 . 𝑑𝑡 + (
!(
!)
¨ If we fix 𝐷𝑒𝑙𝑡𝑎 =
!(
!)
, we then obtain
¨ 𝑑Π =
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' . 𝑑𝑡
¨ This is NOT and SDE anymore !! Does not depend on 𝑑𝑧. Does not depend on 𝜇
¨ This is wonderful, we do not have to worry about Ito, Stratonovitch, and all that stochastic
calculus that no one understands (well we still do because as soon as we change the
function we will still have to deal with Ito lemma)
¨ The portfolio is “riskless”, the change in the value of the portfolio does not depends on the
risk driver 𝑑𝑧
¨ Because the portfolio is “riskless”, it should return the same rate as the risk-free rate 𝑟
¨ 𝑑Π = 𝑟. Π . 𝑑𝑡
186

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The final diffusion equation
¨ 𝑑Π =
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' . 𝑑𝑡
¨ 𝑑Π = 𝑟. Π . 𝑑𝑡
¨ Π = 𝑓 − 𝐷𝑒𝑙𝑡𝑎 . 𝑆 = 𝑓 −
!(
!)
. 𝑆
¨
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' = 𝑟. (𝑓 −
!(
!)
. 𝑆)
¨
!(
!#
+ 𝑟. 𝑆.
!(
!)
+
&
'
!!(
!)! . 𝜎'. 𝑆' = 𝑟. 𝑓
¨ This is the Black-Sholes-Merton equation.
¨ It is a diffusion equation, subject to the proper boundary conditions
¨ for a call, at maturity 𝑡 = 𝑇, 𝑓 𝑆, 𝑇 = 𝑀𝐴𝑋(𝑆 − 𝐾, 0)
187

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An example of going back and checking our assumptions
¨ Hold on a second, 𝐷𝑒𝑙𝑡𝑎 =
!(
!)
, and so is ALSO a function of 𝑆 and 𝑡
¨ Π = 𝑓 − (
!(
!)
). 𝑆
¨ 𝑑Π = 𝑑𝑓 −
!(
!)
. 𝑑𝑆 − 𝑆. 𝑑(
!(
!)
(𝑆, 𝑡))
¨ 𝑑Π = 𝑑𝑓 −
!(
!)
. 𝑑𝑆 − 𝑆. {
!!(
!)! . 𝑑𝑆 +
!!(
!)!#
. 𝑑𝑆. 𝑑𝑡 +
&
'
.
!=(
!)= . 𝑑𝑆. 𝑑𝑆}
¨ 𝑑Π =
!(
!)
. 𝜇. 𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' . 𝑑𝑡 +
!(
!)
. 𝜎. 𝑆. 𝑑𝑧 −
!(
!)
. 𝑑𝑆 − 𝑆. {
!!(
!)! . 𝑑𝑆 +
!!(
!)!#
. 𝑑𝑆. 𝑑𝑡 +
&
'
.
!=(
!)= . 𝑑𝑆. 𝑑𝑆}
¨ 𝑑Π =
!(
!)
. 𝜇. 𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' . 𝑑𝑡 +
!(
!)
. 𝜎. 𝑆. 𝑑𝑧 −
!(
!)
. (𝜇. 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧) −
𝑆. {
!!(
!)! . (𝜇. 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧) +
&
'
.
!=(
!)= . 𝜎'. 𝑆'. 𝑑𝑡}
188

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We just made things worse
¨ 𝑑Π =
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' . 𝑑𝑡 − 𝑆. {
!!(
!)! . (𝜇. 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧) +
&
'
.
!=(
!)= . 𝜎'. 𝑆'. 𝑑𝑡}
¨ Argghh !!!! It is still and SDE with now higher order terms.
¨ In particular
!!(
!)! . (𝜎. 𝑆. 𝑆). 𝑑𝑧, and now also a
!=(
!)= ???
¨ So this is really not working as we hoped
189

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So what is wrong ?
¨ Footnote in Hull p.330
¨ This derivation in equation (15.16) is not completely rigorous. We need to justify ignoring
the changes in
!(
!)
(𝑆, 𝑡) in the time interval between 𝑡 and 𝑡 + ∆𝑡 in equation (15.13). A
more rigorous derivation involves setting up a self-financing portfolio (i.e. a portfolio that
requires no infusion or withdrawal of money)
¨ The bottom line is : 𝑓 𝑆, 𝑡 is “truly” stochastic and the full Ito lemma applies
¨ 𝐷𝑒𝑙𝑡𝑎 =
!(
!)
(𝑆, 𝑡) is not truly stochastic because in practice, it is discrete, it is the amount
of hedge we put in the portfolio at a given point in time, and gets “re-balanced” to the new
value of
!(
!)
(𝑆 + ∆𝑆, 𝑡 + ∆𝑡) , it does not move the way the underlier 𝑆 or the contingent
claim 𝑓(𝑆, 𝑡) evolves
¨ It is still less than satisfying as a rigorous answer
190

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Hull p.329
¨ The stock price follows the process 𝑑𝑆 = 𝜇. 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧
¨ The short selling of securities with full use of proceeds is permitted
¨ There are no transaction costs or taxes. All securities are perfectly divisible
¨ There are no dividends during the life of the derivative
¨ There are no riskless arbitrage opportunities
¨ Security trading is continuous and delta hedging is continuous
¨ The risk free rate of interest is constant and the same for all maturities
191

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Assumption I
¨ The stock price follows the process 𝑑𝑆 = 𝜇. 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧
192

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Assumption II
¨ The short selling of securities with full use of proceeds is permitted
193

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Assumption III
¨ There are no transaction costs or taxes. All securities are perfectly divisible
194

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Assumption IV
¨ There are no dividends during the life of the derivative
¨ That could be incorporated using some simplifying assumptions.
¨ Usual assumption is that the asset (stock) receives a continuous and constant dividend yield
(𝐷𝑌)
¨ This means that over a time period (𝑑𝑡) the holder of the asset 𝑆 receives an amount
𝐷𝑌 . 𝑆. 𝑑𝑡
¨ let’s construct a portfolio Π = 𝑓 − (𝐷𝑒𝑙𝑡𝑎). 𝑆
¨ 𝑑Π =
!(
!)
. 𝑑𝑆 +
!(
!#
. 𝑑𝑡 +
&
'
!!(
!)! . 𝜎'. 𝑆'. 𝑑𝑡 − 𝐷𝑒𝑙𝑡𝑎 . 𝑑𝑆− 𝐷𝑒𝑙𝑡𝑎 .(DY).S.dt
¨ Compared to the previous : 𝑑Π =
!(
!)
. 𝑑𝑆 +
!(
!#
. 𝑑𝑡 +
&
'
!!(
!)! . 𝜎'. 𝑆'. 𝑑𝑡 − 𝐷𝑒𝑙𝑡𝑎 . 𝑑𝑆
195

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Assumption IV - b
¨ 𝑑Π =
!(
!)
. 𝜇. 𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' − 𝐷𝑒𝑙𝑡𝑎 . 𝜇. 𝑆 . 𝑑𝑡 + (
!(
!)
¨ Becomes
¨ 𝑑Π =
!(
!)
. 𝜇. 𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' − 𝐷𝑒𝑙𝑡𝑎 . 𝜇. 𝑆 − 𝐷𝑒𝑙𝑡𝑎 . 𝐷𝑌 . 𝑆 . 𝑑𝑡 + (
!(
!)
. 𝜎. 𝑆 −
𝐷𝑒𝑙𝑡𝑎 . 𝜎. 𝑆). 𝑑𝑧
!(
!)
, we then obtain
¨ 𝑑Π =.
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' −
!(
!)
. 𝐷𝑌 . 𝑆 𝑑𝑡 = 𝑟. Π. 𝑑𝑡 = 𝑟(𝑓 −
!(
!#
. 𝑆). 𝑑𝑡
¨
!(
!#
+ 𝑟. 𝑆.
!(
!)
+
&
'
!!(
!)! . 𝜎'. 𝑆' = 𝑟. 𝑓 becomes
!(
!#
+ [𝑟 − 𝐷𝑌 ]. 𝑆.
!(
!)
+
&
'
!!(
!)! . 𝜎'. 𝑆' = 𝑟. 𝑓
196

Luc_Faucheux_2020
Assumption IV - c
¨ A constant dividend yield is akin to changing the risk-free rate 𝑟 to [𝑟 − 𝐷𝑌 ] but NOT
everywhere, only in the part of the equation that relates to the Delta hedging
¨ So essentially 𝑑𝑆 = 𝜇. 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧 becomes 𝑑𝑆 = (𝜇 − 𝐷𝑌 ). 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧
¨ Or in the risk neutral valuation that was possible from Delta hedging
¨ 𝑑𝑆 = 𝑟. 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧 becomes 𝑑𝑆 = (𝑟 − 𝐷𝑌 ). 𝑆. 𝑑𝑡 + 𝜎. 𝑆. 𝑑𝑧
¨ BUT this is only an adjustment to the stock price process (the portfolio as other riskless
securities will still return the risk-free 𝑟)
197

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Assumption V
¨ There are no riskless arbitrage opportunities
198

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Assumption VI
¨ Security trading is continuous and delta hedging is continuous
¨ It is not clear what “continuous” actually means even though is it being used quite a lot in
textbooks.
¨ We know for a fact that we need to write something like 𝐷𝑒𝑙𝑡𝑎 =
!(
!)
¨ But also we treat 𝐷𝑒𝑙𝑡𝑎 as a ”constant” over the stochastic jump
¨ So we mathematically did NOT treat 𝐷𝑒𝑙𝑡𝑎 as a full stochastic variable, or a continuous
Brownian process even though we are using Ito lemma on 𝑓 𝑆, 𝑡
199

Luc_Faucheux_2020
Assumption VII
¨ The risk free rate of interest is constant and the same for all maturities
200

Luc_Faucheux_2020
Risk-Neutral probabilities
!(
!)
, we then obtain 𝑑Π =
!(
!#
+
&
'
!!(
!)! . 𝜎'. 𝑆' . 𝑑𝑡 = 𝑟. Π . 𝑑𝑡
¨ So the option price follows an equation that does NOT depend on the drift 𝜇 anymore
¨ It will depend on the risk-free rate 𝑟
¨ In a way different people could have different estimate for the drift of the stock, they will
still agree on the same option price, and will have to use the risk-free drift (risk neutral
probabilities) to value the option.
¨ This is seen again in the currency options, where traders in each of their native currencies
(numeraire) will obviously have different local risk free rates, yet will agree on the option
price on the currency pair (“2-countries paradox”)
201

Luc_Faucheux_2020
Delta hedging
¨ Delta hedging does not change the expected profit, it changes the distribution of tose
expected profit
¨ Delta hedging brings the option price into the risk-neutral world (does not depend on the
drift (𝜇. 𝑆) anymore
¨ The Delta hedging argument assumes that “rebalancing” is possible at no cost
202

Luc_Faucheux_2020
Changing the variables in order to simplify the problem
𝜕𝑓
𝜕𝑡
+ 𝑟. 𝑆.
𝜕𝑓
𝜕𝑆
+
1
2
𝜕' 𝑓
𝜕𝑆' . 𝜎'. 𝑆' = 𝑟. 𝑓
¨ Because we are receiving the payoff at time 𝑡 = 𝑇, it seems natural to write
¨ 𝑓 𝑆, 𝑡 = 𝑔(𝑆, 𝑡)𝑒*+(-*#)
¨ This only changes the derivatives with respect to time
¨
!(
!#
=
!/
!#
. 𝑒*+(-*#) + 𝑟. 𝑓 𝑆, 𝑡
¨
!(
!)
=
!/
!)
. 𝑒*+(-*#)
¨
!!(
!)! =
!!/
!)! . 𝑒*+(-*#)
¨
!/
!#
. 𝑒*+(-*#) + 𝑟. 𝑓 𝑆, 𝑡 + 𝑟. 𝑆.
!/
!)
. 𝑒*+(-*#) +
&
'
!!/
!)! . 𝑒*+(-*#). 𝜎'. 𝑆' = 𝑟. 𝑓
203

Luc_Faucheux_2020
Simplifying the Black Sholes equation II
¨
!/
!#
. 𝑒*+(-*#) + 𝑟. 𝑓 𝑆, 𝑡 + 𝑟. 𝑆.
!/
!)
. 𝑒*+(-*#) +
&
'
!!/
!)! . 𝑒*+(-*#). 𝜎'. 𝑆' = 𝑟. 𝑓
¨
!/
!#
+ 𝑟. 𝑆.
!/
!)
+
&
'
!!/
!)! . 𝜎'. 𝑆' = 0
¨ Looks nicer
¨ Because we are looking at something that is going to diffuse “backward” from the payoff at
expiry, again it seems natural to put ourselves in the time frame 𝜏 = 𝑇 − 𝑡
¨
!/
!#
+ 𝑟. 𝑆.
!/
!)
+
&
'
!!/
!)! . 𝜎'. 𝑆' = 0
¨
!/
!0
= 𝑟. 𝑆.
!/
!)
+
&
'
!!/
!)! . 𝜎'. 𝑆'
¨ Starting to look like a diffusion equation except for a first order term (drift)
204

Luc_Faucheux_2020
Simplifying the Black Sholes equation III
𝜕𝑔
𝜕𝜏
= 𝑟. 𝑆.
𝜕𝑔
𝜕𝑆
+
1
2
𝜕' 𝑔
𝜕𝑆' . 𝜎'. 𝑆'
¨ The diffusion coefficient scales as the square of the asset
¨ So clearly we would like to simplify that and have something like
!!/
!)! . 𝑆'~𝑐𝑡𝑒
¨ So
!!/
!)! ~𝑆*', which reminds us of
!/
!)
~𝑆*&, and 𝑔~𝐿𝑛(𝑆)
¨ So let’s have a new variable 𝜉 = 𝐿𝑛(𝑆)
¨
!/
!)
=
!/
!1
.
&
)
=
!/
!1
. 𝑒*1
¨
!!/
!)! =
!
!)
.
!/
!)
=
!
!)
.
!/
!1
.
&
)
=
*&
)! .
!/
!1
+
&
)
.
!
!)
.
!/
!1
=
*&
)! .
!/
!1
+
&
)
.
!!/
!1! .
&
)
¨
!!/
!)! = −
!/
!1
. 𝑒*'1 +
!!/
!1! . 𝑒*'1
205

Luc_Faucheux_2020
Simplifying the Black Sholes equation IV
¨
!/
!0
= 𝑟. 𝑆.
!/
!)
+
&
'
!!/
!)! . 𝜎'. 𝑆' with 𝜉 = 𝐿𝑛(𝑆)
¨
!/
!)
=
!/
!1
.
&
)
=
!/
!1
. 𝑒*1
¨
!!/
!)! = −
!/
!1
. 𝑒*'1 +
!!/
!1! . 𝑒*'1
¨
!/
!0
= (𝑟 −
&
'
. 𝜎').
!/
!1
+
&
'
!!/
!1! . 𝜎'
¨ Note that this is a little more nicely symmetrical around 0 as if 𝑆 ∈ [0, +∞] we now have
ξ ∈ [−∞, +∞]
¨ Note also that now the coefficients of this equation are NOT function of the variable ξ
¨ We are certainly getting somewhere
206

Luc_Faucheux_2020
Simplifying the Black Sholes equation V
𝜕𝑔
𝜕𝜏
= (𝑟 −
1
2
. 𝜎').
𝜕𝑔
𝜕𝜉
+
1
2
𝜕' 𝑔
𝜕𝜉' . 𝜎'
¨ We would like to get rid of the first term (first order or also drift)
¨ If there was no second term (diffusive term), the equation would just read
¨
!/
!0
= (𝑟 −
&
'
. 𝜎').
!/
!1
, which if we note 𝑅2 = (𝑟 −
&
'
. 𝜎'), indicates that we should look for
something like 𝑥 = 𝜉 + 𝑅2. τ for a function ℎ(𝜏, 𝑥)
¨ More formally we want to got from 𝑔(𝜏, 𝜉) to ℎ(𝜏3, 𝑥) where:
¨ d
𝜏3 = 𝜏
𝑥 = 𝜉 + 𝑅2. τ
207

Luc_Faucheux_2020
Simplifying the Black Sholes equation VI
𝜕𝑔
𝜕𝜏
= (𝑟 −
1
2
. 𝜎').
𝜕𝑔
𝜕𝜉
+
1
2
𝜕' 𝑔
𝜕𝜉' . 𝜎'
¨ 𝑔 𝜏, 𝜉 = ℎ(𝜏3, 𝑥)
¨ 𝜏3 = 𝜏
!03
!0
= 1
!03
!1
= 0
¨ 𝑥 = 𝜉 + 𝑅2. τ
!4
!0
= 𝑅2
!4
!1
= 1
¨
!/
!0
=
!5
!03
.
!03
!0
+
!5
!4
.
!4
!0
=
!5
!03
+ 𝑅2.
!5
!4
¨
!/
!1
=
!5
!03
.
!03
!1
+
!5
!4
.
!4
!1
=
!5
!4
¨
!!/
!1! =
!
!1
.
!/
!1
=
!
!1
.
!5
!4
=
!
!0>
!5
!4
.
!03
!1
+
!
!4
!5
!4
.
!4
!1
=
!
!4
!5
!4
=
!!5
!4!
208

Luc_Faucheux_2020
Simplifying the Black Sholes equation VII
𝜕𝑔
𝜕𝜏
= (𝑟 −
1
2
. 𝜎').
𝜕𝑔
𝜕𝜉
+
1
2
𝜕' 𝑔
𝜕𝜉' . 𝜎'
¨ 𝑔 𝜏, 𝜉 = ℎ(𝜏3, 𝑥)
¨
!5
!03
+ 𝑅2.
!5
!4
= 𝑅2.
!5
!4
+
&
'
!!5
!4! . 𝜎'
𝜕ℎ
𝜕𝜏′
=
1
2
𝜕'ℎ
𝜕𝑥' . 𝜎'
¨ That is a nice looking diffusion equation with constant diffusion coefficients, we know how
to deal with it, we know a solution (Gaussian distribution), we can also use linearity
arguments (A linear combination of solutions is also a solution)
209

Luc_Faucheux_2020
Simplifying the Black Sholes equation VIII
𝜕𝑓
𝜕𝑡
+ 𝑟. 𝑆.
𝜕𝑓
𝜕𝑆
+
1
2
𝜕' 𝑓
𝜕𝑆' . 𝜎'. 𝑆' = 𝑟. 𝑓
¨ 𝑓 𝑆, 𝑡 = ℎ 𝜏3, 𝑥 . 𝑒*+(-*#)
¨ 𝜏′ = 𝑇 − 𝑡
¨ 𝑥 = 𝐿𝑛 𝑆 + (𝑟 −
&
'
. 𝜎')(𝑇 − 𝑡)
𝜕ℎ
𝜕𝜏′
=
1
2
𝜕'ℎ
𝜕𝑥' . 𝜎'
210

Luc_Faucheux_2020
Solution of the diffusion equation I
¨ We could use what we know about the diffusion equation and use the Gaussian function as
a solution.
¨ We can also keep changing the variables in order to check one more time that our math is
right
¨
!5
!0
=
&
'
!!5
!4! . 𝜎'
¨ We know that the Gaussian is only a function of a single variable that is itself a function of
the time and space (𝜏 and 𝑥), so we kind of cheat and look for something like :
¨ ℎ 𝜏, 𝑥 = 𝜏6. 𝜑((𝑥 − 𝑥7)/𝜏8)
¨ Where 𝛼, 𝛽and 𝑥7are constant, and we are dealing with a single variable function 𝜑
¨ In that case we hope to be dealing with an ODE (Ordinary Differential Equation) as opposed
to a PDE (Partial Differential Equation)
¨ Note that for sake of simplicity we dropped the ‘ and just use 𝜏 instead of 𝜏′
211

Luc_Faucheux_2020
Solution of the diffusion equation II
212
¨
!5
!0
=
&
'
𝜎' !!5
!4!
¨ ℎ 𝜏, 𝑥 = 𝜏6. 𝜑((𝑥 − 𝑥7)/𝜏8)
¨
!5
!0
= 𝛼. 𝜏6*&. 𝜑
4*4?
0@ − 𝜏6.
!
!0
(
4*4?
0@ ). 𝜑′
4*4?
0@
¨
!5
!0
= 𝛼. 𝜏6*&. 𝜑
4*4?
0@ − 𝜏6. 𝛽.
4*4?
0@AB . 𝜑′
4*4?
0@
¨
!5
!4
= 𝜏6. 𝜏*8. 𝜑′
4*4?
0@
¨
!!5
!4! = 𝜏6. 𝜏*8. 𝜏*8. 𝜑′′
4*4?
0@
¨ 𝛼. 𝜏6*&. 𝜑
4*4?
0@ − 𝜏6. 𝛽.
4*4?
0@AB . 𝜑′
4*4?
0@ =
&
'
𝜎'. 𝜏6. 𝜏*8. 𝜏*8. 𝜑′′
4*4?
0@

Luc_Faucheux_2020
Solution of the diffusion equation III
213
¨ Regrouping the terms we get:
¨ 𝛼. 𝜏6*&. 𝜑
4*4?
0@ − 𝜏6. 𝛽.
4*4?
0@AB . 𝜑′
4*4?
0@ =
&
'
𝜎'. 𝜏6. 𝜏*8. 𝜏*8. 𝜑′′
4*4?
0@
¨ 𝜏6*&. 𝛼. 𝜑
4*4?
0@ − 𝜏6*&. 𝛽.
4*4?
0@ . 𝜑′
4*4?
0@ =
&
'
𝜎'. 𝜏6. 𝜏*8. 𝜏*8. 𝜑′′
4*4?
0@
¨ 𝜏6*&{𝛼. 𝜑
4*4?
0@ − 𝛽.
4*4?
0@ . 𝜑′
4*4?
0@ } = 𝜏6*'8{
&
'
𝜎'. 𝜑′′
4*4?
0@ }
¨ So we would like to remove the terms that are powers of 𝜏, this would make things easier
¨ So if 𝛼 − 1 = 𝛼 − 2. 𝛽, or 𝛽 = 1/2, the equations simplifies to
¨ 𝛼. 𝜑
4*4?
0@ − 𝛽.
4*4?
0@ . 𝜑′
4*4?
0@ =
&
'
𝜎'. 𝜑′′
4*4?
0@

Luc_Faucheux_2020
Solution of the diffusion equation IV
214
¨ We also know from the diffusion part, that we are going to end up with something that is a
probability density function most likely, and so we would like the integral of this over space
for any given time to be a constant (=1), and be conserved.
¨ So this is clearly another constraint that we are at a liberty to enforce
¨ So C = ∫*9
:9
𝜏6. 𝜑
4*4?
0@ . 𝑑𝑥 should be a constant, and should not depend on time
¨ Doing (yet) another change of variable µ =
4*4?
0@
¨ 𝜑
4*4?
0@ . 𝑑𝑥 = 𝜑 𝜇 . 𝑑𝜇.
;4
;<
= 𝜑 𝜇 . 𝑑𝜇. 𝜏8
¨ C = ∫*9
:9
𝜏6. 𝜑
4*4?
0@ . 𝑑𝑥 = ∫*9
:9
𝜏6:8. 𝜑 𝜇 . 𝑑𝜇 = 𝜏6:8 . ∫*9
:9
. 𝜑 𝜇 . 𝑑𝜇

Luc_Faucheux_2020
Solution of the diffusion equation V
215
¨ And so
¨ C = ∫*9
:9
𝜏6. 𝜑
4*4?
0@ . 𝑑𝑥 = ∫*9
:9
𝜏6:8. 𝜑 𝜇 . 𝑑𝜇 = 𝜏6:8 . ∫*9
:9
. 𝜑 𝜇 . 𝑑𝜇
¨ We want to enforce the fact that C is a constant (the solution of the diffusion equation is
such that the density probability is conserved, i.e. we are not losing any particles, which can
happen in systems where for example there are what is called “absorbing boundaries”, or in
other systems like radioactive decays where the number of particles will actually change
with time)
¨ One way to achieve this is to set 𝛼 + 𝛽 = 0
¨ So we now have 𝛽 = 1/2 and α = −1/2
¨ 𝛼. 𝜑
4*4?
0@ − 𝛽.
4*4?
0@ . 𝜑′
4*4?
0@ =
&
'
𝜎'. 𝜑′′
4*4?
0@ can be now written as
¨ −𝜑
4*4?
0@ −
4*4?
0@ . 𝜑′
4*4?
0@ = 𝜎'. 𝜑′′
4*4?
0@

Luc_Faucheux_2020
Solution of the diffusion equation VI
216
¨ We have:
¨ −𝜑
4*4?
0@ −
4*4?
0@ . 𝜑′
4*4?
0@ = 𝜎'. 𝜑′′
4*4?
0@
¨ Using µ =
4*4?
0@ , it is easier to write as
¨ −𝜑 𝜇 − 𝜇. 𝜑′ 𝜇 = 𝜎'. 𝜑′′ 𝜇 , or using the usual notation for ordinary derivatives
¨ −𝜑 𝜇 − 𝜇.
;
;<
𝜑 𝜇 = 𝜎'.
;!
;<! 𝜑 𝜇
¨ −
;
;<
{𝜇. 𝜑 𝜇 } = 𝜎'.
;!
;<! 𝜑 𝜇
¨ 𝜎'.
;!
;<! 𝜑 𝜇 +
;
;<
𝜇. 𝜑 𝜇 = 0

Luc_Faucheux_2020
Solution of the diffusion equation VII
217
¨ We have:
¨ 𝜎'.
;!
;<! 𝜑 𝜇 +
;
;<
𝜇. 𝜑 𝜇 = 0
¨
;
;<
𝜎'.
;
;<
𝜑 𝜇 + 𝜇. 𝜑 𝜇 = 0
¨ 𝜎'.
;
;<
𝜑 𝜇 + 𝜇. 𝜑 𝜇 = 𝑐𝑡𝑒
¨ Let’s make our life easier and set the constant to 0
¨ 𝜎'.
;
;<
𝜑 𝜇 + 𝜇. 𝜑 𝜇 = 0

Luc_Faucheux_2020
Solution of the diffusion equation IX
218
¨ We have:
¨ 𝜎'.
;
;<
𝜑 𝜇 + 𝜇. 𝜑 𝜇 = 0
¨
;
;<
𝜑 𝜇 = −𝜎*'. 𝜇. 𝜑 𝜇 = 0
¨ 𝜑 𝜇 = A. exp
*<!
'%! + 𝐵 is a solution of this ODE
¨ Again to make our life easier we choose B=0
¨ 𝜑 𝜇 = A. exp
*<!
'%!
¨ We now choose A so that ∫*9
:9
𝜑 𝜇 . 𝑑𝜇 = 1
¨ ∫*9
:9
exp −𝜇' . 𝑑𝜇 = 2𝜋

Luc_Faucheux_2020
Solution of the diffusion equation X
219
¨ We have:
¨ ∫*9
:9
exp −𝜇' . 𝑑𝜇 = 2𝜋
¨ So 𝜑 𝜇 =
&
'=%!
. exp
*<!
'%! and µ =
4*4?
0@ with 𝛽 = 1/2
¨ 𝜑 𝑥, 𝑥2, 𝜏 =
&
'=%!
. exp
*(4*4C)!
'%!0
¨ ℎ 𝜏, 𝑥 = 𝜏6. 𝜑((𝑥 − 𝑥7)/𝜏8) with 𝛽 = 1/2 and α = −1/2
¨ ℎ 𝑥, 𝑥2, 𝜏 =
&
'=%!0
. exp
*(4*4C)!
'%!0
!5
!0
=
&
'
𝜎' !!5
!4!
¨ ℎ 𝑥, 𝑥′, 𝑡 =
&
>=?#
. 𝑒𝑥𝑝(−
(4*4>)!
>?#
) with 𝐷 = 𝜎'/2
!5
!#
= 𝐷' !!5
!4!

Luc_Faucheux_2020
Linearity and propagators I : the Dirac peak
220
¨ We have:
¨ ℎ 𝑥, 𝑥2, 𝜏 =
&
'=%!0
. exp
*(4*4C)!
'%!0
!5
!0
=
&
'
𝜎' !!5
!4!
¨ Limiting case of 𝜏 → 0
¨ Again as we saw when looking at the diffusion equation, saying that something “goes to 0” is
sometimes not enough, remember for a diffusive process we need to have to avoid
unbounded behaviors to have the space variable scale as the root square of the time
variable when going to the continuous limit of a discrete process (when the time step and
the jump both go to 0)
¨ BUT here we do not have such issues, because we are in the regular deterministic calculus
framework, we are looking at the solution of a PDE, and a regular function
¨ NEVERTHELESS, when 𝜏 → 0,
&
'=%!0
→ ∞, but exp
*(4*4C)!
'%!0
→ 0 faster than √𝜏 for any
𝑥 ≠ 𝑥2

Luc_Faucheux_2020
Linearity and propagators II : the Dirac peak
221
¨ We have:
¨ ℎ 𝑥, 𝑥2, 𝜏 =
&
'=%!0
. exp
*(4*4C)!
'%!0
→ δ(𝑥 − 𝑥2) when 𝜏 → 0
¨ The δ(𝑥 − 𝑥2) is called the Dirac function.
¨ It is 0 everywhere except at 𝑥 = 𝑥2
¨ Its integral over 𝑥 is still equal to 1
¨ ∫*9
:9
δ 𝑥 − 𝑥2 . 𝑑𝑥 = 1
¨ The value of δ(𝑥 − 𝑥2) at (𝑥 = 𝑥2) is “infinite” (it is the limit of
&
'=%!0
when 𝜏 → 0 so it
should be viewed as a limit of functions indexed by 𝜏)
¨ It is usually viewed as the “starting value” for the probability density function of a Brownian
process starting at 𝑥 = 𝑥2 at time 𝜏 = 0, which will diffuse into being at time

Luc_Faucheux_2020
Linearity and propagators III : the Dirac peak
222
¨ A useful property of the Dirac function is that
¨ ∫*9
:9
δ 𝑥 − 𝑥2 . 𝑑𝑥 = 1
¨ And so for any function Payoff(𝑥) we also have
¨ ∫*9
:9
δ 𝑥 − 𝑥2 . Payoff 𝑥 . 𝑑𝑥 = Payoff(𝑥2)
¨ Or presented in terms of limit
¨ lim
0→2
{∫*9
:9 &
'=%!0
. exp
*(4*4C)!
'%!0
. Payoff 𝑥 . 𝑑𝑥} = Payoff(𝑥2)
¨ lim
0→2
{∫*9
:9
ℎ(𝑥, 𝑥2, 𝜏). Payoff 𝑥 . 𝑑𝑥} = Payoff(𝑥2) with
!5
!0
=
&
'
𝜎' !!5
!4!
¨ In particular, the function {ℎ 𝑥, 𝑥2, 𝜏 . Payoff(𝑥2)} is a solution of the diffusion equation
since Payoff 𝑥2 is a constant

Luc_Faucheux_2020
Linearity and propagators IV : Re-casting the variables
223
¨ We have:
¨ lim
0→2
{∫*9
:9
ℎ(𝑥, 𝑥2, 𝜏). Payoff 𝑥 . 𝑑𝑥} = Payoff(𝑥2) with
!5
!0
=
&
'
𝜎' !!5
!4!
¨ 𝑓 𝑆, 𝑡 = ℎ 𝜏3, 𝑥 . 𝑒*+(-*#)
¨ 𝜏′ = 𝑇 − 𝑡
¨ 𝑥 = 𝐿𝑛 𝑆 + (𝑟 −
&
'
. 𝜎')(𝑇 − 𝑡)
¨
!(
!#
+ 𝑟. 𝑆.
!(
!)
+
&
'
!!(
!)! . 𝜎'. 𝑆' = 𝑟. 𝑓
¨ The boundary conditions were expressed as 𝑓 𝑆, 𝑇 = Payoff(𝑆)
¨ For example for a call struck at strike 𝐾, Payoff 𝑆 = 𝑀𝐴𝑋(𝑆 − 𝐾, 0)

Luc_Faucheux_2020
Linearity and propagators V : Re-casting the variables
224
¨ We have:
¨ lim
0→2
{∫*9
:9
ℎ(𝑥, 𝑥2, 𝜏). Payoff 𝑥 . 𝑑𝑥} = Payoff(𝑥2)
¨ ℎ 𝑥, 𝑥2, 𝜏 =
&
'=%!0
. exp
*(4*4C)!
'%!0
which is our special solution to the diffusion equation
¨ ℎ(𝑥, 𝜏) = ∫*9
:9
ℎ 𝑥, 𝑥2, 𝜏 . Payoff 𝑥2 . 𝑑𝑥2 is the general solution
¨ 𝑓 𝑆, 𝑡 = ℎ 𝜏3, 𝑥 . 𝑒*+(-*#)
¨ 𝜏′ = 𝑇 − 𝑡
¨ 𝑥 = 𝐿𝑛 𝑆 + (𝑟 −
&
'
. 𝜎')(𝑇 − 𝑡)
¨
!(
!#
+ 𝑟. 𝑆.
!(
!)
+
&
'
!!(
!)! . 𝜎'. 𝑆' = 𝑟. 𝑓 with 𝑓 𝑆, 𝑇 = Payoff(𝑆)

Luc_Faucheux_2020
Linearity and propagators VI : Re-casting the variables
225
¨ Note that Payoff 𝑥2 = Payoff 𝑥2, 𝑇 = 𝑡 = Payoff(𝑥2, 𝜏 = 0)
¨ ℎ(𝑥, 𝜏) = ∫*9
:9
ℎ 𝑥, 𝑥2, 𝜏 . Payoff 𝑥2, 𝜏 = 0 . 𝑑𝑥2 is the general solution
¨ 𝑓 𝑆, 𝑡 = ℎ 𝜏3, 𝑥 . 𝑒*+(-*#)
¨ 𝜏′ = 𝑇 − 𝑡
¨ 𝑥 = 𝐿𝑛 𝑆 + (𝑟 −
&
'
. 𝜎')(𝑇 − 𝑡), 𝑥2 = 𝐿𝑛 𝑆2
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9
ℎ 𝑥, 𝑥2, 𝜏 . Payoff 𝑥2, 𝜏 = 0 . 𝑑𝑥2
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9
ℎ 𝑥, 𝑥2, 𝜏 . Payoff 𝑆2, 𝜏 = 0 . 𝑑(𝐿𝑛(𝑆2))
¨ Note that it is natural to have the Payoff function expressed in term of the stock 𝑆2 rather
than the variable 𝑥2 = 𝐿𝑛 𝑆2

Luc_Faucheux_2020
Linearity and propagators VII : Re-casting the variables
226
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9
ℎ 𝑥, 𝑥2, 𝜏 . Payoff 𝑆2 .
;)C
)C
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9 &
'=%!0
. exp
*(4*4C)!
'%!0
. Payoff 𝑆2 .
;)C
)C
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9 &
'=%!(-*#)
. exp
*(AB ) : +*
B
!
.%! -*# *AD )C )!
'%!(-*#)
. Payoff 𝑆2 .
;)C
)C
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9 &
'=%!(-*#)
. exp
*(AB( ED
DC
): +*
B
!
.%! -*# *AD )C )!
'%!(-*#)
. Payoff 𝑆2 .
;)C
)C

Luc_Faucheux_2020
Linearity and propagators VII : Re-casting the variables
227
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9
ℎ 𝑥, 𝑥2, 𝜏 . Payoff 𝑆2 .
;)C
)C
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9 &
'=%!0
. exp
*(4*4C)!
'%!0
. Payoff 𝑆2 .
;)C
)C
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9 &
'=%!(-*#)
. exp
*(AB ) : +*
B
!
.%! -*# *AD )C )!
'%!(-*#)
. Payoff 𝑆2 .
;)C
)C
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9 &
'=%!(-*#)
. exp
*(AB( ED
DC
): +*
B
!
.%! -*# *AD )C )!
'%!(-*#)
. Payoff 𝑆2 .
;)C
)C

Luc_Faucheux_2020
Linearity and propagators IX :
228
¨ 𝑓 𝑆, 𝑡 = 𝑒*+(-*#). ∫*9
:9 &
'=%!(-*#)
. exp
*(AB( ED
DC
): +*
B
!
.%! -*# *AD )C )!
'%!(-*#)
. Payoff 𝑆2 .
;)C
)C
¨ This is the general version of the Black-Sholes equation
¨ Depending on the functional form of Payoff 𝑆2 , the integration can be done easily in closed
form, but sometimes not
¨ This demonstrates the diffusion “backward” of the terminal payoff function, or boundary
condition
¨ If the stock diffuses “forward”, the equivalent problem is the payoff diffusing backward
¨ Note: this is for the canonical Black-Sholes assuming lognormal distribution for the stock
¨ Similar derivation for a normal process (actually easier)
¨ Similar derivation for a jump process (Poisson instead of Gaussian)

Luc_Faucheux_2020
Black-Sholes in the Normal world
¨ For the sake of simplicity we will keep the same notation
¨ 𝑑𝑆 = 𝜇𝑆. 𝑑𝑡 + 𝜎DFGH. 𝑑𝑧
¨ BEAR in mind that the variables 𝜇 and 𝜎 have a very different meaning
¨ For example the units are different
¨ In a Lognormal model, 𝜎AFI is in (%/year) for an annualized volatility
¨ In a Normal model, 𝜎DFGH is in (UNIT(S)/year) for an annualized volatility
¨ For a stock denominated in $, 𝜎DFGH is in ($/year) for an annualized volatility
¨ For a rate denominated in basis points, 𝜎DFGH is in (bp/year) for an annualized volatility
¨ In practice, ALWAYS ask for the units of volatility
229

Luc_Faucheux_2020
Normal Black-Sholes II
¨ 𝑑𝑆 = 𝜇𝑆. 𝑑𝑡 + 𝜎D. 𝑑𝑧
¨ We note 𝑓 the price of a derivative contingent on 𝑆, like the price of a call option.
¨ Applying Ito lemma within Ito calculus,
¨ 𝑑𝑓 𝑆, 𝑡 =
!(
!)
. 𝑑𝑆 +
!(
!#
𝑑𝑡 +
&
'
!!(
!)! . (𝑑𝑆)'+
&
'
!!(
!#! . (𝑑𝑡)'+
!!(
!#!)
. (𝑑𝑡. 𝑑𝑆) + 𝒪 …
¨ Expressing this in terms of the variables 𝑑𝑡 and 𝑑𝑧
¨ 𝑑𝑓 𝑆, 𝑡 =
!(
!)
. 𝜇𝑆. 𝑑𝑡 +
!(
!#
𝑑𝑡 +
&
'
!!(
!)! . 𝜎D
'. 𝑑𝑡 +
!(
!)
. 𝜎D. 𝑑𝑧
¨ 𝑑𝑓 𝑆, 𝑡 =
!(
!)
. 𝜇𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎D
' . 𝑑𝑡 +
!(
!)
. 𝜎D. 𝑑𝑧
230

Luc_Faucheux_2020
Normal Black-Sholes III
¨ 𝑑𝑓 𝑆, 𝑡 =
!(
!)
. 𝜇𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎D
' . 𝑑𝑡 +
!(
!)
. 𝜎D. 𝑑𝑧
¨ This is the SDE that 𝑓 𝑆, 𝑡 follows
¨ The term in front of the stochastic driver is quite complicated:
!(
!)
. 𝜎D
¨ What if we were to create a portfolio composed of this contingent claim 𝑓 𝑆, 𝑡 and some
units of the underlying stock 𝑆?
¨ More precisely let’s construct a portfolio Π = 𝑓 − (𝐷𝑒𝑙𝑡𝑎). 𝑆
¨ Π = 𝑓 − (𝐷𝑒𝑙𝑡𝑎). 𝑆
¨ 𝑑Π = 𝑑𝑓 − (𝐷𝑒𝑙𝑡𝑎). 𝑑𝑆 and 𝑑𝑆 = 𝜇𝑆. 𝑑𝑡 + 𝜎D. 𝑑𝑧
¨ 𝑑Π =
!(
!)
. 𝜇𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎D
' − 𝐷𝑒𝑙𝑡𝑎 . 𝜇𝑆 . 𝑑𝑡 + (
!(
!)
. 𝜎D − 𝐷𝑒𝑙𝑡𝑎 . 𝜎D). 𝑑𝑧
231

Luc_Faucheux_2020
Normal Black-Sholes IV
¨ 𝑑Π =
!(
!)
. 𝜇𝑆 +
!(
!#
+
&
'
!!(
!)! . 𝜎D
' − 𝐷𝑒𝑙𝑡𝑎 . 𝜇𝑆 . 𝑑𝑡 + (
!(
!)
. 𝜎D − 𝐷𝑒𝑙𝑡𝑎 . 𝜎D). 𝑑𝑧
!(
!)
, we then obtain
¨ 𝑑Π =
!(
!#
+
&
'
!!(
!)! . 𝜎D
' . 𝑑𝑡
¨ This is wonderful, we do not have to worry about Ito, Stratonovitch, and all that stochastic
calculus that no one understands (well we still do because as soon as we change the
function we will still have to deal with Ito lemma)
¨ The portfolio is “riskless”, the change in the value of the portfolio does not depends on the
risk driver 𝑑𝑧
¨ Because the portfolio is “riskless”, it should return the same rate as the risk-free rate 𝑟
¨ 𝑑Π = 𝑟. Π . 𝑑𝑡
232

Luc_Faucheux_2020
Normal Black-Sholes V
¨ 𝑑Π =
!(
!#
+
&
'
!!(
!)! . 𝜎D
' . 𝑑𝑡
¨ 𝑑Π = 𝑟. Π . 𝑑𝑡
¨ Π = 𝑓 − 𝐷𝑒𝑙𝑡𝑎 . 𝑆 = 𝑓 −
!(
!)
. 𝑆
¨
!(
!#
+
&
'
!!(
!)! . 𝜎D
' = 𝑟. (𝑓 −
!(
!)
. 𝑆)
¨
!(
!#
+ 𝑟. 𝑆.
!(
!)
+
&
'
!!(
!)! . 𝜎D
' = 𝑟. 𝑓
¨ This is the Black-Sholes-Merton equation in the NORMAL world
¨ It is a diffusion equation, subject to the proper boundary conditions
¨ for a call, at maturity 𝑡 = 𝑇, 𝑓 𝑆, 𝑇 = 𝑀𝐴𝑋(𝑆 − 𝐾, 0)
¨ We can derive the solution (will do that in the Black-Sholes deck)
233

Luc_Faucheux_2020
234
Some Equations (Lognormal Black-Scholes)
ò¥-
-
=
-=
+=
-=
x
dexN
Tdd
T
T
KFLn
d
dNKdNFTKFC
x
p
s
s
s
s
x
..
2
1
)(
2
1)(
)(.)(.),,,(
)
2
1
(
12
1
21
2

Luc_Faucheux_2020
235
Greeks and Scaling in the Lognormal Model
Greeks Definition Black formula Units Incremental P/L Vega scaling
Delta
F
C
¶
¶
=D )( 1dN ($/bp) )( FdD
Gamma
2
2
F
C
¶
¶
=g )('
1
1dN
TFs
($/bp/bp) 2
)(
2
1
Fdg
TF s2
1
Theta
T
C
¶
¶
=Q )('
2
2dN
T
TKs ($/day) )( TdQ
T2
s
Vega
s¶
¶
=
C
Vega )(' 2dNTK ($/%) )(dsVega 1
Vanna
s¶¶
¶
=
F
C
Vanna
2
)('' 2dN
F
K
s
($/%/bp) ))(( dsdFVanna
TF
d
s
2-
Volga
2
2
s¶
¶
=
C
Volga )('' 2
1
dNTK
d
s
-
($/%/%) 2
)(
2
1
dsVolga
s
21dd

Luc_Faucheux_2020
236
Normal Model
{ }
T
KF
d
dNddNTTKFC
N
NN
s
ss
)(
)(.)('),,,(
-
=
+=

Luc_Faucheux_2020
237
Greeks and Scaling in the Normal Model
Delta
F
C
¶
¶
=D )(dN ($/bp) )( FdD
Gamma
2
2
F
C
¶
¶
=g )('
1
dN
TNs
($/bp/bp) 2
)(
2
1
Fdg
TNs
1
Theta
T
C
¶
¶
=Q )('
2
dN
T
TNs ($/day) )( TdQ
T
N
2
s
Vega
N
C
Vega
s¶
¶
= )(' dNT ($/%) )( NVega ds 1
Vanna
NF
C
Vanna
s¶¶
¶
=
2
)(''
1
dN
Ns
($/%/bp) ))(( NFVanna dsd
T
d
Ns
-
Volga
2
2
N
C
Volga
s¶
¶
= )('' dNT
d
Ns
-
($/%/%) 2
)(
2
1
NVolga ds
N
d
s
2

Luc_Faucheux_2020
Shifted Lognormal Model
¨ Shifted Lognormal model with shift 𝛽:
¨ 𝐶 𝐹, 𝐾, 𝑇, 𝜎) = (𝐹 + 𝛽). 𝑁 𝑑& − (𝐾 + 𝛽). 𝑁(𝑑')
¨ 𝑑& =
&
%D -
𝐿𝑛(
$:8
J:8
) +
&
'
𝜎) 𝑇
¨ 𝑑' = 𝑑& − 𝜎) 𝑇
¨ If 𝐶K) 𝐹, 𝐾, 𝑇, 𝜎) is the usual lognormal Black-Sholes formula
¨ And 𝐶)A 𝐹, 𝐾, 𝑇, 𝜎) the shifted lognormal one
¨ 𝐶)A 𝐹, 𝐾, 𝑇, 𝜎) = 𝐶K) 𝐹 + 𝛽, 𝐾 + 𝛽, 𝑇, 𝜎)
238

Luc_Faucheux_2020
Greeks and Scaling in the shifted Lognormal Model
239
Delta ($/bp)
Gamma ($/bp/bp)
Theta ($/day)
Vega ($/%) 1
Vanna ($/%/bp)
Volga ($/%/%)
F
C
¶
¶
=D )( 1dN )( FdD
2
2
F
C
¶
¶
=g
TF
dN
Ssb )(
)(' 1
+
2
)(
2
1
Fdg 2
)(
11
bs +FTS
T
C
¶
¶
=Q )('
2
)(
2dN
T
TK Ssb+
)( TdQ
T
S
2
s
S
C
Vega
s¶
¶
= )(')( 2dNTK b+ )( SVega ds
SF
C
Vanna
s¶¶
¶
=
2
)(''
1
)(
)(
2dN
F
K
Ssb
b
+
+
))(( SFVanna dsd
TF
d
Ssb
1
)(
2
+
-
2
2
S
C
Volga
s¶
¶
= )('')( 2
1
dNTK
d
S
b
s
+
- 2
)(
2
1
SVolga ds
S
dd
s
21

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